Exercise 3, § 3

The topology $\mathscr{T}$ defined by a saturated family $\{f_i\}$ of left-invariant pseudometrics on a group $G$ is compatible with the group structure of $G$ if and only if, for each $a \in G$, each index $i$ and each real number $x > 0$, there exists an index $x$ and a real number $\hat{p} > 0$ such that the relation $f_i(e, x) \leq \hat{p}$ implies $f_i(a, axa^{-1}) \leq x$. If this condition is satisfied, the uniformity on $G$ defined by the family $\mathscr{F}$ is the same as the left uniformity of the topological group obtained by endowing $G$ with the topology $\mathscr{T}$.