Isolated points and Baire category theorem

Question

Let $(X,d)$ be compact. Can the set of isolated points be countably infinite? Are the set of isolated points constitute a discrete set? Also I would like to know whether we can consider "the intersection point" in the proof of Baire Category theorem as an isolated points?

Answer 1

Let $X=\{0\}\cup\left\{\frac1n:n\in\Bbb N\right\}$ with the usual Euclidean metric that it inherits from $\Bbb R$; this is a compact metric space, and all of its points except $0$ are isolated, so it has countably infinitely many isolated points. A compact metric space cannot have uncountably many isolated points, however, because it is separable, and a dense subset of a space must contain every isolated point of that space.

I’m not sure what you mean by ‘the intersection point’ in the proof of the Baire Category Theorem, but none of the points that are constructed in the usual proof is necessarily isolated.