Can anyone provide me with an example of a compact space that has subsets that aren't compact?
Any compact subset of a Hausdorff space with the relative topology is a compact space.
Now any open set is that compact subspace is not compact. WHY?
Example: $\displaystyle [-2,2]\subset \mathbb{R}^1$ with the relative topology is a compact space.
But $\displaystyle (0,1)$ is not compact.