Exercise 7, § 7

A subset $A$ of a topological space $X$ is said to be primitive if it is the set of limit points of an ultrafilter on $X$. Let $\Upsilon_A$ denote the set of ultrafilters on $X$ whose set of limit points is $A$.

a) If $A$ is a primitive subset of $X$, show that every open subset of $X$ which meets $A$ belongs to all the ultrafilters $U \in \Upsilon_A$.

b) Let $A, B$ be two distinct primitive subsets of $X$. Show that there is an ultrafilter $U \in \Upsilon_A$, an ultrafilter $\mathfrak{B} \in \Upsilon_B$ and a subset $M$ of $X$ such that $M \in U$ and $\complement M \in \mathfrak{B}$.

c) Let $f : X \to Y$ be a continuous mapping. Show that the image under $f$ of any primitive subset of $X$ is contained in a primitive subset of $Y$.

d) If $x \in X$ and $\{x\}$ is closed in $X$, show that $\{x\}$ is a primitive subset of $X$.