Can anyone prove this please?
Let $Y$ be regular and $A\subset Y$ any infinite subset. Then there exists a family $\{U_n| n \geq 0\}$ of open sets whose closures are pairwise disjoint and such that $A\cap U_n \neq \emptyset$ for each $n\geq1$.
This is a theorem from James Dugundjis's book of Topology, he has a proof but I can't quite understand it.
His definition of regular spaces is as follows:
A Hausdorff space is regular if each $y\in Y$ and closed set A not containing y have disjoint neighborhoods.
I hope this will be answered. Thanks