The inverse limit of a system of compact Hausdorff spaces is compact (and Hausdorff). It is stated in a comment to this question that if the spaces are compact non-Hausdorff, the inverse limit need not be compact.
What would be an example of an inverse limit of compact spaces that is not compact?
Best Answer
"Stealing" the mathoverflow answer, because it's so nice, and to have a self-contained site "here":
All spaces $X_n$ are on set $\{1,2,\dots\}$. The space $X_n$ has the unique topology that makes $\{1,2,\dots,n\}$ discrete and $\{n+1,n+2, \dots\}$ indiscrete. Of course $X_n$ is compact (the topology is finite) and non-Hausdorff. Map $X_{n+1} \to X_n$ by the bijective map $f_{n+1}(x)=x$, which is continuous (as can easily be checked). The inverse limit is $\{1,2,3,\ldots\}$ in the discrete topology (all "threads" are of the form $(n,n,n,\ldots)$ and have are isolated points in the limit as a subspace of $\prod_n X_n$, as can also easily be checked: we can take $\{n\}$ in the $n$-th component of a basic product open set etc.)
So the inverse limit of compact non-Hausdorff spaces can be very non-compact..