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. 



Is a inverse limit of compact spaces again compact ?
What does this example do ... 


All spaces are on set $\{1,2,\dots\}$. Space $X\_n$ has topology that makes $\{1,2,\dots,n\}$ discrete and $\{n+1,\dots\}$ indiscrete. Of course $X\_n$ is compact non-Hausdorff. Map $X\_{n+1} \to X\_n$ by the "identity". Inverse limit is ... ???


