Definition 9.1.5.label Let $G$ be a topological group, then:
- (L1)
There exists a unique left translation-invariant uniformity $\fU_{L}$ on $G$ that induces its topology.
- (L2)
For each $U \in \cn_{G}(1)$, let $U_{L, V}= \bracsn{(x, y) \in G^2|x^{-1}y \in V}$, then $\fB_{L} = \bracs{U_{L, V}|V \in \cn_G(1)}$ is a fundamental system of entourages for $\fU_{L}$.
and
- (R1)
There exists a unique right translation-invariant uniformity $\fU_{R}$ on $G$ that induces its topology.
- (R2)
For each $U \in \cn_{G}(1)$, let $U_{R, V}= \bracsn{(x, y) \in G^2|yx^{-1} \in V}$, then $\fB_{R} = \bracs{U_{R, V}|V \in \cn_G(1)}$ is a fundamental system of entourages for $\fU_{R}$.
The uniformities $\fU_{L}$ and $\fU_{R}$ are the left and right uniformities of $G$, respectively. The inversion map $G \to G$ with $g \mapsto g^{-1}$ is an isomorphism of the left and right uniformities.
Proof, [I.1.4, SW99]. (L2): For each $V \in \cn_{G}(0)$, $U_{L, V}$ is left translation-invariant.
- (FB1)
For each $V, V' \in \cn_{G}(1)$, $V \cap V' \in \cn_{G}(0)$, so $U_{L, V \cap V'}= U_{L, V}\cap U_{L, V'}$.
- (UB1)
For each $V \in \cn_{G}(1)$, $1 \in V$, so $\Delta \subset U_{L, V}$.
- (UB2)
For each $V \in \cn_{G}(1)$, by (TG1), there exists $W \in \cn_{G}(1)$ such that $WW \subset V$. In which case, $U_{L, W}\circ U_{L, W}\subset U_{L, V}$.
By Proposition 6.1.8, $\fB_{L}$ forms a fundamental system of entourages for a left translation-invariant uniformity $\fU_{L}$ on $G$.
(L1): Let $\fV$ be a left translation-invariant uniformity on $G$. For each symmetric, left translation-invariant entourage $V \in \fV$, and $g \in G$,
so $V = U_{L, V(1)}$. Therefore $\fV \subset \fU_{L}$ by Lemma 9.1.4.
On the other hand, for each $W_{0} \in \cn_{G}(1)$, there exists a symmetric, left translation-invariant entourage $W \in \fV$ such that $W(1) \subset W_{0}$. In which case, $W = U_{L, W(1)}\subset U_{L, W_0(1)}$, and $\fV \supset \fU_{L}$.$\square$
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