Talk:Heine–Borel theorem

message: Why is this page under a title with an en dash, with the hyphenated page redirecting to here? En dashes are for separating members of ranges (like "3–10 years") whereas hypens are for separting words (as in the title of this article), and having the wrong one both looks irksome and makes a google for the theorem show up incorrectly. I would put in one of those move article vote thingys here but I'm not sure how to initiate that. (By the way I also fiddled with the formatting of the below discussion to separate it from this one.) Quietbritishjim (talk) 09:46, 11 March 2008 (UTC)
thread_title:  Page Title 

message: The current (April 21, 2009) proof of the part under If a set is compact, then it must be closed. is wrong.
Let a set S be compact. A finite collection C of open sets which do not intersect at least one neighborhood of an accumulation point \vec a cannot be an open cover of S, because the intersection of those neighborhoods not met forms an open set, which contains a point in S not covered by C.
1) Every set imaginable will intersect at least one neighborhood of any point; just increase the size of the neighborhood until it intersects your set. So that sentence makes no sense. It is possible that the original author meant to say every neighborhood.
2) The last part is hard to figure out, but I think the author intended to say the union of neighborhoods that are not intersected by C contains a point in S not covered by C. The problem is, if S is a closed set, then it contains every accumulation point.
3) I'm not really sure about this, but do we need to assume that we're working in \mathbb R^n in order to prove that compactness => closure? According to the statement of the theorem in the introduction, in the general case we need to prove completeness, not closure.  —Preceding unsigned comment added by 75.18.231.173 (talk) 05:30, 22 April 2009 (UTC)
replies: text: The part of the proof that you mentioned is not wrong, it is just hard to follow. I'll try to make it clearer if I can. --Bdmy (talk) 09:28, 22 April 2009 (UTC)

text: I think unsigned is misinterpreting the text of the proof, because it is harder to be specific in English (and other natural languages) than in symbolic logic. The quote is not saying that the open sets in C don't intersect any neighborhood of a at all. It is saying that, for each set in C, there is some neighborhood of a that it does not intersect. I think that perhaps the current wording ("such that each open set U is disjoint from some neighborhood ...") makes this clearer. --Hccrle (talk) 22:56, 19 March 2012 (UTC)
thread_title: Wrong proof?

message: I don't think this can actually be done. The proofs I have seen, including the one given in the citation, use a filter- or net-based definition of completeness rather than the sequential one given here. The following is of course not conclusive, but on p253 of Mícheál Ó Searcóid's Elements of Abstract Analysis, the author states that "it cannot be shown in ZF that a sequentially closed subset of an arbitrary metric space is necessarily closed, simply because it cannot be proven that an arbitrary infinite set has a countable subset."
I'm talking about the 2002 edition of Searcóid's book and the 1996 edition of Schechter's.
At the risk of being an idiot, just think about it: How am I supposed to use the completeness of a space if I can't even construct a sequence to my specifications?  —Preceding unsigned comment added by 67.175.99.81 (talk) 04:12, 2 September 2010 (UTC)
thread_title:  Provability of uniform space (or metric space) version from ultrafilter principle 

message: I removed the following link from the references section
* A simple and humorous proof of the Heine-Borel theorem using model theory, by M. Macauley et al
First of all because it wasn't a reference for any material on the page, secondly because it isn't notable enough to warrent inclusion as an external link, and finally because at the critical part of the proof (where the completeness of the reals is actually used) they, in their own words, "omit crucial but obvious details". 
To expand on the third point: I haven't thoroughly read the paper, but on skimming it it doesn't make sufficiently clear where the completeness of the reals is used.  Suspiciously, near the beginning the authors say that when proving Heine-Borel one must "summon ... the completeness of the reals, one of those blatantly obvious analysis facts that you had no idea how to prove".  Well, normally you don't prove it, you assume it as an axiom.  You can prove it if you construct the reals, that's really just to check that a set satisfying the axioms exists, not to prove things about the reals; in any case, if that's the view they take, they should state what construction of the reals that they're using, and how their proof uses that. Quietbritishjim (talk) 16:40, 20 October 2010 (UTC)
replies: text: The paper has not been published yet apparently, so removing the link is not inappropriate.  On the other hand, the idea is to use the compactness of the space of binary trees.   You show that the interval [0,1] is compact by showing that it is the image of a compact set.  Where do they use the completeness of the reals?  If they did, their argument would be circular.  Having said that, it is possible that that's the reason it did not get published :)  Tkuvho (talk) 14:07, 21 October 2010 (UTC)
thread_title:  Removed link 

message: The proof of the Heine–Borel theorem uses Cantor's intersection theorem, and the proof of Cantor's intersection theorem uses the Heine–Borel theorem. The two theorems each use the other to prove themselves! Nick Levine (talk) 08:13, 6 August 2013 (UTC)
thread_title:  Circularity? 

message: "Indeed, the intersection of the finite family of sets V_U is a neighborhood W of a in R^n." 212.149.241.8 (talk) 14:51, 7 February 2014 (UTC)
replies: text: Let W be the intersection of the finite family of sets VU. Then W is a neighborhood of a in Rn. 
Quietbritishjim (talk) 15:02, 7 February 2014 (UTC)
thread_title:  This is not a sentence. 

message: The article cites two versions of A Pedagogical History of Compactness by Raman–Sundström, one an arXiv preprint
and one the final published version from American Mathematical Monthly.  I did not want to update the arXiv citation to point to the published version, because I don't know what revisions the arXiv version might have undergone before publication.  But it looks strange to cite two versions of the same paper, and this ought to be cleaned up at some point.  —Mark Dominus (talk) 18:27, 7 December 2015 (UTC)
thread_title:  Two versions of the same paper 

message: In the proof, when showing that a closed subset of a compact space is compact, there's an error...sort of. The article never clarifies what it means by an open cover; since we're only working in \mathbb{R}^n, an open cover of a set X could mean a collection of open subsets of \mathbb{R}^n whose union includes X, in which case the given proof is fine. If, on the other hand, an open cover of X refers to a collection of open subsets of X whose union is X, then the proof given is false. In particular, it claims that, given an open cover C_K of K, then C_K\cup \{U\} is an open cover of T (where U is the complement of K). But an open subset of K is only relatively open in T, it need not actually be open. The correct proof then would be to union U onto every set in C_K.
Given that U is defined as \mathbb{R}^n\backslash K, the author presumably intended an open cover to mean open subsets of R^n - but either this should be made explicit, or the proof should be changed. I'm not sure which is the better option, so I've just put this comment, rather than making any actual changes.  — Preceding unsigned comment added by 130.88.123.219 (talk) 13:24, 11 January 2017 (UTC)
thread_title:  Error/ambiguity in proof? 

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thread_title:  External links modified 

message: This page is in the category «Properties of topological spaces». Can anyone explain why it is the case ? It seems not to be a property, but a theorem with conditions and conclusion. Should we remove this category ?  — Preceding unsigned comment added by Arthur MILCHIOR (talk • contribs) 04:21, 4 March 2018 (UTC)
thread_title:  Property ?