A finiteness condition 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.

If I read it correctly you want $V$ to meet only finitely many of the $A_s$, but every set containing $x$ to meet infinitely many of the closures of the $A_s$.  Without some restriction on $V$ and the $A_s$ this is easy.  Just have all the $A_s$ have x as a limit point but not a member.  Then $V$ is just the point $x$ and any $W$ containing $x$ will meet the closure of all the $A_s$.
