TITLE: Compact open sets which are not closed.
QUESTION [13 upvotes]: Can a nonclosed open subset of a $T_1$ topological space be compact?
I mean an open compact set which is not clopen.

REPLY [14 votes]: 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.