# Talk:Compact space

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## First Paragraph

I'm physicist so be patient. That paragraph is intended as an intuitive approach so a non-mathematician should understand it, but I don't. This are the important parts (copied and pasted) '...whenever one takes an infinite number of "steps" in the space, eventually one must get arbitrarily close to some other point of the space. Thus a closed and bounded subset... of a Euclidean space is compact because ultimately one's steps are forced to "bunch up" near a point of the set..., whereas Euclidean space itself is not compact because one can take infinitely many equal steps in any given direction without ever getting very close to any other point of the space.' First, if those "steps" has to be equal you should mention it at the beginning . Second, if you are in a point and you take infinite steps towards another point (which I guess is not the infinite itself) how can you not get to that second point? Paranoidhuman (talk) 21:49, 1 April 2010 (UTC)

I agree. This intuitive approach, as written, is neither very helpful nor rigorous. Someone needs to rewrite it, or possibly just delete it entirely. --seberle (talk) 22:27, 22 August 2010 (UTC)
Flash out the definition first. Spend a sentence or two on intuition, if there is any. Wikipedia should be concise: fast for retrieving information. The first paragraph is chatty and misleading and should be deleted. — Preceding unsigned comment added by 50.129.229.145 (talk) 11:26, 11 August 2012 (UTC)
Also, it's an intuitive description of "limit point compactness", which isn't the same thing as compactness in non-metric spaces.--DealPete (talk) 20:20, 31 October 2010 (UTC)
As the lead already amply say. Sławomir Biały (talk) 22:16, 31 October 2010 (UTC)
I do not agree. I am studying physics and I keep coming across concepts like Distributions and generalized functions which I never encountered before. It is a good thing to provide an intuitive introduction for mathematical concepts. The limitations of intuitive mode pf thinking can be stated in the article and a more rigorous treatment can be provided subsequently. There are many of us who use Wikipedia as our starting point in the (self)study of physics and mathematics and it has been very helpful. I think intuitiveness is being opposed only for 'ideological' reasons. —Preceding unsigned comment added by 210.212.179.137 (talk) 09:58, 23 January 2011 (UTC)

The fact the opening paragraph does not feature the phrase "Every open cover has a finite subcover" is surely a weakness? 130.88.174.147 (talk) 22:20, 1 February 2011 (UTC)

By which I mean: compactness is a topological concept. Suck it up. 130.88.174.147 (talk) 22:22, 1 February 2011 (UTC)
We don't customarily start encyclopedia articles with abstract definitions. This definition is notoriously difficult to explain to a general audience, and so the first paragraph discusses sequential compactness (historically the first, and also the simplest notion). A formal definition appears later on, in the appropriately labeled section. Sławomir Biały (talk) 22:27, 1 February 2011 (UTC)

I've tried to explain to myself the notion of compactness by saying that a "compact" area on a flat surface could be completely filled by a finite number of arbitrary splats of paint. This is possible only if you allow the paint to touch the boundaries of your area. Likewise, a finite number of arbitrary line segments could obscure a compact line, which is only possible if you allow the pen to touch the ends of the line. Sparaig2 (talk) 06:03, 21 May 2011 (UTC)

I am a computer science student in the second semester and I must say "wow". This was one of the best first paragraphs I have ever read on a mathematical topic.--129.132.45.232 (talk) 10:08, 14 July 2012 (UTC)

I too think the first parts of the article are among the best I have seen, but the first statement threw me off:
a compact space is a mathematical space in which any infinite collection of points sampled from the space must — as a set — be arbitrarily close to some point of the space.
On first reading it sounds like there is some fixed point where all infinite sets come arbitrarily close. (This ambiguity is hard to avoid in English, or in any language for that matter, unless resorting to the kind of formulations that put non-mathematical readers off, e.g. "for every X there is Y such that...") On second reading I thought it said that for every infinite set there would be some (at least one) point in the space arbitrarily close to the set [which is trivial as every point is arbitrarily close to itself, and points in the set are also in the space]. The expression "as a set" in the present lead did not work for me. Only after reading the formal definition I could see what the statement was saying.
Now, to make the statement intuitive, some rigor must be sacrificed, but how? I would perhaps prefer something like
... any infinite collection of points sampled from the space must contain some "cluster",
where "cluster", with quotes, is meant to be an intuitive, possibly non-standard concept. What I actually mean by a cluster, you may imagine, is that what makes a cluster point into a cluster point - sans the cluster point itself: e.g. an infinite sequence of infinite subsets of the set, such that there is nearby some point of the space not necessarily in the set, whose neighborhoods all contain a "tail" of that sequence of subsets. (And by "tail" I mean a sequence except for a finite initial sequence.) But I hope that it would not be necessary to spell it out, the statement would lead the imagination of the reader in the right direction. (If the word cluster has established meanings that conflict with this plan, perhaps "cluster of points", "clustering of points", "concentration", "accumulation", "density peak" or some other word could do the job. I agree that the lead should accommodate the largest possible set of readers. Those needing rigor can find it in the first section below the index, in the definition. Most math articles are like that.Cacadril (talk) 15:21, 7 October 2012 (UTC)
Point above remedied? Instead of "in which any infinite collection of points sampled from the space must — as a set — be arbitrarily close to some point of the space", why not "in which for any infinite collection of points sampled from the space, there must be some point of the space to which these points are arbitrarily close"? (i.e just disambiguate the scope ambiguity!). 82.46.70.132 (talk) 05:03, 14 March 2013 (UTC)

I don't mean to offend whoever wrote this article, and I know it is easy to criticize. Whoever wrote it probably knows more than I do about topology, and it appears to contain a lot of valuable information, once one gets past the definition. But I think that in mathematics, there is no excuse for writing something that is imprecise or incorrect, even if the goal is to make something easier to understand. In this case, I don't think that goal was achieved.Gsspradlin (talk) 23:10, 29 October 2013 (UTC)

Formal details can be found in the "Definition" section for those who have enough mathematical background to understand the open cover definition of compactness. The lead is meant to be readable by a broad audience, and it is not incorrect as currently written. It is already asserted that there are different, inequivalent, notions of compactness and that the version described in the first sentence is sequential compactness. Sławomir Biały (talk) 01:30, 30 October 2013 (UTC)

On the contrary, the very first sentence has at least two errors. It defines compactness incorrectly as sequential compactness, and in the attempt to define sequential compactness, it is suggested that the entire sequence, not just a subsequence, must converge to a point. If you write something later like "Oh yeah, that definition was actually sequential compactness", you have still started with a bad definition. If you want to convey the "flavor" of a compact set to people with little mathematical background, then why not give some properties and examples, such as finite sets, close and bounded subsets of Euclidean space, and give the fact that a continuous real-valued function on a compact space has a maximum and minimum value. That fact may be the reason why any non-mathematicians are interested in compact spaces/sets at all. Then give a real, correct, definition of compact space for those who can handle it. I don't think the open-cover definition is really any more difficult than the sequential-compactness definition. One must know what an open set is to understand the former, and what a subsequence is to understand the latter. If a reader doesn't know either, then you can't expect them to understand a definition of either. You wouldn't try to define, say, the convergence of an infinite series to someone who can't compute 2 + 3. This article seems to attempt to define compact spaces without requiring the reader to know anything at all about topology, even topology of metric spaces. I think that's a bad idea. Gsspradlin (talk) 19:56, 30 October 2013 (UTC)

Sequential compactness is among several different types of compactness, and was actually historically the first to be introduced by mathematicians. In fact, as late as the first few decades of the 20th century "compact" meant "sequentially compact". This is discussed in both the lead and the article body. There is absolutely nothing wrong with presenting this notion of compactness first, as long as the article indicates clearly that it is about sequential compactness. The reason that the authors of the article decided to present the material this way is that the notion of sequential compactness is much easier to explain in a manner that is accessible to as wide an audience as possible (see WP:MTAA and WP:LEAD). Commentators on this discussion page have included readers from a wide variety of backgrounds, not all of whom would be comfortable with the notion of compactness defined in terms of covers (though that more usual form of compactness is also discussed in the lead of the article). In short, it is simply wrong to declare that "sequential compactness" is not "compactness": it is one among many definitions of compactness that mathematicians use that are equivalent in good cases, but inequivalent in general.
Also, contrary to your statement, the lead does not suggest that the sequence itself must converge. (This was the subject of a recent discussion on this very page, where an amenable wording was reached in fact.) Moreover, the first example given in the lead is not itself a convergent sequence: it has two convergent subsequences that converge to different limits.
Finally, both the lead and the article itself already do convey the "flavor" as you suggest, by including a discussion of closed and bounded subsets of Euclidean space. Sławomir Biały (talk) 20:48, 30 October 2013 (UTC)
But the default meaning of compact in a modern mathematical context is not in fact sequential compactness; it's "every open cover has a finite subcover". I agree that the lead paragraph should be as accessible as possible, but it should not mislead, and as it stands it is misleading. With the current text, there's a very real risk that a naive reader could read the first sentence, skim the rest, and come to the conclusion that compactness is generally defined as sequential compactness, which is not true.
I might be OK with starting with sequential compactness provided it's done in such a way that you can't miss that it's not the general definition. That would have to be made very very very explicit. I mean you really have to HIT THE READER OVER THE HEAD with it very very very hard. --Trovatore (talk) 00:13, 31 October 2013 (UTC)
Agree. Paul August 01:52, 31 October 2013 (UTC)

### How about starting with closed and bounded in R^n?

The problem with the existing text is that it suggests (no, actually, it flat out asserts) that a compact set is a sequentially compact one, which is just simply not standard mathematical usage. Sławomir's justification is that it is easier to explain, which may be true, but while we want to be as accessible as possible, we must never indulge in lies to children; I hope everyone agrees that those are abominable and beyond the pale.

We could try to find wording that would start with sequential compactness but make it more explicit that it's not the general definition, but it's going to be awkward.

So how about this? We start with an even simpler-to-explain definition, that of being closed and bounded, but specify that the real concept is a generalization thereof. Just off the top of my head; this can no doubt be much improved:

In the mathematical discipline of general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, contained within some finite cube of the appropriate dimension). This notion is generalized to more general topological spaces in various ways, including sequential compactness, which means (...). The standard meaning of the unqualified word compact is (here put every open cover has a finite subcover, but with appropriate explanations and glosses).

