Let $X$ be Hausdorff and $C$ is countable discrete in $X$ and $x \in cl(C)$.
Does there exist a subset $D$ of $C$ such that $x \in cl(D)$ and there is a point-finite family $\{U_n\}$of open sets of $X$ such that $D \subset \cup U_n$?
If $X$ is regular, such $D$ does really exist. However I don't know such case that $X$ is Hausdorff. If the answer to the above question is negtive, I want to know what property we add that could make the answer is Yes. (Of course, I don't want to consider that $X$ is regular.)
The reference on this is also very welcome.
Thanks ahead.