Exercise 3, § 8

a) Let $X$ be an accessible space, $A$ a subset of $X$, and $x$ a point of $\overline{A}$ which does not belong to $A$. Show that every neighbourhood of $x$ contains an infinite number of points of $A$.

b) Let $X$ be an accessible space and let $A$ be a subset of $X$. Show that the set of points $x \in \overline{A}$ such that every neighbourhood of $x$ contains a point of $A$ other than $x$ is closed in $X$.

c) Let $X$ be an accessible space. Show that the intersection of all neighbourhoods of any subset $A$ of $X$ is equal to $A$.