TITLE: Is a inverse limit of compact spaces again compact ?
QUESTION [17 upvotes]: Then one can construct a model for the inverse limit by taking all the compatible sequences. 
This is a subspace of a product of compact spaces. This product is compact by Tychonoff. If all the spaces are Hausdorff, then this is even a closed subspace. 
However, if the spaces are not Hausdorff, it needn't be a closed subspace. If you take a two point space with the trivial topology as $X_n$ and constant structure maps, you will get as the inverse limit the space of all constant sequences, which is not a closed subspace of the infinite product, as the infinite product also has the trivial topology. 
But the space is again compact. So I am wondering, whether there is a generalization of the proof of Tychonoff's theorem, that applies directly to inverse limits.

REPLY [11 votes]: Although the question is quite old I just ran into it. You might still want to (if you haven´t done so yet) look at the article:
A.H. Stone, Inverse limits of compact spaces, General Topology and its Applications 10 Issue 2 (1979) pp 203–211 https://doi.org/10.1016/0016-660X(79)90008-4.
It contains some positive results.