Proposition 1

Let $G$ be a group and let $\mathfrak{B}$ be a filter on $G$ satisfying the axioms $(GV_1)$, $(GV_{II})$ and $(GV_{III})$. Then there is a unique topology on $G$, compatible with the group structure of $G$, for which $\mathfrak{B}$ is the neighbourhood filter of the identity element $e$. For this topology the neighbourhood filter of any point $a \in G$ is the same as each of the two filters $a.\mathfrak{B}$ and $\mathfrak{B}.a$.

If there is a topology with the required properties, then by what has been said above the neighbourhood filter of $a$ coincides with each of the filters $a.\mathfrak{B}$ and $\mathfrak{B}.a$; hence the topology is unique, if it exists. Its existence will be established if we show 1) that the filters $a.\mathfrak{B}$ are the neighbourhood filters of a topology on $G$, and 2) that this topology is compatible with the group structure of $G$.

1) The filter $a.\mathfrak{B}$ satisfies axiom $(V_{III})$ (see Chapter I § 1, no. 2) by reason of $(GV_1)$ and $(GV_{II})$, as we have already seen; hence to show that $a.\mathfrak{B}$ is the neighbourhood filter of $a$ in a topology on $G$, we have to verify axiom $(V_{IV})$. Let then $V$ be any set of $\mathfrak{B}$, and $W$ a set of $\mathfrak{B}$ such that $W.W \subset V$; then for any $x \in a.W$ we have $x.W \subset a.W.W \subset a.V$, so that $a.V$ belongs to the filter $x.\mathfrak{B}$; hence $(V_{IV})$ is satisfied.

2) Let us now show that the topology defined by the neighbourhood filters $a.\mathfrak{B}$ satisfies $(GT')$. Let $a, b$ be any two points of $G$; if we put $x = au$ and $y = bv$, then we have to show that $xy^{-1}$ is as near as we please to $ab^{-1}$ whenever $u$ and $v$ are close enough to $e$. Now $(ab^{-1})^{-1}(xy^{-1}) = buv^{-1}b^{-1}$; let $U$ be any neighbourhood of $e$, then we shall have $buv^{-1}b^{-1} \in U$ if $uv^{-1} \in b^{-1}Ub = V$, and $V \in \mathfrak{B}$ by reason of $(GV_{III})$. But by $(GV_1)$ and $(GV_{II})$ there exists $W \in \mathfrak{B}$ such that $W.W^{-1} \subset V$; hence it is enough to take $u \in W$ and $v \in W$ in order to have $xy^{-1} \in (ab^{-1})U$. This completes the proof.

A common method of defining a topology compatible with a group structure on $G$ consists in giving a filter satisfying the axioms $(GV_1)$, $(GV_{II})$ and $(GV_{III})$. The corresponding conditions for a filter base $\mathfrak{B}$ are as follows:

$(GV'_1).$ Given any $U \in \mathfrak{B}$, there exists $V \in \mathfrak{B}$ such that $V.V \subset U$.

(GV_{II}). Given any $U \in \mathfrak{B}$, there exists $V \in \mathfrak{B}$ such that $V^{-1} \subset U$.

(GV'_{III}). Given any $a \in G$ and any $U \in \mathfrak{B}$, there exists $V \in \mathfrak{B}$ such that $V \subset a . U . a^{-1}$.

A neighbourhood of $e$ which coincides with its image under the symmetry $x \to x^{-1}$ is said to be symmetric. If $V$ is any neighbourhood of $e$, then $V \cup V^{-1}$, $V \cap V^{-1}$ and $V . V^{-1}$ are symmetric neighbourhoods. By (GV_{II}), the symmetric neighbourhoods form a fundamental system of neighbourhoods of $e$. Also it follows from (GV_{I}) that when $V$ runs through a fundamental system of neighbourhoods of $e$, the sets $V^n$ (where $n$ is a fixed integer $\neq 0$) form a fundamental system of neighbourhoods of $e$.