Projective limits of compact groups

Question

Is the projective limit of an exact sequence of projective systems of compact (not necessarily Hausdorff) topological groups exact?

Is the projective limit of nonempty compact (not necessarily Hausdorff) spaces compact?

Answer 1

No: just take any counterexample in ordinary groups, and give all the groups the indiscrete topology. Explicitly, for instance, you could take the natural projective system formed by the short exact sequences $$0\to p^n\mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}/(p^n)\to 0$$ whose limit is $$0\to 0\to \mathbb{Z}\to \mathbb{Z}_p\to 0$$ which is not exact.

For the second question, you can use a related example. Let $G_n$ be $\mathbb{Z}$ with the topology where the open sets are unions of cosets of $p^n\mathbb{Z}$. The identity maps make these into an inverse system, and each $G_n$ is compact. However, the inverse limit is $\mathbb{Z}$ with the $p$-adic topology, which is not compact.