Topology: Example of a compact set but its closure not compact Can anyone gives me an example of a compact subset such that its closure is not compact please? Thank you.
Write $\tau$ for the standard topology on $\Bbb R$. Consider now $\tau_0=\{U\cup\{0\}\mid U\in\tau\}\cup\{\varnothing\}$.

It's not hard to verify that $\tau_0$ is a topology on $\Bbb R$. It's also not hard to see that $\overline{\{0\}}=\Bbb R$. However one can easily engineer an open cover without a finite subcover.
Consider $\Bbb R$ with topology generated by basis $B=\{(-\infty,a):a \in\Bbb R\}$ then the set $K=\{1/n : n \in\Bbb N\}$ is compact in this topology but its closure $[0, \infty)$ is not compact. Is this answer correct?