Exercise 11, § 3

Let $X$ be a topological space, $\mathfrak{T}$ the topology of $X$, $A$ a dense subspace of $X$. Show that the set of topologies on $X$ which are finer than $\mathfrak{T}$, in which $A$ is dense in $X$, and which induce the same topology as $\mathfrak{T}$ on $A$, has at least one maximal element; such a maximal element is called an $A$-maximal topology. Show that a topology $\mathfrak{T}_0$ on $X$ is $A$-maximal if and only if every subset $M$ of $X$, such that $A \cap M$ is dense in $M$ (with respect to $\mathfrak{T}_0$) and open in $A$, is open in the topology $\mathfrak{T}_0$, and that the subspace $CA$ is then discrete in the topology induced by $\mathfrak{T}_0$, and is closed in $X$. Show also that if $\mathfrak{T}_0$ is $A$-maximal and if the topology induced by $\mathfrak{T}_0$ on $A$ is quasi-maximal (§ 2, Exercise 6) then $\mathfrak{T}_0$ is quasi-maximal.