It is known that a $T_2$ space $X$ is $KC$, i.e. every compact subset of $X$ is closed. The space $[0, \omega_1]$ is $T_5$ but not $T_6$ and the subset $[0, \omega_1)$ is sequentially compact (and thus countably compact) but not closed. Is there an example of a $T_6$ space with a sequentially compact (or countably compact) subspace which is not closed?
Remark: Spaces $X$ in which countably compact subsets are closed were investigated by Ismail and Nyikos. They called such space C-closed. Examples for C-closed spaces are $T_2$ sequential spaces or even $T_2$ spaces in which every countably compact subspace is sequential (e.g. $C([0,1])$ with pointwise convergence). From above it follows that $[0, \omega_1]$ is not a C-space (it is moreover not sequential).
The Arens-Fort space is $T_6$ and not sequential and I hope that one can show that it is also not C-closed.