Compact open sets which are not closed.
Consider cofinite topology on an infinite set, it is $T_1$ and every set is compact. In particular nonempty open sets which are not closed.
Another example: $[0,1]\cup\{0_2\}$, where $0_2$ is "another copy of $0$". A basis for the topology consists of all open subsets of $[0,1]$ with its standard topology, along with all sets of the form $\{0_2\}\cup(0,a)$ with $0<a<1$. (This is a modification of the line with two origins.)
In this space, $[0,1]$ compact, open, and not closed.