Remark

If $G$ is commutative, we have $x . A . x^{-1} = A$ for every subset $A$ of $G$ and every $x \in G$, and therefore (GV_{III}) [resp. (GV'_{III})] is automatically satisfied for every filter (resp. filter base) on $G$. On the other hand, if $G$ is not abelian, then (GV_{III}) is not a consequence of (GV_{I}) and (GV_{II}) [see Exercise 5].

If $G$ is a commutative group, written additively, the axioms which characterize the filter $\mathfrak{B}$ of neighbourhoods of the origin for a topology compatible with the group structure of $G$ are therefore the following:

(GA_{I}). Given any $U \in \mathfrak{B}$, there exists $V \in \mathfrak{B}$ such that $V + V \subset U$.

(GA_{II}). Given any $U \in \mathfrak{B}$, we have $-U \in \mathfrak{B}$.