Finite topological space

Question

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?

score 3 · Answer 1 · edited Apr 13 '17 at 12:20

3

Hint: every infinite Hausdorff space has an infinite discrete (in itself) subset.

The cofinite topology on $\mathbb{N}$ shows that being $T_1$ is not enough.

edited Apr 13 '17 at 12:20

Community

answered Nov 14 '15 at 19:18

Henno Brandsma

I don't understand how this answers the Q. Maybe I'm not seeing something obvious. – DanielWainfleet Nov 14 '15 at 23:16
For that infinite discrete subset, ask yourself: what are its components? And those of its closure? – Henno Brandsma Nov 15 '15 at 06:00
The Q is about components of closed subsets – DanielWainfleet Nov 15 '15 at 07:02
Consider the closure. – Henno Brandsma Nov 15 '15 at 07:13
yes ok it's obvious – DanielWainfleet Nov 15 '15 at 07:35

1 Answers1