---
tag: A54T86KS63
kind: concept
name: Uniform space
msc: ["54E15"]
sources:
  - url: https://en.wikipedia.org/wiki/Uniform_space
same_as:
  wikidata: Q652446
  wikipedia: Uniform space
---

## Summary

A uniform space is a set with a uniform structure, a family of [[entourages|FVFFKNKEDF]] that say when two points are close to a given degree.
It is the natural setting for [[uniform continuity|0M4JJT780X]], Cauchy filters and completeness, without a metric.
Metric spaces and topological groups carry uniform structures, and every uniform space has an underlying topology.

## Explanations

### L1 human

In a metric space one can say that $x$ and $y$ are as close as $u$ and $v$: both distances are below $\varepsilon$.
A topology alone cannot compare closeness at different places, but a uniform structure can.
It lists the relations "being $\varepsilon$-close" as sets of pairs of points, without numbers.
For example, on $\R$ the sets $\set{(x, y) : \abs{x - y} < \varepsilon}$ for $\varepsilon > 0$ generate the usual uniform structure.