[Math] The number of isolated points in a compact space

Question

Let $X$ be a compact topological space( it is Hausdorff). Can $X$ have infinitely many isolated points?

Answer 1

Yes. Take $X=\{0\}\cup\{2^{-n}:n\in\mathbb{N}\}$ with the topology given by the distance $d(x,y)=|x-y|$. Then $X$ is compact (it is bounded and closed) and it contains infinitely many isolated points, namely $\{2^{-n}:n\in\mathbb{N}\}$.