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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.

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    Yeah, no, don't do that please.2013-03-02
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    If $V$ and $W$ are meant to be *open* sets, this cannot happen. The set of closures of a locally finite family is locally finite (which is not true for point-finite).2013-03-02

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