# 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?

No. Consider $$X=[0,1]$$, $$A=\{0\}\cup\{\frac1n:n\in\mathbb{N}\}$$.
• Thanks. What can say about $A$, if $A$ has no isolated point? – user479859 Dec 13 '18 at 21:14