Exercise 6, § 2

A topology on a set $X$ is said to be quasi-maximal if it is maximal in the set of topologies in which $X$ has no isolated points.

a) Show that the following properties of a topology $\mathfrak{T}$ on $X$ are equivalent: $\alpha)$ $\mathfrak{T}$ is quasi-maximal; $\beta)$ if $X$ carries the topology $\mathfrak{T}$, then $X$ has no isolated point and every subset of $X$ which has no isolated points is open.

b) Let $X$ be a Kolmogoroff space ($§ 1$, Exercise 2) in which every non-empty open set is infinite. Show that there exists a quasi-maximal topology on $X$ which is finer than the given topology (show that the set of topologies on $X$ in which all the non-empty open sets are infinite is inductive, and use Exercise 2 d) of $§ 1$).