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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
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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"
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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
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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
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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.
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only a finite amout of members from the sequence is missing, so it contains infinite number of members of the subsequence a
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Does the topological proof actually use
Hausdorff anywhere? I think not. Can we remove the Hausdorffness assumption?
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Czech Wiki proves the first part of the theorem this way: Have a sequence a
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is a closed and bounded subset of
Euclidean space (see for example
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which has a limite "a", then "a" must be contained in every C
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section. The second statement assumes (implicitly) that
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