# Is it true that every perfect set of a compact Hausdorff space is uncountable?

Let $$(X, \mathcal{U})$$ be a compact Hausdorff space and it has no isolated point. Let $$A\subseteq X$$ is a closed and infinite set with no isolated point . Is it true that $$A$$ is uncountable set?

Thanks for your help.

• What are your thoughts? – Math1000 Dec 13 '18 at 20:52
• @Math1000,I know that in a complete metric space, every perfect set is uncountable. But The set of rationals with the usual subspace topology is a countable perfect set. In recently, I study uniform spaces and I need to know that when a closed and infinite set is uncountable. – user479859 Dec 13 '18 at 21:00

## 1 Answer

No. Consider $$X=[0,1]$$, $$A=\{0\}\cup\{\frac1n:n\in\mathbb{N}\}$$.

EDIT: However, any (non-empty) compact Hausdorff space with no isolated points is uncountable. Show the given space is uncountable. And if you have a closed subset, it is also a compact Hausdorff space, so if it (is non-empty and) has no isolated points, it must be uncountable.

• Thanks. What can say about $A$, if $A$ has no isolated point? – user479859 Dec 13 '18 at 21:14
• Made an edit to answer that question. – SmileyCraft Dec 13 '18 at 21:24