Talk:Compact space

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 ${\displaystyle \forall n,i_{n}<i_{n+1}}$, 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, . 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)

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.

In the modification by @TakuyaMurata: 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)

In a compact Hausdorff space, each closed subset is compact and each compact subset is closed. (Here, "subset" means proper or improper.) Is there an example of a compact non-Hausdorff space where each subset is closed if and only if it is compact? If so, let's add it to the article. —Quantling (talk | contribs) 19:08, 1 June 2023 (UTC)

This article uses collection and subcollection many times, in contradistinction to "set", "subset", "class" or "subclass". I'm not sure why... collection is an informal term (more a part of the language of naive set theory than topology) and it's appearance here is strange. Is this an attempt to make the article more readable? Ross Fraser (talk) 00:49, 24 August 2023 (UTC)

It's used when there are "sets of sets"; instead one says "collections of sets" even though there is no change in meaning. That way when one says, let A be a collection or let A be a set, one knows instantly whether A is one of the containing sets or one of the contained sets. I guess that means that it is for readability.

In contrast "class" is not synonymous with "set", so it would be a poor choice in the present article. —Quantling (talk | contribs) 01:29, 24 August 2023 (UTC)

I see at the Family of sets article that "family" (and "subfamily") could be used instead of "collection" (and "subcollection") in this compact space article. I prefer the latter because it is what I was taught, but if you feel strongly then you could change it to the former and I would live with that. —Quantling (talk | contribs) 13:31, 24 August 2023 (UTC)

Full disclosure: I have just edited the Family of sets article, so that article no longer is an independent voice in this discussion. —Quantling (talk | contribs) 16:02, 24 August 2023 (UTC)

This doesn't seem to be correct. In the cocountable topology, the only compact sets are the finite sets. These are closed, as the cocountable topology is T1. But the cocountable topology on an uncountable set is not Hausdorff. 2A02:A03F:8CEC:D00:CC18:E805:F3B7:DD52 (talk) 16:56, 3 January 2024 (UTC)

Good observation, but I think the cocountable topology is not a TVS topology because addition is not continuous (any T1 topological group is hausdorff). A reference would be nice though. Tito Omburo (talk) 17:38, 3 January 2024 (UTC)

After further reflection, the singleton ${\displaystyle \{0\}}$ is compact, and is closed iff the topology is hausdorff. Tito Omburo (talk) 17:47, 3 January 2024 (UTC)

Link: https://imgur.com/a/y59lw01

Note: this proof depends on the axiom of choice (AC). Without it, I am not sure it is possible to prove that condition 12 is equivalent to compactness.

Now, is it possible to prove AC from the equivalence of compactness to condition 12? I don't know.

If so, it may be related to the collection of partial well orderings of a set X. A couple of observations:

1. the collection of partial well orderings of a set X is partially ordered by initial segment relation, x <= x' if x is an initial segment of x'
2. it may be possible to endow this partially ordered set with a topology and thereby obtain a link between AC, the well ordering theorem, and equivalence of compactness to condition 12

Cheers,

24.43.142.162 (talk) 04:07, 25 March 2024 (UTC)

Hi 24.43.142.162, and thanks for your contribution.

Could you provide an external reference for the proposition you added to the article? E.g, a book where the proof (or at least the statement) can be found? Otherwise, per Wikipedia's policies, it will be considered "original research" (the term has a special meaning on Wikipedia, see WP:OR) and will eventually be removed.

Note that it does not matter that you provided a proof: the core principle of Wikipedia is that it is not an authoritative source, but a well-structured synthesis that provides links to external sources (also note that since contributors are anonymous and anybody can contribute, you can't trust contributors to check the logic behind scientific arguments). So, even though that might be annoying to you, your proof is worthless in that context. If you want to contribute proofs to a collaborative encyclopedia, you can check ProofWiki.

Cheers, Malparti (talk) 13:37, 25 March 2024 (UTC)

Fine, then remove it, write a letter to American Mathematical Monthly with the proof included, tell them that you've submitted my proof, in care of yourself, and the publication should bear that authorship (by X (me) in care of you). When they publish it, you can add this proof back and cite the publication, year, month, issue, page, etc...

If you get started quick you might have it done by the end of the year!

24.43.142.162 (talk) 14:51, 25 March 2024 (UTC)

Here is a shorter, more intuitive proof: https://imgur.com/a/BIv9O0w — Preceding unsigned comment added by 47.44.0.75 (talk) 22:33, 13 January 2025 (UTC)

It is false that this is equivalent to compactness.