What do you think? --Trovatore (talk) 00:34, 31 October 2013 (UTC)

I like this idea a lot. --JBL (talk) 00:43, 31 October 2013 (UTC)

Would this be acceptable? I have moved your suggested edit regarding the standard unqualified use of the term "compact" to the third paragraph, where that definition is actually addressed. Sławomir Biały (talk) 01:03, 31 October 2013 (UTC)

 In the mathematical discipline of general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, contained within some finite cube of the appropriate dimension). This notion is generalized to more general topological spaces in various ways. For instance, a space is sequentially compact if any infinite sequence of points sampled from the space must eventually, infinitely often, get arbitrarily close to some point of the space. The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sense if and only if it is closed and bounded. Examples include a closed interval or a rectangle. Thus if one chooses an infinite number of points in the closed unit interval, some of those points must get arbitrarily close to some real number in that space. For instance, some of the numbers 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, … get arbitrarily close to 0. (Also, some get arbitrarily close to 1.) Note that the same set of points would not have, as an accumulation point, any point of the open unit interval; hence that space cannot be compact. Euclidean space itself is not compact since it is not bounded. In particular, the sequence of points 0, 1, 2, 3, … has no sub-sequence that ultimately gets arbitrarily close to any given real number. Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points but of functions. The term compact was introduced into mathematics by Maurice Fréchet in 1906 as a distillation of this concept. Compactness in this more general situation plays an extremely important role in mathematical analysis, because many classical and important theorems of 19th century analysis, such as the extreme value theorem, are easily generalized to this situation. A typical application is furnished by the Arzelà–Ascoli theorem and in particular the Peano existence theorem, in which one is able to conclude the existence of a function with some required properties as a limiting case of some more elementary construction. Various equivalent notions of compactness, including sequential compactness and limit point compactness, can be developed in general metric spaces. In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, involves the existence of certain finite families of open sets that "cover" the space in the sense that each point of the space must lie in some set contained in the family. The standard unqualified use of the term compact in mathematics usually means compactness in this latter sense. This more subtle definition exhibits compact spaces as generalizations of finite sets. In spaces that are compact in this sense, it is often possible to patch together information that holds locally—that is, in a neighborhood of each point—into corresponding statements that hold throughout the space, and many theorems are of this character.

Hmm. It's not too bad. Certainly I think it's a huge improvement over the existing text. My only worry is that the standard definition is maybe a little hard to find, for people who are actually looking for it. --Trovatore (talk) 01:06, 31 October 2013 (UTC)

I agree that the proposed change (what is in the box) is a huge improvement over what is there now. As several people have pointed out, the current article starts with a false assertion because it attempts to give an easy definition for a concept that is not easy (though not terribly difficult). It's better to begin with a description of the idea or flavor of compactness, and then give a good definition. Gsspradlin (talk) 01:37, 31 October 2013 (UTC)

Since there have been no strong objections so far, and the consensus seems to be that this is an improvement, I have gone ahead and pasted this in. Sławomir Biały (talk) 11:29, 31 October 2013 (UTC)
Suggestion: early use of a second style of example, closed manifolds. Naturally after Whitney we know that these examples are not more general than compact subsets of Euclidean space. Less general, more geometric, can have pictures. Charles Matthews (talk) 08:15, 8 November 2013 (UTC)

## Examples for spaces which are NOT compact

This article lacks examples of spaces which are not compact, which in my opinion are important to the understanding of the definition. I would suggest adding some simple examples (maybe with an explanation or a proof) - such as the open unit interval, the real line, and C(0,1).

—The preceding unsigned comment was added by 132.69.230.37 (talk) 07:58, 26 March 2007 (UTC).

Well in the first few lines it says that in Euclidean space a subset is compact is it is closed and bounded. So for a counter example chose a subset that is not closed, say ${\displaystyle (0,1)\subset \mathbb {R} }$, one that is not bounded, say ${\displaystyle [0,\infty )}$, or one that is neither closed nor bounded, say ${\displaystyle \mathbb {R} }$ itself. Dharma6662000 (talk) 01:22, 26 August 2008 (UTC)
R is closed in R. In any case, I don't think there's much point giving so many examples of non-compact subsets of Rn when we already have two in the lead, plus the full characterization of compact Euclidean sets in its own section. We could maybe use some more interesting examples of noncompactness in the article though (there's already two hidden at the bottom of the examples list). Algebraist 01:31, 26 August 2008 (UTC)
Yes, it is absolutely true that R is closed, because it clearly contains all of its limit points. A good example of a set that is not compact is a bounded set of the rational numbers Q. It is not compact because it is not closed. I also cannot understand people who belly-ache about "too many examples" - because as a math student in college-level and graduate-level courses, examples was what "brought the subject to life". For those of you who subsist on abstract thinking all the time - maybe go reside with Osama bin Laden?98.67.103.56 (talk) 18:15, 8 August 2009 (UTC)
As with my above example of an intuitive way of looking at compact sets, the alternate to a compact set is a set that can't be covered by a finite number of splats of paint, simply because you aren't allowed to touch the boundaries. Because of that, there is always a gap that isn't covered, no matter how many times you add a splat. This seems pretty darned intuitive. Is there something I am missing?Sparaig2 (talk) —Preceding undated comment added 03:27, 23 May 2011 (UTC).

## Notions

Zundark, sorry that I screwed up the implications between the various compactness notions. --AxelBoldt

That's okay. I should have noticed it at the time. --Zundark, 2001 Dec 15

I'm new to Wikipedia, so I don't know if this is a good place to present a question. Prove or disprove: There exists a compact space X which can be covered by two (intersecting) open sets U and V so that no two compact sets K and F, with K a subset of U and F a subset of V, cover X. The example, if standard set theory can provide one, will be more complex than it seems at first glance. This is connected with properties of the compact-open topology. --Roman.

For a non-hausdorff space, it's easy. Just take an indecomposable space (e.g., the reals with the Zariski topology). For a hausdorff space, it's impossible. Apply normality to the complements of U and V, then take the complements of your resulting open sets. These are K and F. —Preceding unsigned comment added by 24.240.36.203 (talk) 21:02, 11 May 2010 (UTC)

## brackets

Why is it important to nowiki the half-open interval, but leave the closed interval as is??? Revolver 11:46, 9 Nov 2004 (UTC)

## help

Can someone help me understand this:

"The modern general definition calls a topological space compact if...any collection of open sets whose union is the whole space has a finite subcollection whose union is still the whole space."

Does [0, 1] fit the definition of a compact space simply because there exist no collections of open sets whose union is [0, 1]? At least I cannot think of any--if you can think of some, please help me. Or should the word "is" be replaced by "contains"?

There are collections of open sets whose union is [0,1]. For example, { [0,.6), (.5,1] }. Remember that the open sets of [0,1] are the sets of the form [0,1]∩U, where U is an open set of R. --Zundark 07:32, 12 May 2005 (UTC)
I'm confused about this one too: taking the link open set's under metric space (R is a metric and topological space), we see that: A subset U of a metric space (M,d) is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with d(x,y) < ε, y also belongs to U. But the '1' in (.5,1] break that statement, no ?
Update: Ok, so the key is relative topology. I think someone should point that out in compact space, cover and possibly open set. The way it is now (4th August, 2005), we can deduce that a subset of a topological space is compact if it has a finite open subcover of its open cover. Then if you look up (open) cover, you conclude that a cover (of a subset S of a topology X) is a union of *open sets* in S which gives (hence the name "cover") the subset. Now, 'open' might be taken (which would be incorrect) to be from the original topology (and not the subspace one).
Neither [0, 0.6) nor (0.5, 1] are open sets, due to the elements 0 and 1 respectively. Also, the word "is" should indeed be replaced by "contains", as per the definition of "open cover." Hence, {(-0.1, 0.6), (0.5, 1.1)} would suffice as an open cover of [0,1] which has a finite subcover of [0,1]. --Tekhnofiend (talk) 22:15, 10 January 2010 (UTC)
Nevermind, I think I've confused the discussion to be about subsets of topological spaces rather than the spaces themselves. You should disregard my earlier comment. --Tekhnofiend (talk) 23:50, 10 January 2010 (UTC)

## merge with Compact set proposal

It seems that compact set and compact space are rather duplicate. Shouldn't we merge these articles, taking care that both "unbounded and closed" and "finite subcover" are well treated and keeping the relation between the different definitions clear? I'm willing to do the merge myself, but comments/suggestions/objections are welcome. --Lenthe 11:29, 27 July 2005 (UTC)

You are right. It would however be preferable if compact set were merged into compact space, in such a way as to preserve the elementary flavour. That is, to be friendly first to those who need the idea on the real line, rather than the generality of topological spaces. Charles Matthews 12:04, 27 July 2005 (UTC)
I've been meaning to do this merge for a long time, but have never gotten around to it. Actually, it looks like much of the material in compact set belongs in the article on the Heine-Borel theorem. The rest of it should go in compact space. -- Fropuff 13:46, 27 July 2005 (UTC)
OK, I've moved the content of compact set (now a redirect) to compact space and Heine-Borel theorem and started polishing both articles a bit. Help on polishing is very welcome. --Lenthe 14:59, 3 August 2005 (UTC)
For the elucidation of foreigners and everyone else, "OK", is always written with All Capitals. The best that historians can deduce, it is an abbreviation for "Old Kinderhook", a nickname that President Martin Van Buren of the United States used.98.67.103.56 (talk) 18:21, 8 August 2009 (UTC)

## Name Change Proposal

It might be more appropriate to name this page "compactness", especially considering the "history" section. It isn't a history of compact spaces, but rather of compactness. It seems a more intuitive name as well. Fell Collar 01:51, 2 March 2006 (UTC)

