Exercise 11, § 2

A topological space is said to be solvable if it is the disjoint union of two dense subsets.

a) If a topological space $X$ is solvable, show that $X$ has no isolated points and that every open subspace of $X$ is solvable. Show also conversely that if $X$ is a topological space and $\mathfrak{B}$ is a set of non-empty solvable open subspaces of $X$ such that every non-empty open subset of $X$ contains a set belonging to $\mathfrak{B}$, then $X$ is solvable (consider a maximal set of mutually disjoint open sets belonging to $\mathfrak{B}$).

b) If a Kolmogoroff space $X$ is solvable, then every non-empty open subset of $X$ is infinite [use Exercise 2 d) of § 1].

c) Let $X$ be a topological space such that 1) the smallest cardinal $\mathfrak{d}$ of a non-empty open subset of $X$ is infinite, and 2) there is a base $\mathfrak{B}$ of the topology of $X$ such that $\mathrm{card}(\mathfrak{B}) \leq \mathfrak{d}$. Show that $X$ is solvable. (Use transfinite induction to construct two disjoint dense subsets of $X$). By the method of Set Theory, Chapter III, § 6, Exercise 24 a). In particular, the rational line is solvable.

d) A topological space $X$ which has no isolated points is said to be isodyne if $\mathrm{card}(U) = \mathrm{card}(X)$ for every non-empty open subset of $X$. Show that in a space $X$ with no isolated points, every non-empty open set in $X$ contains a non-empty open subspace which is isodyne. Deduce that if every isodyne open subspace of $X$ is soluble, then $X$ is soluble.