4:40 AM
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I was reading III vol. of Princeton lectures on analysis. Proposition 1.4: "If $\Omega_{1}\supset\Omega_{2}\supset\ldots\supset\Omega_{n}\supset\ldots$ is a sequence of non-empty compact sets in $\Bbb C$ with the property that: $$\operatorname{diam}(\Omega_{n})\to 0\text{ as } n\to\infty,$$ th...

Looking at the question at that question it seems natural to ask whether Cantor's intersection theorem holds without AC.
In real analysis, a branch of mathematics, Cantor's intersection theorem, named after Georg Cantor, is a theorem related to compact sets of a topological space S {\displaystyle S} . It states that a decreasing nested sequence of non-empty compact subsets of S {\displaystyle S} has nonempty intersection. In other words, supposing {Ck} is a sequence of non-empty, closed and totally bounded sets satisfying C 0 ⊇ ...
We have two versions of Cantor's theorem (compact spaces, complete metric spaces). And we may ask about various classes of spaces.
It seems that proof from Asaf's answer would work for Polish spaces.
And the proof from vow lacks forte's answer would work for compact spaces.
(I hope I am not missing something there.)