Yet "compactness" can refer to a large number of things (even in mathematics), while "compact space" relates more clearly to the topological property. --Lenthe 08:40, 2 March 2006 (UTC)
What other things are you thinking of? Fell Collar 18:11, 2 March 2006 (UTC)
Nobody ever answered this question. "Compactness" refers to automobiles, pistols, portable radios, the nuclear reactors in submarines, the small varieties of wrestlers and boxers, white dwarf and black dwarf starts, and 101 other things.98.67.103.56 (talk) 18:29, 8 August 2009 (UTC)
At any rate, the article isn't just about compact spaces. It's also about compact subsets of metric spaces or Rn. Moreover, while "compactness" may refer to a number a things in mathematics, it seems to me that this is by far the most common. To clear up any confusion, we could always add a disambiguation link at the top of the article. --Fell Collar 15:40, 9 March 2006 (UTC)
First of all, didn't we just finish merging compact set into this article? It almost sounds like you want to split again (which I don't support). Second of all, the definition of compact sets in Rn or metric spaces is simply that they are compact spaces in their subspace topologies. So the name "compact space" covers everything. I do not support a name change. -lethe talk + 16:43, 9 March 2006 (UTC)
I didn't mean to suggest we split the article again; I was suggesting that we keep all the content here, but change the name to something more inclusive of perspectives outside of topology that would justify the section about history as well. Compactness is also important in analysis, but one rarely refers to "compact spaces" there, nor does one reference topology in the definition of compact (Wheeden and Zygmund's "Measure and Integral", for example, refers only to "compact sets" and gives the finite open covering definition). This is a valid perspective as well. Rather than pick one point of view over the other, I'm suggesting we simply name the article "compactness" and list both perspectives. This seems rational to me, and in keeping with Wikipedia's NPOV guidelines. --Fell Collar 19:08, 9 March 2006 (UTC)

The appropriate name for this page is compact space. Please don't rename it. If you want to start a new article on compactness which discusses the various forms of compactness (sequential, countable, locally, para-) and their histories that would be fine. You could move some of the material from this page there. -- Fropuff 05:08, 10 March 2006 (UTC)

I'd be okay with that, if other people are as well. I expect there would be a considerable amount of redundancy with this article, which might be a concern given the recent merge. What do other people think? --Fell Collar 19:58, 10 March 2006 (UTC)
The only reason I would think that a separate page was needed would be if this page were too long. This page is currently not very long, so I do not support such a split, though I would change my mind if someone showed up who wanted to write all the stuff about those other forms of compactness which would make this article too long. Are you that person? -lethe talk + 22:59, 10 March 2006 (UTC)
I'm not particularly interested in writing all that, and I don't want this to be any more controversial than it already has been, so I'm just going to leave it alone. --Fell Collar 03:55, 11 March 2006 (UTC)

## Mnemonical rule to remember Compact space

This rule came from Russia in a form of a joke: Mathematician is talking to a pretty girl: - You are so compact... Girl fondly specified his answer: - Do you mean well-shaped and thin? - No. Closed and bounded! --Yuriy Lapitskiy 22:26, 9 March 2006 (UTC) hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

## Compactness vs. completeness

(Copied from user talk:AxelBoldt:) Do you happen to know the proof (or where it can be found) of the statement

A metric space X is compact if and only if every metric space homeomorphic to X is complete.

that you added to the compact space entry in 2002? Slawekk 17:41, 26 May 2006 (UTC)

I'm not so sure anymore that the statement is correct and I have removed it for now. The one direction is clear: if X is compact, then every space homoeomorphic to X is compact, and every compact metric space is complete. For the other direction, I wanted to use the Stone–Čech compactification βX of X: if βX is metrizable, then the subspace X of βX is a metric space homeomorphic to X and is therefore complete by assumption, and a complete subspace of a compact metric space is itself compact. Problem is, I'm not sure whether βX is metrizable. So I'm missing the following statement:

• The Stone–Čech compactification of every complete metric space is metrizable.

AxelBoldt 21:16, 26 May 2006 (UTC)

I don't think the missing statement is true. X has a metrizable compactification iff X is 2nd countable and tychonoff [1]. Of course not every complete metric space is second countable. Slawekk 22:24, 26 May 2006 (UTC)

You are right, thanks a lot! AxelBoldt 23:54, 27 May 2006 (UTC)

## Compact space vs. complete algebraic variety

An algebraic variety ${\displaystyle X}$ is complete iff (by definition):

for all algebraic variety ${\displaystyle Y}$, the second projection ${\displaystyle p_{2}:X\times Y\to Y}$ is a closed map.

Is something like this true for topological spaces? (According to Dieudonne this should be true) That is: Let be ${\displaystyle X}$ a topological space. Then

${\displaystyle X}$ is compact iff (for all topological space ${\displaystyle Y}$, the second projection ${\displaystyle p_{2}:X\times Y\to Y}$ is a closed map ).

Could someone help me? Thanks in advance. —Preceding unsigned comment added by Pepe 986 (talkcontribs) 00:23, 12 March 2008 (UTC)

Yes. A Hausdorff space X is compact if and only if for every Hausdorff space Y, the projection X × YY is a closed map. I believe that Hausdorffness is required. Ryan Reich (talk) 20:53, 28 April 2009 (UTC)

## What should be done here?

This page is labelled as being "barely B class" and of high importance. I don't see anything terribly wrong with it, although it could use some editing for internal consistency (a few things get mentioned several times seemingly needlessly, and there is a lack of parallelism in certain symbols and fonts used).

Of course the article does not say everything that one could possibly say about compactness (no single article could), but I wonder what people think is missing here?

One thing that springs to mind is that it would be nice to mention the compactness theorem in model theory and how it does, in fact, assert the compactness of a certain Boolean space. Plclark 10:09, 8 October 2007 (UTC)Plclark

I think this article should give much more of an overview of the concept of compactness and how it is used in various parts of mathematics. It should also summarise the articles on the various variant definitions. There are probably plenty of other things that are missing. I am sure that an entire book can be written on compactness, and that this article has the potential to be featured. --Hans Adler (talk) 10:29, 21 April 2009 (UTC)

## compact vs. connected

can someone clarify the differences between compact & connected? 76.68.228.196 (talk) 16:29, 24 October 2008 (UTC)

They are totally different concepts. I really can't see how you could confuse them. Algebraist 17:51, 24 October 2008 (UTC)
See compact space and connected space. Maybe you were just looking at the wrong definitions.

Topology Expert (talk) 21:22, 7 December 2008 (UTC)

Simple answer: The union of [0,1] and [2,3] is compact but not connected. (0,1) and [0,∞) are each connected but not compact.
I think I have encountered this confusion before. (And personally I sometimes have problems distinguishing words that start the same.) Perhaps the article should give these simple examples. But then, perhaps it's enough to have them here on the talk page. --Hans Adler (talk) 10:25, 21 April 2009 (UTC)
Reading the Talk page is usually my last resort to sort out an issue with the Article. I think that a note in the beginning of the Article about compact spaces and the above mentioned example in the example section should solve this issue.
TomyDuby (talk) 16:28, 21 April 2009 (UTC)
I am very forgetful, and I hope there is a way, when I need it, I can find it easily. The simple answer in "Compact vs. connected" section is very nice. It is much easy for readers as me to get the idea. By reading the article, I didn't get anything but headache.

## Compact vs. Tight

Is there anyone can elaborate the difference between compact vs. tightness? Jackzhp (talk) 16:50, 10 July 2010 (UTC)

seems that tightness is compact+connected. Jackzhp (talk) 16:29, 9 August 2010 (UTC)

## Help for Math Student

I am very confused by the definition of compactness in conjunction with the definition of Cover, and I figured that this might be something that should be explained better in the article. Either I am understanding something terribly incorrectly, and there is nothing in the article that addresses or clears my misunderstanding, or the Heine-Borel theorem is wrong and [0,1] is not compact.

If [0,1] is compact, according to the definitions on the articles I mentioned above, there should be no open cover without a finite subcover. Then is A={(-0.1, 0.5), (0.4, 1.1)} an open cover for [0,1]? If not, why not? If so, how can I make a subcover for A? This seems very similar logic used in the set (n-1,n+1) that was used to prove that there exists a cover of the integers that has no subcover. Can someone clear this up either in this article or in the article Cover (topology)?

Thanks 24.7.179.171 (talk) 01:39, 20 August 2009 (UTC)

I do not quite understand your confusion. The cover A you have given is indeed an open cover of [0, 1] (each element of [0, 1] is in one of the elements of A). In fact, it is a finite cover because there are only two elements in A. Therefore, it is a finite subcover of itself, and this does not contradict compactness of A.
An open cover of [0, 1] is a collection of open sets whose union is [0, 1]. To create a finite subcover, one must delete some open sets in this collection so that only finitely many are remaining, and prove that this finitely many does indeed constitute a cover. If the sets in the open cover themselves have infinitely many elements, it does not change the number of sets in the cover (in other words, the number of elements of the open cover does not relate to the number of elements of a particular open set that is one of many which constitutes the cover). The key point here is the notion of "a set of sets". When one studies sets of objects, it does not matter what the objects are, or how many elements they have. In the case of open covers, one considers a set of open sets (so the objects are open sets). With compactness, what matters is not what the open sets are (as long as they constitute a cover), but how many there are. Hope this helps. --PST 07:32, 20 August 2009 (UTC)
Oops! I made a stupid mistake: I forgot that a set is a subset of itself. Very silly of me. Then finite covers are trivial when discussing compactness, and one only needs to consider whether or not infinite covers have finite subcovers. Thank you. 24.7.179.171 (talk) 03:38, 23 August 2009 (UTC)
It's my understanding, from the definition of an open cover, that it is incorrect to say that an open cover of [0, 1] is a collection of open sets whose union is [0, 1]. By definition, an open cover of [0,1] is a collection of open setes whose union CONTAINS [0,1]; equality is not required. This confusion arose earlier in this talk page and even creeped it's way into the main article. I'll make the change in the main article. Apologies if I'm somehow mistaken, in which case let me know! --Tekhnofiend (talk) 22:15, 10 January 2010 (UTC)
I was referring to sets open in the subspace topology on [0,1], when defining an open cover of [0,1]. --PST 02:34, 11 January 2010 (UTC)
The definition in the article is correct for an open cover of a topological space. It doesn't say anything about the slightly different question of what an open cover of a subset of a topological space is. In fact, the article doesn't seem to cover subsets properly at all; there should probably be a note somewhere saying that a subset A of a space X is compact iff it's a compact space when endowed with the subspace topology. Algebraist 22:34, 10 January 2010 (UTC)
I see what you mean now. Thanks for the clarification. --Tekhnofiend (talk) 23:50, 10 January 2010 (UTC)

## clarify

I understood this stuff 30 years ago, but I've forgotten most of it. I have a question/comment about the lead.

Suppose you have a closed disc of radius 2 centered at the origin. Suppose your infinite number of steps go (1,0), (0,1), (-1,0), (-1,-1), (1,0), etc. What point is it approaching?

