Uniform space

Builds on Cartesian product Binary relation Composition of relations

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:

Every $U \in \mathcal{U}$ contains $\Delta$ .
If $U \in \mathcal{U}$ and $U \subseteq V \subseteq X \times X$ , then $V \in \mathcal{U}$ .
If $U, V \in \mathcal{U}$ , then $U \cap V \in \mathcal{U}$ .
If $U \in \mathcal{U}$ , then there is $V \in \mathcal{U}$ with $V \circ V \subseteq U$ .
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 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. 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. The uniform topology on $X$ declares $O \subseteq X$ open when for each $x \in O$ there is $U \in \mathcal{U}$ with $\set{y : (x, y) \in U} \subseteq O$ .

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Tag	`YWW42G978D`
URI	`m2://YWW42G978D`
Kind	definition
Writing	W0 stub
MSC	54E15
Defines	Uniform space
Uses	Cartesian product, Binary relation, Composition of relations
Changed	2 November 2026

Uniform space

Definition

Remarks

Reading path

Further reading

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