Exercise 27, § 5

a) Let $G$ be a topological group, and let $A$ be an almost open set (Exercise 6) in $G$. Show that if $A$ is not meagre then $AA^{-1}$ is a neighbourhood of the identity element in $G$. [Note first that $G$ is a Baire space by virtue of Exercise 3. Let $A^*$ be the union of all open sets $U \subset G$ such that $U \cap CA$ is meagre. Show that $A^*$ is not empty and that if $x \in G$ is such that $xA^*$ meets $A^*$, then $xA$ meets $A$. Hence show that $AA^{-1} \supset A^*(A^*)^{-1}.$]

b) Deduce from a) that an almost open subgroup of a topological group $G$ is either meagre or else open (and closed). In particular, a countable topological group which is a Hausdorff Baire space is discrete.

c) Show that there exist subgroups $H$ of the topological group $R$ such that $R/H$ is countably infinite (use a Hamel base); such a subgroup is dense but not meagre nor almost open.

d) Let $G$ be a metrizable topological group whose left and right uniformities coincide. Show that if there exists a metric on $G$, compatible with the topology of $G$, and with respect to which $G$ is a complete metric space, then $G$ is a complete metrizable group [use b); § 3, Exercise 2; and § 2, Exercise 24].