Find an example of a compact space which is not locally compact.

Question

I know that every $T_2$ compact space is locally compact.So I need to find a space $X$ that is compact but not $T_2$ , then prove that the there exist a point $x$ that is not in $A^o$ for $A$ is compact subset of $X$. So I guess a finite set will guarantee that I have a compact set, but it may also lead to the locally compact as well.

I use this definition: A spcae $X, \tau$ is locally compact if $\forall x \in X$ and any neighborhood $U$ of $x$, there is a compact set $A$ such that

$x \in A^o \subset A \subset U$

Answer 1

Let $\hat{\mathbb{Q}}$ be the Alexandrov compactification of $\mathbb{Q}$ with the usual topology. Then $\hat{\mathbb{Q}}$ is (quasi) compact, being an Alexandrov compactification. But it is not locally compact, because (with your definition, equivalent to what I was saying earlier) $(-1,1)$ is a neighborhood of $0$ but it contains no compact set $A$ such that $0 \in A^\circ \subset A \subset (-1,1)$ (in fact every compact subset of $(-1,1)$ has empty interior).