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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)
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)
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)
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)
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:
While this is true for subsequences of with , the article subsequence specifically states that subsequences are defined in terms of instead of :
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)
References
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 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)
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)
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)
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:
Cheers,
24.43.142.162 ( talk) 04:07, 25 March 2024 (UTC)
![]() | This ![]() It is of interest to the following WikiProjects: | ||||||||||
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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)
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)
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)
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)
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:
While this is true for subsequences of with , the article subsequence specifically states that subsequences are defined in terms of instead of :
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)
References
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 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)
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)
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)
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:
Cheers,
24.43.142.162 ( talk) 04:07, 25 March 2024 (UTC)