Talk:Cantor's intersection theorem

Czech Wiki proves the first part of the theorem this way: Have a sequence an such that its n-th member is inside Cn. Then the whole sequence is inside C1, and because C1 is compact, there is a convergent subsequence ank inside C1 with some limite "a". Since every Cl contains all following Cl+1, Cl+2,..., that means it contains al,al+1,... Thus, in every Cl only a finite amout of members from the sequence is missing, so it contains infinite number of members of the subsequence ank. Therefore, one can choose a sub-subsequence from ank that will converge in the said Cl. But since this sub-subsequence is chosen from ank which has a limite "a", then "a" must be contained in every Cl, because they are compact... Hope this helps M. Pos — Preceding unsigned comment added by 185.116.77.206 (talk) 12:40, 10 June 2018 (UTC)[reply]

I'm unsure the statement of the theorem as it currently stands is correct, it should either be clear that the spaces in question are metric. Or if the article is intended to deal in more generality it should be more precise as if we take the statement to be about topological spaces it is not correct (we need to assume Haussdorfness). The article did used to make sense when it only dealt with subsets of the reals, but was edited in February and is confusing as stands. Alex J Best (talk) 12:13, 23 May 2013 (UTC)[reply]

My problem with this is that 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. Nick Levine (talk) 08:12, 6 August 2013 (UTC)[reply]

Yeah, there are definitely problems as the article stands right now. The article asserts that the statements "a decreasing nested sequence of non-empty compact subsets of $S$ has nonempty intersection" and " a (nested) sequence of non-empty, closed and bounded sets (has nonempty intersection)" are equivalent statements of the theorem, but these two statements are not equivalent. The first statement is true for any compact topological space and is proved in the "proof" section. The second statement assumes (implicitly) that $S$ is a metric space (so that "bounded" means anything), and is the one used to prove the Heine-Borel theorem, but it is not true without additional assumptions.

Completeness is basically always assumed, but even this isn't good enough. One additional sufficient additional assumption is that the diameters of the nested sets approach 0 (see for example [1]), though some sources even explicitly assume that $S$ is a closed and bounded subset of Euclidean space (see for example [2]). Perhaps we can say that Cantor's Intersection Theorem refers to two different theorems, and state both of them? Abramorous (talk) 16:08, 28 August 2013 (UTC)[reply]

Nice proof! The last line is very informative. — Preceding unsigned comment added by 163.1.98.15 (talk) 15:25, 8 October 2013 (UTC)[reply]

Does the topological proof actually use Hausdorff anywhere? I think not. Can we remove the Hausdorffness assumption? — Preceding unsigned comment added by Agnishom (talk • contribs) 06:31, 18 February 2018 (UTC)[reply]