Can a countably infinite compact topological space have isolated point? Can it admit a minimal subsystem?

Question

Examples I could think of are all sequences with their limit. But is every countably infinite compact space admit atleast one isolated point?

Answer 1

Let $(X, \mathcal T)$ be a compact topological space (countable or not, it does not matter). For $x \notin X$ let $Y = X \cup \{x\}$ be a new topological space with the topology generated by $\mathcal T \cup \{ \{x\} \}$.

Then $Y$ is compact and $x$ is an isolated point in $Y$.