Suppose the statement was true that this equals compactness. Now consider an open interval (0, 1). Take any nested sequence of closed subsets from it. The first subset is compact (closed and bounded), hence according to the statement in question the sequence must have a non-empty intersection. We have proved than an open interval (0, 1) is compact, per the statement, which is clearly wrong.

Tito Omburo, do you agree?

I see that exactly the same criterion is written below in the section for ordered spaces. It may be correct there, I am not familiar with those, but for metric spaces it is wrong. 2A00:79E0:2820:7:C4B9:FEB8:ABE0:377A (talk) 16:17, 27 August 2025 (UTC)

The closed subsets ${\displaystyle S_{n}=(0,1/n]}$ in your example have empty intersection. Tito Omburo (talk) 16:32, 27 August 2025 (UTC)

A statement that is correct is the contrapositive and with the complement of the sets ... of the statement "Every open subcover has a finite subcover", which is the archetypal definition of compactness of any topological space. A set is compact if and only if "For every collection of closed subsets that has the property that the intersection of any finitely many of them is non-empty, it will be the case that the intersection of the entire collection is non-empty."

Although it is a bit of an aside, maybe there is no harm in also insisting that the collection of closed sets be a sequence and that the sequence be nested, which, in terms of opens sets would be like insisting that the open cover be a sequence and that the sequence be of ever larger open sets. —Quantling (talk | contribs) 17:38, 27 August 2025 (UTC)

Apologies, I see we have a link to finite intersection property in the article already. Nvm. —Quantling (talk | contribs) 17:45, 27 August 2025 (UTC)

It's only equivalent for nested sequences in a space whose topology is sequential (e.g., metric spaces). For arbitrary sequences with the finite intersection property, it characterizes countable compactness rather than compactness. Tito Omburo (talk) 17:55, 27 August 2025 (UTC)

The lede gives the following as intuition for compact spaces:

The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact.

This is the intuition for closed sets, not compact sets, and compact sets need not be closed (so in my opinion this intuition is wrong). Of course, in a metric space compact sets are closed (and this is mentioned later on), but the two concepts are not the same in general. A better intuition would be that it somehow generalises "finiteness" (functions from finite sets have nice properties, continuous functions from compact sets have similar nice properties, see ). Also, this aligns with the main reason why compactness is useful in proving things. I'm not entirely sure how to rewrite it in an approachable way right now... ★Maxman013★^(talk) 04:33, 17 November 2025 (UTC)

I think the current lede paragraph is largely is disimprovement over this revision, but I think trying to explain what compactness is "from within" in the first paragraphs is the wrong approach in any case. I also think would be better to say that compact spaces are generalizations of finite sets. Finite sets have the property that every function is bounded, every function assumes a maximum and minimum value, and every infinite sequence must assume the same value infinitely many times (pigeonhole principle). Compact spaces satisfy the first two, with functions replaced by continuous functions, and every sequence has a convergent subsequence. Tito Omburo (talk) 11:43, 17 November 2025 (UTC)

I agree that the revision quoted by Omburo is much better and should be restored. For the momemt, I have fixed the current version, by removing the confusing concept of "punctured set" and replacing all "would be" by "be". Nevertheles, the quoted version remains better. D.Lazard (talk) 12:31, 17 November 2025 (UTC)

I think "a generalization of closed and bounded" is the way forward. Obviously "closed and bounded" is equivalent to the definition when it comes to Euclidean space, and readers will be more familiar with Euclidean space than most any other topological spaces. But also, it would be good if the lede somehow expresses why people care whether a space is compact. So, I'd like to see both that simpler definition and some motivation for it. —Quantling (talk | contribs) 14:23, 17 November 2025 (UTC)

I sort of disagree. "Closed and bounded subsets of Euclidean space" is too technical for the first paragraph. We can actually say what compact spaces are "from outside" more easily than "from within", and this would give casual readers a clearer idea why compact spaces are important and useful, as opposed to what they "are". Tito Omburo (talk) 00:31, 18 November 2025 (UTC)

The article says "This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929". Pavel Urysohn died in 1924, so this cannot be correct. Kung Midas (talk) 21:19, 25 February 2026 (UTC)

That is because the relevant seminal article is "Mémoire sur les espaces compacts", by Alexandrov and Urysohn, which was indeed published in 1929. I do not know what's the best way to solve the issue. Malparti (talk) 21:50, 25 February 2026 (UTC)