Let $X$ be a Hausdorff topological space such that every closed subset has finitely many connected component. How can I verify that $X$ is finite?
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Hint: every infinite Hausdorff space has an infinite discrete (in itself) subset.
(E.g. see this answer)
The cofinite topology on $\mathbb{N}$ shows that being $T_1$ is not enough.
answered Nov 14, 2015 at 19:18
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$\begingroup$ I don't understand how this answers the Q. Maybe I'm not seeing something obvious. $\endgroup$ Nov 14, 2015 at 23:16
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$\begingroup$ For that infinite discrete subset, ask yourself: what are its components? And those of its closure? $\endgroup$ Nov 15, 2015 at 6:00
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$\begingroup$ The Q is about components of closed subsets $\endgroup$ Nov 15, 2015 at 7:02
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