On nowhere dense sets

Question

Let $A$ be a countably infinite set of non-isolated points of a Hausdorff topological space which is also discrete (i.e., $A\cap A’ = \emptyset$). Is it true that $A$ is a nowhere dense?

Answer 1

Yes.

By definition, a set $A$ is nowhere dense if every nonempty open set $U$ has a nonempty open subset which is disjoint from $A.$

Let $U$ be a nonempty open set. If $U\cap A=\emptyset,$ there is nothing to prove. Suppose $U\cap A\ne\emptyset.$ Choose $a\in U\cap A.$ Since $A$ is discrete, there is an open set $V$ such that $V\cap A=\{a\}.$ Then $U\cap V$ is a nonempty open set containing $a.$ Since $a$ is not isolated in the space $X,$ we can choose a point $x\in U\cap V,\ x\ne a.$ Since $X$ is $\mathrm T_1,$ there is an open set $W$ which contains $x$ but not $a.$ Then $U\cap V\cap W$ is a nonempty open subset of $U$ which is disjoint from $A.$