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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.

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  • $\begingroup$ I don't understand how this answers the Q. Maybe I'm not seeing something obvious. $\endgroup$ – DanielWainfleet Nov 14 '15 at 23:16
  • $\begingroup$ For that infinite discrete subset, ask yourself: what are its components? And those of its closure? $\endgroup$ – Henno Brandsma Nov 15 '15 at 6:00
  • $\begingroup$ The Q is about components of closed subsets $\endgroup$ – DanielWainfleet Nov 15 '15 at 7:02
  • $\begingroup$ Consider the closure. $\endgroup$ – Henno Brandsma Nov 15 '15 at 7:13
  • $\begingroup$ yes ok it's obvious $\endgroup$ – DanielWainfleet Nov 15 '15 at 7:35

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