---
tag: YWW42G978D
kind: definition
name: Uniform space
msc: ["54E15"]
defines: A54T86KS63
uses: [09CEDJYBKP, 05AVARPXA2, 0PYSR20HGF]
sources:
  - url: https://en.wikipedia.org/wiki/Uniform_space
---

## Definition

Let $X$ be a set and let $\Delta = \set{(x, x) : x \in X}$ be the diagonal of $X \times X$.
For relations $U, V \subseteq X \times X$, write $U^{-1} = \set{(y, x) : (x, y) \in U}$ and $V \circ U = \set{(x, z) : \text{there is } y \text{ with } (x, y) \in U \text{ and } (y, z) \in V}$.
A *uniformity* on $X$ is a nonempty family $\mathcal{U}$ of subsets of $X \times X$ such that:

1. Every $U \in \mathcal{U}$ contains $\Delta$.
2. If $U \in \mathcal{U}$ and $U \subseteq V \subseteq X \times X$, then $V \in \mathcal{U}$.
3. If $U, V \in \mathcal{U}$, then $U \cap V \in \mathcal{U}$.
4. If $U \in \mathcal{U}$, then there is $V \in \mathcal{U}$ with $V \circ V \subseteq U$.
5. If $U \in \mathcal{U}$, then $U^{-1} \in \mathcal{U}$.

A *uniform space* is a pair $(X, \mathcal{U})$ of a set $X$ and a uniformity $\mathcal{U}$ on $X$.

## Remarks

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $U_\varepsilon = \set{(x, y) : d(x, y) < \varepsilon}$.
The sets that contain some $U_\varepsilon$ with $\varepsilon > 0$ form a uniformity.
Clauses 1, 2, 3 and 5 follow from $d(x, x) = 0$, from $U_\varepsilon \cap U_\delta = U_{\min(\varepsilon, \delta)}$ and from symmetry of $d$.
Clause 4 holds with $V = U_{\varepsilon/2}$, by the [[triangle inequality|01M6CW9VM6]].
On $\R$, the family of sets that contain $\set{(x, y) : \abs{x - y} < 1}$ satisfies clauses 1, 2, 3 and 5 but not 4.
For $V$ in this family, $V \circ V$ contains $\set{(x, y) : \abs{x - y} < 2}$, which does not lie in $\set{(x, y) : \abs{x - y} < 1}$.
The members of $\mathcal{U}$ are its [[entourages|FVFFKNKEDF]].
The uniform topology on $X$ declares $O \subseteq X$ open when for each $x \in O$ there is $U \in \mathcal{U}$ with $\set{y : (x, y) \in U} \subseteq O$.

## Explanations

### L1 human

A uniformity is a list of "closeness relations" on a set.
Each relation is a set of pairs of points that count as close.
The clauses say that every point is close to itself and that closeness relations can be enlarged and combined.
Each relation can be turned around, and inside each relation lies a smaller one that, applied twice, stays inside the first, like $\varepsilon/2$ and $\varepsilon$ in a metric space.