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If $X$ is a first countable space can I then somehow show that $X$ is also locally compact? Or are there counterexamples?

Thanks for your time and best regards.

Adrian Keister
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  • or does first countable at least implies the T_0 Axiom ? –  May 17 '18 at 13:12
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    Consider $\mathbb{Q}$. It also doesn't imply $T_0$, for the indiscrete topology on any non singleton set is first countable but not $T_0$. – John Griffin May 17 '18 at 13:12

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The rational numbers equipped with the subset topology from the reals is first countable but not locally compact (since every compact set will have empty interior, so it will not be a neighborhood of any point).

SlipEternal
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