Can any of you guys think of a topological space $(X,\tau)$ and a family of subsets {As}${s \in S}$ of $X$ such that for a certain $x \in X$ you can find a subset $V$ such that $x \in V$ and {$s \in S: V \cap A_{s} \neq \emptyset$} is finite, whereas for every $W \subseteq X$, with $x \in W$, we have that {$s \in S: W \cap \mathrm{cl}(A_{s}) \neq \emptyset$} is never finite?
I thank you in advance for your replies.