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?
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2026-02-22 23:10:47.1771801847
Is it true that every perfect set of a compact Hausdorff space is uncountable?
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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.