In Royden 3rd P192,
Assertion 1: Let $K_n$ be a decreasing sequence compact sets, that is, $K_{n+1} \subset K_n$. Let $O$ be an open set with $\bigcap_1^\infty K_n \subset O$. Then $K_n \subset O$ for some $n$.
Assertion 2: From this, we can easily see that $\bigcap_1^\infty K_n$ is also compact.
I know this is trivial if $K_1$ is $T_2$ (Hausdorff). But is it true if we assume only $T_0$ or $T_1$?
Any counterexample is greatly appreciated.