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?


1 Answer 1


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.

  • $\begingroup$ I don't understand how this answers the Q. Maybe I'm not seeing something obvious. $\endgroup$ Nov 14, 2015 at 23:16
  • $\begingroup$ For that infinite discrete subset, ask yourself: what are its components? And those of its closure? $\endgroup$ Nov 15, 2015 at 6:00
  • $\begingroup$ The Q is about components of closed subsets $\endgroup$ Nov 15, 2015 at 7:02
  • $\begingroup$ Consider the closure. $\endgroup$ Nov 15, 2015 at 7:13
  • $\begingroup$ yes ok it's obvious $\endgroup$ Nov 15, 2015 at 7:35

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .