Exercise 4, § 1

A paratopological group is a group $G$ endowed with a topology satisfying axiom $(\mathrm{GT}_1) (*)$.

a) A filter $\mathfrak{B}$ on a group $G$ is the neighbourhood filter of $e$ for a topology which makes $G$ a paratopological group if and only if $\mathfrak{B}$ satisfies axioms $(\mathrm{GV}_1)$ and $(\mathrm{GV}_{\mathrm{III}})$ and $e \in V$ for all $V \in \mathfrak{B}$. The corresponding topology is then unique; it is Hausdorff if and only if the intersection of the sets $V.V^{-1}$, where $V$ runs through $\mathfrak{B}$, consists of the point $e$ alone.

b) On the group $\mathbf{Z}$ of integers, let $V_n$ be the set consisting of $0$ and all integers $m \geq n$, for each $n > 0$. Show that the $V_n$ form a fundamental system of neighbourhoods of $0$ for a non-Hausdorff topology on $\mathbf{Z}$, and that with this topology $\mathbf{Z}$ is a paratopological group in which $\{0\}$ is closed.

c) A paratopological group $G$ is a topological group if and only if, for each subset $A$ of $\overline{G}$, the intersection of the sets $AV$ (or of the sets $VA$) is the closure $\overline{A}$ of $A$ (as $V$ runs through the neighbourhood filter of $e$) (to see that the condition is sufficient, take $A$ to be the complement of an open neighbourhood of $e$).