Exercise 22, § 8

a) With the notation of Exercise 20 c), show that the set $E(\mathfrak{T}_0)$ of topologies $\mathfrak{T}$ such that $\mathfrak{T}^* = \mathfrak{T}_0$ is inductive with respect to the relation "“$\mathfrak{T}$ is coarser than $\mathfrak{T}'$". If $\mathfrak{T}_0$ is a semi-regular topology on $X$, a maximal element of $E(\mathfrak{T}_0)$ is called a submaximal topology, and the set $X$ with such a topology is called a submaximal space. Thus every topology of $E(\mathfrak{T}_0)$ is coarser than some submaximal topology.

b) A topology $\mathfrak{T}$ on $X$ is submaximal if and only if every subset of $X$ which is dense in the topology $\mathfrak{T}$ is open in $\mathfrak{T}$. A non-empty submaximal space is therefore not solvable (§ 2, Exercise 2).

c) Show that every subspace of a submaximal space is locally closed and submaximal (consider first the open subspaces and the closed subspaces).

d) Show that the space associated with a filter which is finer than the filter of complements of finite subsets of an infinite set is a submaximal space.

e) If $M$ is a subset of a submaximal space $X$, show that the subspace $\overline{M} - \dot{M} = Fr(M)$ is discrete.

f) Conversely, let $X$ be a topological space which has a dense open subset $U$ such that $U$ is submaximal and the topology of $X$ is $U$-maximal (§ 3, Exercise 11). Then $X$ is submaximal.