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 

Since Qiaochu's example is almost too trivial to give much intuition, here is another good example to think about.
Take the real line minus the origin. Now take a disjoint union with 2 points and declare the neighborhoods of both points to be exactly the neighborhoods of the origin. This topological space should be thought of as "the real line with 2 origins". It is definitely not Hausdorff.
Now take $x \in \mathbb{R} - \{0\}$ and consider the subspace $[-x, 0) \cup (0, x] \cup *$, where $*$ is one of the origins.
This is a compact subspace because it is basically the space $[-x,x] \subseteq \mathbb{R}$, but it isn't closed (the other origin is isolated in the complement).
