sequence of decreasing compact sets 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.   

Here's a T_1 space for which Assertion 2 fails.  Take the set of integers.  Say that a set is open iff it is either a subset of the negative integers or else is cofinite.  Then let K_n be the complement of {0, 1, ..., n}.  Then each K_n is compact, but the intersection of K_n from n=1 to infinity is the set of negative integers, which is open and noncompact.