Am I misunderstanding it or is the description in the lead not complete? Shouldn't it be that for every infinite series of points in the space that converges, it converges to a point in the space? Bubba73 (Who's attacking me now?), 18:56, 16 December 2009 (UTC)

The description in the lead is not intended to be completely rigorous. Sequential compactness, which is properly defined later on, states that any sequence whatsoever has a convergent subsequence. Obviously any convergent sequence converges (...to a point...in the space), so the alternative you offer is vacuous as stated. In spirit it is similar to that of a closed set (a subset of a metric space which contains the limits of every sequence that converges to a limit in the larger space) or a complete space (a metric space in which every Cauchy sequence converges). Both of these conditions are strictly weaker than compactness. Sławomir Biały (talk) 21:42, 16 December 2009 (UTC)
Hey, I've come here to ask that same question... ― ___A._di_M. (formerly Army1987) 21:35, 4 April 2010 (UTC)

Well, Sławomir Biały, you're right about the theorems. But although as Paranoidhuman had said, the first paragraph should be easy to understand for laymen, however, it should be relatively strict as well. I think the metaphor the article gives is definitely not clear enough and also not nature-revealing. Trvoldemort (talk) 07:09, 8 May 2010 (UTC)

I'm certainly open to suggestions about better ways of expressing it for laymen, but this is the best I've found so far. The lead should be understandable by laymen: it does not need to transmute laymen into experts on the subject. Paranoidhuman seemed to think that the steps had to be equal (but they don't) and had some trouble with a notional paradox of infinity (Zeno's paradox). Well, we can't really do much about that. I have no idea what "theorems" you are referring to. Nowhere in this discussion page have I referred to any theorems. The above post was simply referring to different definitions of related ideas: closed set and complete space. Sławomir Biały (talk) 11:33, 8 May 2010 (UTC)

## Peanut gallery challenge

There seem to be lots of criticisms from the "peanut gallery" that the present lead fails to give a notion of compactness that is both understandable to laypersons and is completely rigorous. While that seems like something of an oxymoron, the criticism has been made—quite explicitly in recent posts. So I've kind of lost patience with the peanut gallery here, and I am about to throw the WP:SOFIXIT gauntlet. The version of compactness that, to me, seems amenable to lay understanding is that of sequential compactness. So the challenge to the hecklers here is to please explain the following in a way that satisfies all requirements for the lead:

• A space X is sequentially compact if every sequence in X has a subsequence that converges to a point of X.

Please refrain from further unconstructive criticism (especially from folks who have not even read the article) until the above challenge is met. I, for one, think the present version of the lead explains this concisely and intuitively, but obviously I am in the minority. Sławomir Biały (talk) 15:07, 8 May 2010 (UTC)

I heavily edited the lead's first paragraph. I decided to say the "steps" one takes have to be distinct ("one takes an infinite number of "steps" to different points in the space"), since if they're not distinct sequential convergence is trivial, and the layman's description doesn't need to split that hair. Hopefully this takes care of the "clarify" criticism above. The original also implied the sequence converges, instead of that a subsequence converges. I understand it wasn't supposed to be rigorous, but that's a pretty big mistake. I also noted right off the bat that there are several versions of compactness--I imagine people reading the page will want to know they have to figure out which version fits their needs, and they'd prefer that information before they skip the rest of the long intro. Finally, I turned the Euclidean space comments into two examples (if you have a better sequence than 1, 1/2, 1, 1/3, 1, 1/4, ..., feel free to suggest it), since a representative example is worth tons of pure exposition. It could certainly be improved, but I find it difficult to do so knowing what compactness is without having more space. 67.158.43.41 (talk) 22:26, 5 October 2010 (UTC)
Overall, I like the edit. I don't agree that the original wording necessarily implied convergence of the sequence, but the new edit eliminates this ambiguity nicely. Also, I am not big on having detailed examples in the lead. This was the original reason that I created the "Introduction and definition" section: the lead had spun out of control with the number of examples and different issues it tried to present. However, I'll be the first to admit that the example of sequences in the unit interval should help to instill an appropriate mathematical understanding of "steps" in a space, and probably is needed. Finally, I hope there is a way to move the discussion of the other definitions of compactness later, so that the first paragraph gets directly to what a compact space is, rather than leave it hanging as an "abstract mathematical space". In fact, it's almost better to go down this route:
a compact space is an abstract mathematical space in which, intuitively, whenever one takes an infinite number of "steps" to different points in the space, some of the steps must get arbitrarily close to a particular point of the space. (More precisely, this is the description of a sequentially compact space, and there are other distinct notions of compactness covered later.)
But I feel that parenthetical comments are a mark of poor writing (which is not to say that I don't use them). Sławomir Biały (talk) 23:30, 5 October 2010 (UTC)
I prefer the first sentences the way they are, unfortunately. I certainly agree it would be nice if the article could immediately say what compactness is instead of "leaving it hanging". However, I think the most agreed-upon definition (finite subcovers of all open covers) is far too abstract to use, and it just doesn't flow well to say "here's a definition" followed immediately by a comment that says "but wait there are many others that sometimes are equivalent but sometimes aren't (and, though I'm not telling you, this probably isn't even the most common one)". Feel free to rework any parenthetical comments. If it were up to me, I would probably rework the intro quite a bit more to first clearly overview the different notions of compactness by naming them but not going into details, noting that sometimes they're equivalent and sometimes they're not. The second paragraph might then explain sequential compactness in an intuitive manner, using Bolzano-Weierstrass and the examples I've given, and more importantly noting that this formulation is equivalent to the others in metric spaces so it's not random to use it over the others. The existing 3rd paragraph would be gutted to create the first paragraph, and the existing 2nd paragraph would become the new third. But, I understand wanting to give a definition right away, so I won't make the changes without discussion. 67.158.43.41 (talk) 10:56, 6 October 2010 (UTC)
I'd like to highlight the issues as I see them. The purpose of a lead is to give context for the topic and summarize the rest of the article in a manner that can be grasped as much as possible without a detailed knowledge of the subject. I think we're in agreement that, at least, the usual definition using open covers is simply too abstract to include in a general-purpose introduction to the subject. This belongs further down in a definition section. The current structure of the lead consists first of a paragraph saying what a compact space is in the most simple and rarefied way possible followed by a paragraph that attempts to give a summary of the "History" section, highlighting the history and importance of compactness, and finally a third paragraph giving some slightly more technical background on the usual notion of compactness, theorems, etc. This may not be the perfect structure, but I think that gutting the first paragraph to create a new first paragraph runs the risk of making the lead much more technical. It seems that the first paragraph would then need to bring in general topological spaces, metric spaces, open sets—very abstract ideas that the typical reader will have no idea about. Of course, a lot of this depends on execution as much as anything else, and so I look forward to future edits with anticipation. Sławomir Biały (talk) 11:35, 6 October 2010 (UTC)

I think the recent edit to the lead is precisely what was needed. My first reaction was that it was too long, and that it should somehow be rearranged to push some of the details down into the "Introduction" section. But perhaps instead the lead should be left as it currently stands, modulo minor edits, and rather the first paragraphs of the introduction section should be revised. I'd like to get some input on this. Sławomir Biały (talk) 13:28, 9 January 2011 (UTC)

## Compactness (topology)

As Compactness (topology) redirects here, this article ideally needs a main section dedicated to the property of compactness, as opposed to a compact space Brad7777 (talk) 13:30, 6 February 2012 (UTC)

I don't understand. This whole article is about compactness as it is defined in topology. A compact space is a topological space (or a subset of a given topological space in the subspace topology) with the property of compactness. Arguably, the article could be renamed from Compact space to Compactness (topology), but I see little benefit in that. — Tobias Bergemann (talk) 14:02, 6 February 2012 (UTC)

## "Smallness" Analogy

I think that describing compactness as a measure of, in some sense, "smallness", is a bit misleading to the uninformed reader, especially since compact spaces can very often be supersets of non-compact spaces (long line vs. real line, etc). Perhaps a change of wording is in order? Thoughts? — Preceding unsigned comment added by 207.239.94.2 (talk) 20:31, 23 March 2012 (UTC)

I don't think a change in wording is needed. The lead already goes to extraordinary lengths to convey that there is no simple obvious characterization of compactness. It also says that sometimes compact spaces can be quite "large". Sławomir Biały (talk) 01:28, 24 March 2012 (UTC)
What we have in mind, even when we say something like smallness, is really something like handlebleness or niceness or havingthepropertiesthatweneedness, but these are unsayable words. 2A01:CB0C:CD:D800:C12E:287B:1E8A:E532 (talk) 20:48, 1 February 2021 (UTC)

## Abominable first paragraphs!!!

The first paragraphs, supposed to introduce the reader to the concept of compactness, could hardly be worse. (At least they are grammatical.)

No information whatsoever is conveyed by saying that a compact space is one possessing the quality of compactness.

Also, virtually no information whatsoever is conveyed by saying that the concept of a compact space is hard to describe intuitively. That very same space on the page could be used to at least attempt to describe the intuitive idea of a compact space. In the most common settings like Euclidean space, compactness is not hard to convey intuitively. That is true only in the most general settings, and there is no need to discuss the most general settings in an introductory paragraph.

(Here is one way to do this could be something like: "For the simplest kind of compact space -- one that is a subspace of some Euclidean space -- a compact space is one in which any infinite set of points must have a limit point." This could be followed by the essence of the Heine-Borel argument for the closed unit interval, and then even for a closed and bounded set in R^n.)

