---
tag: B7PZV00FP4
kind: definition
name: Uniform structure of a topological group
msc: [22A05, "54E15"]
defines: P89SJ7JWJE
uses: [92700QAB5Y, A54T86KS63, FVFFKNKEDF, 0ZCY0Z80B6]
sources:
  - url: https://en.wikipedia.org/wiki/Uniform_space
---

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] with identity $e$, and let $\mathcal{N}$ be the set of [[neighbourhoods|0ZCY0Z80B6]] of $e$.
For $U \in \mathcal{N}$ put
$$L_U = \set{(x, y) \in G \times G : x^{-1} y \in U}, \qquad R_U = \set{(x, y) \in G \times G : y x^{-1} \in U}.$$
The *left uniform structure* of $G$ is the [[uniform structure|A54T86KS63]] on $G$ whose [[entourages|FVFFKNKEDF]] are the subsets of $G \times G$ that contain some $L_U$ with $U \in \mathcal{N}$.
The *right uniform structure* of $G$ is defined in the same way with the sets $R_U$.

## Remarks

Both uniform structures induce the given topology of $G$: the neighbourhoods of $x$ are the sets $xU$ for the left one and $Ux$ for the right one, with $U \in \mathcal{N}$.
Example: for $(\R, +)$, the set $L_U$ with $U = (-\varepsilon, \varepsilon)$ is $\set{(x, y) : \abs{y - x} < \varepsilon}$, so both uniformities are the uniformity of the usual metric.
If $G$ is abelian, then $L_U = R_U$ for every $U$, so the two structures coincide.
In general they differ; the inversion map $x \mapsto x^{-1}$ carries one to the other.
Some authors also use the two-sided uniformity, generated by the sets $L_U \cap R_U$.