Exercise 2, § 1

A semi-topological group is a group $G$ endowed with a topology such that, for each $a \in G$, the translations $x \to ax$ and $x \to xa$ are continuous on $G$, and such that the symmetry $x \to x^{-1}$ is continuous on $G$ (*).

a) A filter $\mathfrak{B}$ on a group $G$ is the neighbourhood filter of $e$ in a topology which makes $G$ a semi-topological group if and only if $\mathfrak{B}$ satisfies axioms $(\mathrm{GV}_{\mathrm{II}})$ and $(\mathrm{GV}_{\mathrm{III}})$ and the family of filters

satisfies axiom $(\mathrm{V}_{\mathrm{IV}})$ of Chapter I, § 1, no. 2.

b) If $G$ is an infinite group, show that the topology in which the open sets are $\varnothing$ and the complements of finite subsets of $G$ makes $G$ a non-Hausdorff semi-topological group in which $\{e\}$ is closed, but is not compatible with the group structure of $G$.

c) Define a topology on $\mathbf{R}$, finer than the usual topology, for which $\mathbf{R}$ is a semi-topological but not topological group [consider a decreasing sequence $(r_n)$ of numbers $> 0$, tending to $0$, and take the intersections of symmetric intervals ]—$a, a[$ and of the complement of the set of points $\pm r_n$. \

(*) If $G$ is locally compact these conditions imply that the topology of $G$ is compatible with its group structure (Chapter X, § 3, Exercise 25).