Let $X$ be a finite set. Prove or disprove: $(X,\tau)$ is separable topological space.

Question

My attempt:

$X$ is finite, so countable, and $\overline{X}=X$, so $(X,\tau)$ is separable. Is this correct?

Answer 1

Sure. It’s totally obvious. We can also state it’s second countable, compact, Lindelöf, countably compact, first countable,all for the same reason.