Non-closed compact subspace of a non-hausdorff space I have a topology question which is:
Give an example of a topological (non-Hausdorff) space X and a a non-closed compact subspace.
I've been thinking about it for a while but I'm not really getting anywhere. I've also realised that apart from metric spaces I don't really have a large pool of topological spaces to think about (and a metric sapce won't do here-because then it would be hausdorff and any compact set of a metric space is closed)
Is there certain topological spaces that I should know about (i.e. some standard and non-standard examples?)
Thanks very much for any help 

Here are some examples that work nicely.


*

*The indiscrete topology on any set with more than one point: every non-empty, proper subset is compact but not closed. (The indiscrete topology isn’t good for much, but as Qiaochu said in the comments, it’s a nice, simply example when it actually works.)

*In the line with two origins, the set $[-1,0)\cup\{a\}\cup(0,1]$ is compact but not closed: $b$ is in its closure.

*The set $\{1\}$ in the Sierpiński space is compact but not closed.

*For each $n\in\Bbb N$ let $V_n=\{k\in\Bbb N:k<n\}$; then $\{V_n:n\in\Bbb N\}\cup\{\Bbb N\}$ is a topology on $\Bbb N$, in which every non-empty finite set is compact by not closed.

*Let $\tau$ be the cofinite topology on an infinite set $X$. Then every subset of $X$ is compact, but the only closed subsets are $X$ and the finite subsets of $X$.
In terms of the common separation axioms: (1) is not even $T_0$; (2) and (5) are $T_1$; and (3) and (4) are $T_0$ but not $T_1$.