This is an encyclopedia, whose purpose is to convey information. Sentences and paragraphs that convey no information will merely frustrate the reader.Daqu (talk) 18:00, 11 September 2012 (UTC)

Couldn't agree more. 129.187.147.189 (talk) 12:20, 20 September 2012 (UTC)

I assume important changes have been made since then. To me its the opposite problem now. Nothing is even mentioned about the general notion of compactness in the introduction, one gets the impression the introduction and most of the article is about Euclidean spaces where "compact" is equivalent to "bounded" (because by definition of a topology, a topological space is always closed! the page is titled "compact space", not "compact subset"). This page should NOT focus (so much) about Euclidean spaces, because it deviates the attention on the important differences of "compact" from bounded. AT LEAST, it should mention "normed vector spaces" (mentioned nowhere outside 3 subsequent lines) more often than "Euclidean space" (mentioned nearly everywhere). There would be no need for a page on "compact spaces" if it's to 90% about Euclidean space where everything (otherwise interesting) is trivial. — MFH:Talk 09:39, 24 May 2017 (UTC)
The third paragraph is about compactness in function spaces, the fourth paragraph is about compactness in general topological spaces. The concept of compactness was originally introduced (by Frechet) as a generalization of sequential compactness in Euclidean spaces, so the article begins by discussing that. Sławomir Biały (talk) 10:57, 24 May 2017 (UTC)

## First line

"In mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite sequence of points sampled from the space must eventually get arbitrarily close to some point of the space." [0, 1] is a compact space. (0.2, 0.8, 0.2, 0.8, ....) is an infinite sequence. It does not eventually get arbitrarily close to any point in [0, 1]. Surely this is not a definition of compact? Veltas (talk) 19:03, 4 June 2013 (UTC)

Pointing out what is hopefully obvious: 0.8 is in [0,1]. Sławomir Biały (talk) 20:12, 4 June 2013 (UTC)
Either you can take "eventually arbitrarily close" to mean eventually the rest of the sequence is a certain distance, or as you have done "there is a point in the sequence that is arbitrarily close to a point in the space", which, in fact, since the sequence is sampled from the space is implied. So either it's the wrong definition, or not a definition at all. Veltas (talk) 20:53, 4 June 2013 (UTC)
Perhaps the confusion is that compactness isn't convergence: it doesn't require every element of that sequence to converge arbitrarily close to the same point. In your sequence above, the the odd elements get arbitrarily close to 0.2 and the even elements get arbitrarily close to 0.8. There are other examples of non-convergent infinite sequences in the article that nonetheless have a subset that gets arbitrarily close to some point in the associated compact space. --Mark viking (talk) 20:50, 4 June 2013 (UTC)
However they have stated specifically "any infinite sequence of points sampled from the space must eventually get arbitrarily close to some point of the space" when it is only true to say "any infinite sequence of points sampled from the space must have a subsequence that gets arbitrarily close to some point of the space eventually." So the 'rough' definition is wrong. It's unfortunate because they're obviously attempting to give an intuitive notion of compactness, but the definition implies something that doesn't really make sense. Veltas (talk) 20:59, 4 June 2013 (UTC)
No, you're reading the quantifier wrong. There's a point in the space that the sequence eventually gets arbitrarily close to. Sławomir Biały (talk) 21:10, 4 June 2013 (UTC)
What quantifier? You seem to suggest that the notion of a sequence eventually getting arbitrarily close to a point is the notion of a convergent subsequence, but intuitively to me a sequence eventually getting arbitrarily close to a point is convergence. When a property of a sequence is 'eventual' in a sequence, I take it to mean it applies to all points beyond a certain position in the sequence. Combined with 'arbitrarily close' we have the literal definition of convergence. The only way I can read this sensibly and garner a sense of the idea of compactness is as: "a compact space is a mathematical space in which any infinite sequence of points sampled from the space must have an infinite subsequence eventually get arbitrarily close to some point of the space," or perhaps better yet; "in a compact space any sequence of the space must have a subsequence that converges to a point in the space", since a sequence may as well be finite (and then it's equivalent to an infinite sequence with the end element repeated at the end), or something of the sort. Veltas (talk) 21:29, 4 June 2013 (UTC)
While it's not wrong, you have a point--it could be made more clear. I have edited the sentence to read: "In mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite sequence of points sampled from the space has a convergent subsequence that must eventually get arbitrarily close to some point of the space." This is consistent with the definition given in Sequentially compact space an in the online sources I've skimmed. Thanks, --Mark viking (talk) 21:21, 4 June 2013 (UTC)
I agree with User:Veltas that the sentence is really ambiguous. What about something like
"must eventually come arbitrarily close, infinitely many times, to some point of the space"
"must eventually come arbitrarily close, at some times, to some point of the space"
or anything of the sort, in a better English than mine.
Bdmy (talk) 21:37, 4 June 2013 (UTC) Bdmy (talk) 21:44, 4 June 2013 (UTC)
Yes, I think what you're trying to describe is a convergent subsequence. Veltas (talk) 21:47, 4 June 2013 (UTC)

I've just realised the whole first line is wrong, it's the definition of sequential compactness. Unless it's made clear that this is a 'watered-down' version for an intuitive glance at its meaning it should be exchanged for the actual definition. That is, for all open covers there is a finite subcover, otherwise the first line is in fact wrong regardless of our different understandings. Sequential compactness is equivalent for metric spaces, at least, but it's crucial that the difference is recognised. Veltas (talk) 21:54, 4 June 2013 (UTC)

I've changed it so that we don't give the wrong meaning anymore, and at least it's clear that there is a difference at least, and maybe the example helps give a 'feel' for what compactness is. It's clear (as has been noted by many others in this talk page) that there is still much to do to clean up the start of this article. Veltas (talk) 22:34, 4 June 2013 (UTC)
The first paragraph of the lead is quite clear that there are different notions of compactness, which are only equivalent in good cases. Sławomir Biały (talk) 15:00, 5 June 2013 (UTC)

The lead of the article has multiple issues; some of them have not been quoted in the lengthy preceding discussions. The lead is too long. It too technical: to understand the first sentences, one needs to well know what is a cover and understand what means "must eventually, infinitely often, get arbitrarily close". It give an undue weight to one definition of compactness. It is confusing by using a language which is wrongly supposed to be more accessible to the layman. For example "mathematical space": somebody who knows what is a topological space will ask "what is this strange object", and people who do not know anything on topological spaces do not understand also this term (as well as the remainder of the lead). The main issue is probably that nothing is said about why compactness means in practice, why it is important and why it is useful (the context). IMHO, to solve all these issue the lead has to be completely rewritten. I suggest something like the following:

In mathematics, a compact space or compact set is a topological space satisfying an important property, the compactness, which is used in many branches of mathematics. Roughly speaking, a compact space is a space where every limit that may be hoped for exists in the space. This means that there is no "lacking point" (the set of the rational numbers is not compact as ${\displaystyle {\sqrt {2}}}$ is a limit of rational numbers and is not rational) and there is no "endless path" in the space (the number line is not compact).
The compact subspaces of Rn are the bounded closed subsets. For more general cases, compactness may be defined intrinsically, that is without referring to a larger space in which the compact space is embedded. There are several definitions, which need some further, commonly satisfied, conditions to be satisfied.
The importance of compactness lies in the number of useful properties that may be deduced from it. For example, every continuous function from a compact space into the real numbers reaches its maximum and its minimum; this means that there exist two points in the compact set, where the value of the function is, respectively, maximum and minimum.

This is only a first draft, but I believe that it solves most of the issues of the present lead. D.Lazard (talk) 10:55, 31 October 2013 (UTC)

P.S. The proposition in #How about starting with closed and bounded in R^n? has been posted when I was writing mine. They are thus completely independent. D.Lazard (talk) 11:05, 31 October 2013 (UTC)
I have some issues with an attempt to completely rewrite the lead. First, the current version of the lead is intended to be accessible to laymen. Indeed, many non-mathematicians have commented here that the current version of the lead is one of the most understandable introductions to the topic that they were able to find. That is clearly an important data point. Most of the negative comments come from students who are upset that the definition described in the lead does not agree exactly with the definition given in their topology textbook. But our aim must be to write "one level down". Compactness is usually introduced in an upper level undergraduate course, so the lead should be accessible to a student of calculus. Second, the current version of the lead summarizes the article, including a historical perspective on the concept and summary of the article. Finally, the lead as currently written was the product of many discussions here. I think it would be inappropriate to disregard these efforts entirely. Sławomir Biały (talk) 11:21, 31 October 2013 (UTC)
The virtues of the lead, as it stands, are those Slawomir Bialy mentions. It is also all these virtues that makes it way too long. After the lead comes a lengthy introduction (essentially an extension of the lead), and then we get to the definition of compactness. YohanN7 (talk) 15:08, 8 November 2013 (UTC)
I feel like the lead is as long as it needs to be to provide an accessible overview that defines the topic of the article and summarizes it. Some trimming might be possible, but it is three paragraphs long per the recommendations of WP:LEAD. However, I would agree that with the present form of the lead, the introduction section is almost entirely redundant. It might be better to replace this with a section including more examples of compact spaces. Charles Matthews makes an interesting suggestion that would allow for the inclusion of images as well in such a section. I wouldn't object if someone wants to start work on such an enterprise. Sławomir Biały (talk) 15:22, 8 November 2013 (UTC)
Off topic: Under Theorems I find "The pre-image of a compact space under a proper map is compact." Isn't this trivially true by definition of proper? I see now that there are several definitions of "proper map". YohanN7 (talk) 21:41, 8 November 2013 (UTC)

## Gloss for "bounded" (cube/ball/sphere)

So while I reverted this edit, it did kind of raise a good point. What's the best way to gloss the word "bounded", assuming we want "closed and bounded" to have a prominent place in the lead?

I put it as "contained within some finite cube of the appropriate dimension"; the IP editor changed "cube" to "sphere". The problem with that is that a sphere, technically, is just the surface of a ball. So it would be correct to put "ball" or "spherical ball", but that would itself need explanation, because not everyone knows what a "ball" is. And if you have to explain your explanations too many times, it gets tiresome to read.

But it can't be denied that the "ball" definition is probably the more common one in the literature.

Suggestions? --Trovatore (talk) 01:06, 13 November 2013 (UTC)

The phrase "inside a sphere" means the interior of the ball, so I don't see a problem with that expression. My reason for not changing it originally was that I wanted to avoid "ball", and I anticipated some objection to the word "sphere". I would like to find an east enough wording that avoids "of the appropriate dimension", regardless of the choice here. "Inside a hypersphere" possibly? Sławomir Biały (talk) 16:20, 13 November 2013 (UTC)
I suppose you could read the word "inside" in that way, but I submit that it's confusing. It's a natural reading in English but not a very natural one in mathematical English — the more "mathematically natural" one would be "is a subset of the sphere", which is not what we want. So I prefer to avoid "sphere". --Trovatore (talk) 19:27, 13 November 2013 (UTC)
I'd personally read "interior of the ball" as theopen ball. Probably best to avoid such a phrase. YohanN7 (talk) 20:41, 13 November 2013 (UTC)
I think that is what Sławomir meant, and that's just fine. Open ball or closed ball makes no difference here. The problem is explaining what a "ball" is in the first place. Maybe "spherical ball" is clear enough without further explanation? --Trovatore (talk) 20:44, 13 November 2013 (UTC)
I would read "inside a sphere" as "within a fixed distance to the origin". But perhaps this is what the article should say? Sławomir Biały (talk) 11:58, 14 November 2013 (UTC)
What origin? We've mentioned Euclidean space; we haven't said it has a distinguished origin. And I still think "inside" is problematic. But I'm starting to think "spherical ball" is not too bad — correct for those who understand the distinction, reasonably intuitive to those who don't. --Trovatore (talk) 23:14, 14 November 2013 (UTC)
Using "spherical ball" seems good to me. Paul August 00:28, 15 November 2013 (UTC)
I've tried out an alternative wording that seems simpler than all of the options presented thus far since it avoids the awkwardness of "the appropriate dimension". Sławomir Biały (talk) 00:45, 15 November 2013 (UTC)
I can live with that. --Trovatore (talk) 00:51, 15 November 2013 (UTC)

## Edit/revert over "fixed distance"

With regard to my reverted edit by User:Joel B. Lewis:

It is false that a bounded set on a metric space is a space where "there is a fixed distance between any two points". That would make any metric space a bounded set.

