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A proof can be found here, but it seems that it uses AC. I would like to know if there is a proof for this fact without AC.

I came up with this question after seeing a proof of sequential compactness $\implies$ Compactness for metrizable topological spaces without using generalized Lebesgue's Number Lemma for sequentially compact metric spaces. Both proofs can be found here.

Eric Wofsey
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William Sun
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The result cannot be proved without the axiom of choice. For instance, let $X$ be an infinite set which contains no countably infinite subset (the existence of such a set is consistent with ZF), and give $X$ the discrete metric. Then $X$ is sequentially compact since there are no sequences in $X$ which take infinitely many different values so every sequence has a constant subsequence. But $X$ is not Lindelöf, since the open cover of $X$ by singletons has no countable subcover.

Eric Wofsey
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  • May I know how do we construct such X? Thanks. – Arctic Char Mar 28 '19 at 04:57
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    That's a very long story if you're not familiar with forcing. See https://math.stackexchange.com/a/1396679/86856 for a sketch of the construction; you can find the details, for instance, in Example 15.52 of Jech's Set Theory (3rd edition). – Eric Wofsey Mar 28 '19 at 05:18
  • That is indeed a good answer, but just curious, does the statement "sequential compactness implies compactness for metric spaces" rely on the axiom of countable choice? Is there a proof without it? – William Sun Mar 28 '19 at 05:27
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    @WilliamSun: Since compact is stronger than Lindelöf, if you need choice to prove sequentially compact metric spaces are Lindelöf you also need it to prove they are compact. – Eric Wofsey Mar 28 '19 at 05:29
  • @WilliamSun In fact, for pseudometric spaces the implication "sequentially compact $\Rightarrow$ compact" has been shown to be equivalent to countable choice by Herrlich and Bentley https://www.sciencedirect.com/science/article/pii/S0166864197001387 For metric spaces, the situation is more difficult. I think Kyriakos Keremedis has a paper out that treats this but I'm not sure. – Cloudscape Mar 26 '24 at 21:12