First the definitions:
The point $p$ is an $\omega$-cluster point for a subset $A$ of a topological space $X$ if every neighbourhood of $p$ contains infinitely many points of $A$.
A space is countably compact if every countable open cover has a finite subcover.
Now the question:
How does one prove that every countable infinite subset $A = \{ p_i : i \in \mathbb N \}$ of a countably compact space $X$ has an $\omega$-cluster point?
I know how to prove it if in addition $X$ is assumed to be Hausdorff: by contradiction assuming that there is no $\omega$-cluster point and considering the open cover $\{U_i : i \in \mathbb N \} \cup \{ \mathcal{C}A \}$, where $U_i$ is a neighbourhood of $p_i$ containing at most finitely many points of $A$, and $\mathcal{C}A=X\setminus A$. However $\mathcal{C}A$ is not necessarily open if $X$ is not Hausdorff and I do not know how to modify the proof.