If you don't like the word "measurable", I have no problem with it being removed. (EDIT: removed my inappropriate assumption.) TricksterWolf (talk) 21:34, 5 August 2014 (UTC)

The relevant text in question reads, "a subset of Euclidean space being ... bounded (that is, all points lie within some fixed distance of each other)." The assertion "a set X is bounded if all points [in X] lie within some fixed distance of each other" is correct; it is also correct when X happens to be a subset of some larger set (like Euclidean space). Your paraphrase (edited slightly) is that "a bounded set X in a metric space M is a space where there is at most a fixed distance between any two points" is not particularly close to what is written; it is also not false, at least if read in the natural way so that the second copy of the word "space" refers to the set X rather than to the ambient space M. I would be happy to discuss alternate phrasings, but the sentence is at risk of becoming unreadable due to its length. --JBL (talk) 02:36, 6 August 2014 (UTC)
At any rate (whether mathematically correct or not), the lead, and in particular, its opening sentences, is awful not good. For a start, why have two parentheses in the very beginning? YohanN7 (talk) 03:29, 6 August 2014 (UTC)
And, yes, I will do something about it, just not right now. Feel free to beat me at it. YohanN7 (talk) 03:35, 6 August 2014 (UTC)
Joel, I agree that we are reading "fixed distance" in a different manner, and hope we can come up with a wording we agree conveys the correct point. I wouldn't be picky about something like this were the definition not so important to the topic.
My view: as worded, the quantifier binding can be read in more than one direction. This is a common problem with any natural language. If one is not explicit that the for-all binds before the there-exists, the statement is ambiguous. In common speech I would generally interpret as true the statement that "any two points in any metric space lie at a fixed distance from one another", because I read the binding of "any" before "fixed", and this is essentially the definition of a metric space. This motivates me to rephrase it by saying "there exists a (measurable) distance" first, thus making the binding clear.
As for "ambient space", this need not exist. A metric space can be bounded all by itself. While the lede suggests that "compact space" is based on subsets of a metric space, I believe this is intended more as an aid to visualizing the concept than how the concept is used in mathematical proofs. I admit I don't see how "ambient space" really changes the English-suggested binding, however. TricksterWolf (talk) 03:51, 6 August 2014 (UTC)
YohanN7, you'll note directly above this conversation several earlier discussions about the lead; I suggest not making any drastic changes before taking them in. (Incidentally, the reason for the two parentheticals is totally clear -- they include common-language rephrasings of technical terms, in order to help make the sentence understandable without requiring reading outside the article.) --JBL (talk) 03:50, 6 August 2014 (UTC)
Oh, I understand why the parentheses are there. There are hundreds of ways of conveying the same message, all of them better than the present version. But I'll carefully read everything in the earlier discussions. I am aware that this is a vital article. Therefore, it shouldn't be unreadable. To be honest, I don't think I've ever been able to read the whole lead in one go, yet, I understand the subject well enough, so I am not here to learn it. I can't imagine a total novice getting anything useful out of the lead, it would even be worse for someone whose native language isn't English. YohanN7 (talk) 04:36, 6 August 2014 (UTC)

@Tricksterwolf: It's true that often natural language can be hard to parse, but I think that in this case the intended reading of the quantifier is clear to someone who puts forth the effort. I don't think it is necessary to rephrase it. Sławomir Biały (talk) 11:49, 6 August 2014 (UTC)

I understand this topic well, yet the reading of the quantifier wasn't remotely clear to me. This is not because I "didn't put forth the effort". Reading the sentence in question led me to spend about five minutes double-checking the article to see whether or not it was actually suggesting that any metric space is bounded.
More generally, I would not expect a reader unfamiliar with the topic to correctly guess the binding. I don't see that clarity in this case has anything to do with the effort expended. Even if it did, is that really a good excuse for ambiguity in a vital mathematics article? TricksterWolf (talk) 12:28, 6 August 2014 (UTC)
Obviously the reason we have mathematical definitions is to avoid ambiguity. But that is not a reason to exterminate natural language descriptions of things. A formal definition already appears later in the article; the term "bounded" is wikilinked with a formal definition appearing in that article as well. Also, the phrase "all points are within some fixed distance of each other" is not ambiguous: it means that there is some fixed distance such that all points are within that distance of each other. This does not mean that all metric spaces are bounded. Sławomir Biały (talk) 13:15, 6 August 2014 (UTC)
Sławomir, I appreciate you sharing your point of view, but "(blah) is not ambiguous" is an opinion based on your reading. It's clearly ambiguous enough that I mistook the meaning, and I am a native English-speaker who is neither stupid nor a stranger to metric spaces. My read might well be the minority, but I have a hard time believing it is rare. I believe some users coming to this article are likely to misread that sentence which is why I would like to suggest an alternate wording. Implying that I can't read English (or mathematical terminology) properly does not really add to the discussion. TricksterWolf (talk) 20:27, 6 August 2014 (UTC)
I stand by my assertion that it is not ambiguous. The passage reads "all of its points are within some fixed distance of each other". The fixed extends the scope of the quantifier some. What else could this mean, pray tell? A genuinely ambiguous use of the existential quantifier is "everyone loves someone", not "all persons love some fixed person". So, congratulations on being a native speaker of English, but you still got it wrong. Sławomir Biały (talk) 22:44, 6 August 2014 (UTC)
It could mean that for each pair of points, there exists a fixed distance between those two points: this is not always true for a metric space (e.g. consider a non-Abelian distance metric). Also, I apologize for my wording previously as I did not intend it to be aggressive or pedantic. I would like to deescalate this to a more neutral tone to avoid hurting feelings.TricksterWolf (talk) 04:04, 9 August 2014 (UTC)
I do not know what a non-abelian distance metric is. In the English language, there will always be scopal ambiguities, but that is not a reason to eliminate English language descriptions of things. In the case under discussion "fixed" is intended to extend the scope of the quantifier. In the unlikely event that the reader is confused about the scope, we have helpfully provided a link to an article containing a formal definition. However, short of a much more awkward construction involving constructions "there exists" and "for all", I am unable to think of a construction in plain English that does not admit the possibility of a marginal misreading of the scope. For example, "there is a number that is larger than the distance between every pair of points." Surely some responsibility does lie with the reader to interpret correctly what is being said. Sławomir Biały (talk) 16:32, 12 August 2014 (UTC)
I don't think it is ambiguous, but I admit that I had to think twice the first time I saw the formulation. There is a difference between within a fixed distance and at a fixed distance. If you, by a mental slip, miss the within, then you can misunderstand. I think that this is what I did. YohanN7 (talk) 17:43, 12 August 2014 (UTC)
Just to add to my point, even combinations of the constructions "for all" and "there exists" admit different grammatically valid scopal variants because the semantics of these constructions is not prescribed in English. In fact, the only reason that such quantifiers are unambiguous in "plain English" is through their specific use in logic, where they have been declared non-commutative. English is, of course, not the only language in which scopal ambiguity is inherent in the grammar. This is true of all actual natural languages. Structural linguists have made very detailed attempts at codifying a grammar for such expressions as they are used in natural languages; see for example the survey here and a completely different perspective here. However, even attempts to codify such rules are for the most part an ad hoc descriptions of how such sentences are understood in practical language. Sławomir Biały (talk) 19:24, 12 August 2014 (UTC)
I think you may misunderstand. I'm not saying "English is worthless" or "scoping is unfixable", I'm saying that "fixed distance" can be taken two ways. Even if it means one specific thing, I think it can be mistaken for the other. This is why I had to review more of the article to be certain it didn't mean something ridiculous. Also, you can search Google for "non-Abelian distance metric" and find several examples of groups to which this applies (I won't LMGTFY here because I don't want to be rude).
In the meantime, I will think about a possible change of phrasing that might be less confusing without being awkward. TricksterWolf (talk) 19:53, 12 August 2014 (UTC)
No results found for "non-Abelian distance metric". I still have no idea what you mean by "non-Abelian distance metric". Sławomir Biały (talk) 19:57, 12 August 2014 (UTC)

@Yohan: The present version of the lead was the product of much discussion, so major changes should probably also be discussed—ideally in a separate section of the talk page. Personally, I don't care for it much either (although I would not apply the adjective "awful"). However, the problem is not the parentheses in the first sentence, but the rest of that paragraph.

