Talk:Compact space

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)

`Subsequence' in this article is different from the linked subsequence

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.

While this is true for subsequences $(s_{i_{n}})_{n}$ of $(s_{n})_{n}$ with $\forall n,i_{n}<i_{n+1}$, the article subsequence specifically states that subsequences are defined in terms of $\leq$ instead of ${<}$:

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 ${<}$ as well as, for example, [1]. 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 $\forall n,i_{n+1}>i_{n},$ and your second quotation means that one may have $i_{n+1}>1+i_{n}.$ 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

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 @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 $X\times Y\to Y$ 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)

Each subset is closed if and only if it is compact

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)

Strange use of "collection" and "subcollection"

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)

"every non-Hausdorff TVS contains compact (and thus complete) subsets that are not closed"

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

Proof for equivalent condition 12 in "Characterizations"

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)