Compact set whose closure isn't compact?

Question

I've been killing myself trying to think of a compact set (in the Borel-Lebesgue sense) whose closure isn't compact. Obviously not considering a Hausdorff space, but any topological space in the most general possible way

Answer 1

Consider the topology $\tau$ on $\mathbb{N}$ generated by $$\{\{1, i\}: i\in\mathbb{N}\}.$$ Then:

• Is $\tau$ compact?

• What is the closure of $\{1\}$?

• Is $\{1\}$ compact? (HINT: it's finite . . .)

Answer 2

Let $X$ be an infinite set, and fix $p\in X$. Let $\tau=\{\varnothing\}\cup\{U\subseteq X:p\in U\}$; then $\tau$ is a topology on $X$, $\{p\}$ is compact, $\operatorname{cl}\{p\}=X$, and

$$\big\{\{p,x\}:x\in X\big\}$$

is an open cover of $X$ with no finite subcover.