At one point, when the article introduced the notion of compactness by discussing sequential compactness first, it made sense to discuss sequential compactness and the Bolzano-Weierstrass theorem. I would still defend the idea of introducing sequential compactness first: this was historically the first notion of compactness, as evidenced by certain famous theorems of the 19th century, and the version of the lead that introduced compactness sequentially was highly regarded by lay persons. Now I think either everything in the first paragraph following the first sentence needs to be deleted (or rewritten entirely), or the lead needs to go back to the earlier version prior to the last "consensus" forming discussion. Sławomir Biały (talk) 11:49, 6 August 2014 (UTC)

Yes, awful is too strong. My apologies. The math content is, of course, correct and voluminous.
I'm, of course, not going to edit against consensus. I'll simply write a draft (time table unknown), and propose it for discussion somewhere.
My main objection against the current version is that it is so massive. Then, admittedly, I have some objections against sequential compactness getting much attention here. It is mostly (Willard puts it this way) of historical interest. Also, it is equivalent to compactness in the spaces studied most back then. But I don't think it should go out entirely. When (and if) I'm done, I'll open a section here for discussion. Right now I have a skeleton in my sandbox, and will move it to a user page if it gets any substance. YohanN7 (talk) 13:20, 6 August 2014 (UTC)
I would suggest a rewrite of the first paragraph from the second sentence onward. The focus should be on which sets are closed/not closed, and bounded/not bounded (rather than on sequential compactness). The focus of the Bolzano-Weierstrass theorem coming at the end of that paragraph should then be something like the assertion that if a set is closed and bounded, then any sequence has a limit point. Thus the aim should be to give a flavor of how closed and bounded is a nice property to have.
In any case, the earlier parts of the article have gone through many somewhat radical shifts throughout the ages, so it might be worth your time to peruse its history so as to avoid reinventing the wheel. Sławomir Biały (talk) 13:41, 6 August 2014 (UTC)
Length, in and by itself, isn't a problem. But now it is also very dense, and the combination makes it "massive" and hard to penetrate. This is what I mean.
Come to think of it, the concept of limit point compactness is also mixed in, but not mentioned in the current lead.
One goal is to get into the lead how compactness can be used to prove existence theorems (like existence of solutions to integral/differential equations).
Thanks for the hint to look at the article history. I also appreciate the warning about reinventing the wheel. (You say "don't spend too much time on this".) I'll use Willard's General Topology as my main reference when I get to doing this, I can also get my hands on Munkres' book if needed. YohanN7 (talk) 14:25, 6 August 2014 (UTC)
Those are good sources, but they are written very much from the perspective of point-set topology. So that may or may not be an important consideration. I would also look at Bourbaki's "General topology" and Dunford and Schwartz "Linear operators" for possibly a balancing perspective and thorough historical survey. Sławomir Biały (talk) 14:37, 6 August 2014 (UTC)

## Proposed structural changes

This article, in its current state, is badly organised and difficult to use for reference and non-introductory learning. It seems that the efforts that were intended to make it less technical and better as an overview have resulted in this misguided notion that the article should predominantly be accessible to non-topologists.

While I agree that it is worthwhile introducing compactness in a less general, more intuitive way like with sequential compactness, this should not be forced at the expense of making the article clear and concise for anybody who wishes to learn compactness in an order that is natural from the perspective of topology (let alone for anybody who wants to check particular details). Sequential compactness is missing the point; the lead of this article gives a misleading description of what compactness semantically is in its usual usage.

In that regard, this is what I propose:

1. A short, explanation-free lead that focuses on the open cover definition and the closed/bounded characterisation in Euclidean space (this is the essence of compactness to a mathematician, is it not?), as well as mentioning similar notions like sequential compactness (etc.) and giving indications of the importance of compactness.
2. A section entitled Intuitive explanation, Introduction or similar, where compactness is introduced using whatever is considered the most intuitive way.
3. A subsection of the above section containing simple explained examples of compact sets and spaces, free of any more advanced concepts.
4. A section on Definitions, containing formal definitions, as well as relationships (including equivalences) with other definitions, grouped by the subclass of spaces being considered (e.g. metrisable spaces), counterexamples of potential converses, etc.
5. A concise section on Examples, containing all of the examples in the article, perhaps duplicating the simple ones in the "Intuitive explanation" section, free of any elementary explanations (perhaps including some harder proofs for more sophisticated examples).
6. A section on Properties, containing notable general theorems about compact spaces.
• The Historical development section can go anywhere, really – perhaps between "Intuitive explanation" and "Definitions".

FMašić (talk) 21:07, 16 September 2014 (UTC)

I disagree strongly with the characterization of making the article accessible to a wide audience as "misguided". This is an article that is likely to be read not just by professional mathematicians, but undergraduate mathematics majors, physicists, computer scientists, and possibly even high school students. The primary consideration should be to make this article accessible to as wide an audience as possible. It is easy to write an article to be read by mathematicians, but substantially harder to write an article aimed at a general audience. We must bear this in mind with all edits that we make to the article.
1. I would argue that closed and bounded is not the essence of compactness, because it is rather useless by itself. Insofar as there is an essence of compactness, it is that closed and bounded subsets of Euclidean space are sequentially compact! So, as I see it, the lead of the article should say that compactness generalizes closed and bounded subsets of Euclidean spaces, then discuss the Bolzano-Weierstrass theorem and how this is a general feature of compact spaces. This is how the first paragraph is already structured. People like Arzela and Frechet were interested in sequential compactness of spaces of functions rather than subsets of Euclidean space, and these were the first compact spaces that were studied by mathematicians, so they should also be mentioned in the lead of the article, and that brings us to the second paragraph.
2. There used to be an "Introduction" section. At some point it was reduced to the existing "Basic examples" section, but I would happily see this expanded to another introduction section, assuming that can be competently done.
3. See 2 and 5.
4. I think the "Definitions" section should be shorter than it currently is (compare with this old revision of the article). I think it was better to put all of the equivalent characterizations of compactness into a separate section, rather than including them alongside the definition.
5. I've gone ahead and moved the list of examples down to the "Examples" section. I think any listing of more than just one or two examples for illustration should not be given before the definition. Even the examples now in the "Basic examples" section are of limited value, I think. Really to justify their prominent position in the article, someone needs to write a decent introductory section there. But unfortunately most edits to the body of the article recently have been just rearranging things (and actually I think this version of the article prior to a lot of the rearrangements was actually better).
6. I agree about "Properties", although in the past this section was labeled "Theorems" and had quite a bit more content. It might be helpful to mine that older revision of the article for anything useful that may have been removed. Sławomir Biały (talk) 11:53, 17 September 2014 (UTC)
(Please excuse the wall of text). I didn't characterise accessibility as misguided, but rather, I don't agree that the primary consideration should be accessibility in the sense of teaching the topic to newcomers. The article should be easily accessible and pleasing to everyone, and there is no reason that it can't be; ideally, newcomers will be able to understand the basics and some intuition, and more experienced people will be able to find and understand information easily as per their needs. I believe the current lead fails the latter group by mixing useful general information with an explanation that is irrelevant to them.
I also believe that the lead fails everyone by mis-characterising compactness as sequential compactness, which it is semantically not. What I mean by this is that compactness, as a concept, usually refers to the open cover definition (and perhaps the closed and bounded equivalent definition for those who don't seek to generalise from Euclidean space). The goal of the article is to provide information that pertains directly to the standard interpretation of its title; the goal is not to try to explain the concept at the expense of obscuring basic facts, like the definition – don't forget, sequential compactness is not even a characterisation in the general topological case. Even in the case of metric spaces, compactness is usually defined with the open cover definition, which is why we have the semantically-distinct concept of sequential compactness.
That sequential compactness is much more intuitive to explain for metric spaces justifies its use in explaining the concept in an introduction but does not allow us to use it to obscure the lead, which should directly deal with how the word is usually used. I think that the lead should first and foremost present both a descriptive open-cover definition and an indication of what it is intended to generalise. We can then refer to related concepts, and to the fact that sequential compactness is intuitively easier to understand as an introduction and is equivalent for metric spaces (at this point, those confused by the open-cover definition or not seeing what it's trying to capture have been referred to the Introduction, which contains an explanation akin to the current first paragraph; everyone else has not been bothered by the explanation).
The moral is that I think the lead should be concise (also not presenting too much more-fundamental information, like definitions of closed and bounded sets), free of lengthy explanations, and present the most directly relevant and important information on the topic, while the introduction should focus on making the article accessible to those new to it. A candidate lead:
 In mathematics, more specifically general topology, a topological space is compact if every open cover of the space has a finite subcover; that is, from every open cover (arbitrary collection of open sets such that each point in the space is a member of at least one of them), finitely many sets can be chosen to still form an open cover. A subset of a topological space is compact if it is a compact subspace. Compactness generalises the notion of a subset of Euclidean space (which is a topological space) being closed and bounded, and indeed, a compact subset of Euclidean space is often defined as a closed and bounded subset if no generalisation is necessary. The general notion of compactness was introduced by Pavel Alexandrov and Pavel Urysohn in 1929. Compactness captures the property of a topological space being small or tightly-knit in some sense, particularly in the sense of having networks of neighbourhoods that can be represented by finite collections of open sets, capturing some of the character of a finite set despite being generally infinite. It is related to the concept of sequential compactness, which has the advantage of conveying this intuition more clearly; a topological space is sequentially compact if every sequence in it contains a subsequence converging to a point in it. This is equivalent to compactness for metric spaces, so a compact metric space is "small" in the sense of each of its sequences having a point that it must get arbitrarily close to, though it may not stay close to the point. Indeed, the term compact was originally introduced by Maurice Fréchet in 1904 to convey the meaning of sequential compactness, and a number of similar, though not generally equivalent, notions of smallness have arisen that inherit their name from compactness, but compactness itself has prevailed as the most useful notion to work with. Compactness is useful in that it often enables the patching together of information that holds locally—that is, in a neighbourhood of each point—into corresponding statements that hold throughout the space, and many theorems are of this character, such as the passage from continuity to uniform continuity in compact metric spaces. Hence, it plays an extremely important role in mathematical analysis, as many important classical theorems of 19th century analysis, such as the extreme value theorem, are easily generalized to compact spaces. Other than closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points but of functions. A typical application in this situation is furnished by the Arzelà–Ascoli theorem and the Peano existence theorem, in which one is able to conclude the existence of a function with some required properties as a limiting case of some more elementary construction.
I suggest building the introduction out of the current explanation of sequential compactness (which, incidentally, it might be worth copying into the bespoke article!) and the basic examples (thank you for moving the others). On your point about equivalent definitions, the reason I think they should be placed after the definition is that I tend to think of them like this – what characterises a compact space? Which definition should I start from? Though it's probably best to move any theorem not relating types of space to each other into the Theorems section. — FMašić (talk) 03:47, 18 September 2014 (UTC)
That is not bad. I would add the explanations of closedness and boundedness to the first paragraph. But, modulo minor details like this, it seems to me like a good starting point for further edits. I know that Yohan was planning on editing the lead, and it might be best to wait for his (or others') approval. Otherwise, I say go with it. Sławomir Biały (talk) 11:48, 18 September 2014 (UTC)
That rewrite looks reasonable. It is good to retain the sequential compactness in the lead as perhaps one of the more intuitive approaches to compactness. I'm with Sławomir Biały in that explanations of closedness and boundedness in the first para would be good, along the lines of the parenthetical versions in the current lead. --Mark viking (talk) 14:45, 18 September 2014 (UTC)
I completely agree to these critics and proposals, see also my other reply above. This should not be to 90% about Euclidean spaces where most of the most interesting aspects of compactness becomes trivial. If it were up to me I'd replace the intro by the above proposal at once without hesitation... - — MFH:Talk 09:43, 24 May 2017 (UTC)
In my view, this article should be titled compact set, not compact space. Occasionally you have only one topological space around and you want to know whether it's compact, but much more often you're interested in lots of different sets in a common context (that is, subsets of a common topological space). --Trovatore (talk) 10:06, 24 May 2017 (UTC)
I think there should be a top level article compactness (mathematics) that discusses the general concept of compactness, like the lead of this article does. There should also be a separate article compact topological space that focuses on the Aleksandrov–Urysohn open covering definition. I'm not sure where compact set fits into that scheme. Sławomir Biały (talk) 11:28, 24 May 2017 (UTC)
I have no particular opinion except that it is probably not important, since a compact set is compact in whichever space it is embedded. YohanN7 (talk) 12:09, 24 May 2017 (UTC)
It's not "90% about Euclidean spaces". It begins with a discussion of Euclidean spaces, then discusses function spaces, and finally topological spaces. This is correct from a historical perspective, and close to how these concepts are introduced in the mathematics curriculum. Personally, I think we should have two separate articles: one for compactness as it is used more broadly in mathematics. This is not necessarily the same thing as Aleksandrov-Urysohn compactness. (Indeed, others have already observed problems with the title of the present article in this regard.) The latter kind of compactness should be in a separate article, unambiguously titled compact topological space. The above proposed replacement lead can probably be trimmed down to serve as the lead of the new article, while the lead of this article could serve as the introduction to compactness (mathematics). Sławomir Biały (talk) 18:56, 25 May 2017 (UTC)

