uniform continuity

In this entry, we extend the usual definition of a uniformly continuous function between metric spaces to arbitrary uniform spaces.

Let (X,𝒰),(Y,𝒱) be uniform spaces (the second component is the uniformity on the first component). A function f:XY is said to be uniformly continuous if for any V𝒱 there is a U𝒰 such that for all xX, U[x]f-1(V[f(x)]).

Sometimes it is useful to use an alternative but equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath version of uniform continuity of a function:

Proposition 1.

Suppose f:XY is a function and g:X×XY×Y is defined by g(x1,x2)=(f(x1),f(x2)). Then f is uniformly continuous iff for any VV, there is a UU such that Ug-1(V).


Suppose f is uniformly continuous. Pick any V𝒱. Then U𝒰 exists with U[x]f-1(V[f(x)]) for all xX. If (a,b)U, then bU[a]f-1(V[f(a)]), or f(b)V[f(a)], or g(a,b)=(f(a),f(b))V. The converseMathworldPlanetmath is straightforward. ∎

Remark. Note that we could have picked U so the inclusion becomes an equality.

Proposition 2.

. If f:XY is uniformly continuous, then it is continuousMathworldPlanetmathPlanetmath under the uniform topologies of X and Y.


Let A be open in Y and set B=f-1(A). Pick any xB. Then y=f(x) has a uniform neighborhood V[y]A. By the uniform continuity of f, there is an entourage U𝒰 with xU[x]f-1(V[y])f-1(A)=B. ∎

Remark. The converse is not true, even in metric spaces.

Title uniform continuity
Canonical name UniformContinuity
Date of creation 2013-03-22 16:43:15
Last modified on 2013-03-22 16:43:15
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 54E15
Related topic UniformlyContinuous
Related topic UniformContinuityOverLocallyCompactQuantumGroupoids
Defines uniformly continuous