## Wrong definition

The open cover definition on this page is wrong. Currently it says the following:

Formally, a topological space X is called compact if each of its open covers has a finite subcover. Otherwise, it is called non-compact. Explicitly, this means that for every arbitrary collection

${\displaystyle \{U_{\alpha }\}_{\alpha \in A}}$

of open subsets of X such that

${\displaystyle X=\bigcup _{\alpha \in A}U_{\alpha },}$

there is a finite subset J of A such that

${\displaystyle X=\bigcup _{i\in J}U_{i}.}$

However, those equals signs should be subsets, ie.:

${\displaystyle X\subseteq \bigcup _{\alpha \in A}U_{\alpha },}$
${\displaystyle X\subseteq \bigcup _{i\in J}U_{i}.}$

Apparently, this gross mistake has stood since 2009.

SBareSSomErMig (talk) 17:46, 25 May 2017 (UTC)

It is correct as stated. The ${\displaystyle U_{\alpha }}$ are open subsets of the topological space ${\displaystyle X}$, so one inclusion is automatic. Sławomir Biały (talk) 18:49, 25 May 2017 (UTC)

You are right. But should we not consider the more general case of ${\displaystyle Y\subseteq X}$ being closed, in which the ${\displaystyle U_{\alpha }}$ will not in general be subsets of ${\displaystyle Y}$? Obviously this would also require a restatement of the rest of the definition, which I neglected. SBareSSomErMig (talk) 19:41, 25 May 2017 (UTC)

I now realize that the definition that I had in mind is stated later on. Perhaps a slight reordering would be meaningful to avoid confusing slightly rash indiduals like myself:)SBareSSomErMig (talk) 20:02, 25 May 2017 (UTC)

## Is "hyperreal definition" specific to metric spaces?

I'm concerned that the "Hyperreal definition" subsubsection is not stated as being specific to metric spaces. It's not clear to me what being "infinitely close to a point of X" might mean, if there is no metric around. Should this section be merged into the "Metric spaces" section? What about the paragraph just above it, the second paragraph of "Characterization by continuous functions", which by the way also seems to be redundant with "Hyperreal definition"? --Trovatore (talk) 07:02, 12 July 2018 (UTC)

## The most standard definition is not prominent enough

I think it is not too bold to say that the nearly universal agreement among mathematicians is that the word "compact" by itself, in the most general context and without further qualification, means that every open cover has a finite subcover. I am not saying that definition should appear in the first sentence; we've discussed this before and I agree with opening more gently with the example of closed and bounded subsets of a Euclidean space.

But I am saying that the third sentence of the six-sentence fourth paragraph is way too deeply buried.

I know that Sławomir has strong feelings about the importance of sequential compactness, and that's fine; I don't care to argue that point. Just the same, when the distinction is made explicitly, sequential compactness is the one that takes the adjective; compactness simpliciter is the cover definition. This is entirely standard in the literature.

I don't have an immediate proposal but I do think the standard general definition needs to be treated earlier and more prominently. --Trovatore (talk) 07:17, 12 July 2018 (UTC)

## Heine-Cantor?

The introduction to the Definitions section mentions "Dirichlet's theorem". It seems to me that this is actually meant to refer to the Heine–Cantor theorem, but I don't feel competent enough to edit the article. — Preceding unsigned comment added by 2A02:8070:A191:7D00:AB:87BA:3A49:1844 (talk) 12:23, 20 January 2019 (UTC)

## What does it mean?!

I think this part is wrong or vague:

"typical examples of compact spaces include spaces consisting not of geometrical points but of functions."

Mojtabakd (talk) 09:22, 8 November 2019 (UTC)

I agree, and I have completely rewritten the paragraph. D.Lazard (talk) 10:13, 8 November 2019 (UTC)

## "Since a continuous image of a compact space is compact, the extreme value theorem"

That is not grammatical English, at the very least, a verb must be adduced, e.g. the extreme value theorem follows, or since... we have... etc. 2A01:CB0C:CD:D800:C12E:287B:1E8A:E532 (talk) 20:44, 1 February 2021 (UTC)

The introduction states:

One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space.[1]

While this is true for subsequences ${\displaystyle (s_{i_{n}})_{n}}$ of ${\displaystyle (s_{n})_{n}}$ with ${\displaystyle \forall n,i_{n}, the article subsequence specifically states that subsequences are defined in terms of ${\displaystyle \leq }$ instead of ${\displaystyle {<}}$:

Subsequences can contain consecutive elements which were not consecutive in the original sequence.

That does not make sense here, since it would allow one to prove that every subset of a metric space is compact via the constant "subsequence" (or, equivalently, every topological space is sequentially compact). The non-strict comparison is also in contrast to the linked reference that explicitely defines the index sequence via ${\displaystyle {<}}$ as well as, for example, [2]. In my opinion, it should be made clear - either here or in subsequence - that (non-)strictness matters depending on the context. --217.251.39.96 (talk) 23:23, 9 May 2021 (UTC)

Apparently you confuse "consecutive" and "equal", or you confuse elements of the sequence with their indexes. Said otherwise, the definition of "subsequence" implies ${\displaystyle \forall n,i_{n+1}>i_{n},}$ and your second quotation means that one may have ${\displaystyle i_{n+1}>1+i_{n}.}$ For example, the positive even integers form a subsequence of the positive integers; the integers 6 and 8 are consecutive in the subsequence of even integers, but not in the integers, since 7 is between them. D.Lazard (talk) 08:49, 10 May 2021 (UTC)

References

1. ^ "sequentially compact topological space in nLab". ncatlab.org. Retrieved 2019-11-25.

## Pavel Urysohn

The article mentions a notion introduced by Alexandrov and Urysohn in 1929. However Urysohn drowned in 1923. The works attributed to both at later dates is due to Alexandrov.

## Not a universal property

In the modification by he refers to the following characterisation of a compact space :

X is compact if and only if for every topological space Y, the projection ${\displaystyle X\times Y\to Y}$ is a closed mapping

as a universal property. This is not the case : a universal property is a categorical construction, not the definition of a class of objects in a category. A compactification of a topological space is defined by a universal property but the notion of a compact space is not. Unless this is satisfactorily explained by Taku I'm going to revert this again.

I also see no reason to separate this characterisation from the others: of course it is distinct from the others, otherwise there'd be no reason to have it on the list. Either change the presentation to have a series of paragraphs for each characterisation, perhaps grouping those which are most similar to each other as the various ones using filters or limit points, or have a list for all characterisation without singling out one over the others. jraimbau (talk) 14:13, 13 September 2022 (UTC)

Of course, a construction in category theory can be characterized by a universal property but we can also talk about a universal property of non-construction. So, I think it’s a matter of how one uses the term "universal property" (only for constructions or more generally). Anyway, the reference (Bourbaki) doesn’t use the term "universal property" so I can agree to avoid the term, since you think it’s a wrong usage.
As for separating it from the list: another reason I didn’t add it to the list was that this characterization is not a definition of a compact space, as far as I can tell. On the other than, some others like filter ones are one of the definition of a compact space; for example, Bourbaki defines a compact (or rather a quasi-compact space) in terms of a filter. We shouldn’t give a misleading impression that the projection characterization is a definition of a compact space.
Oh and one more reason: one of a definition of a proper map is that it is a universally closed map. So, a compact space is an absolute version of a proper map or the latter is a relative version of the former. That type of discussion will be awkward to do in the list. -- Taku (talk) 15:44, 13 September 2022 (UTC)
Regarding universal property : there is a proper definition for what this is (of which most people studying "pure" mathematics have an intuition through various examples), so i strongly think that informal usage of the term should be avoided when discussing math formally.
I don't like the list format much here either but if it is used i think it should be used consistently. If one of the characterisations is of particular interest and you want to have a more detailed discussion of it i think you'd be better off putting it in a separate subsection. In fact, the list given here seems flawed in terms of presentation : a few paragraphs above it the usual general definition of compacity (open covers) is given, and it is repeated as item #2 in the list. Maybe a better way to present it would be to split it into several subsections each giving an alternate definition (maybe called "characterisation" if it does not feel like a definition, as seems to be the case with #12). Looking briefly at the list it seems that #2-3 and #6-11 should be grouped together (and so #3 put in the "open cover definition") ; #6-11 could be put in a section called "sequential definition and generalisations" ; that leaves #4, 5 and 12 which could each have their own subsection if puffed up, or for the moment just put into a section called "alternate characterisations" or something else to this effect, avoiding the list format. jraimbau (talk) 15:35, 14 September 2022 (UTC)