proof that uniformly continuous is proximity continuous

Let f:XY be a uniformly continuous function from uniform spaces X to Y with uniformities 𝒰 and 𝒱 respectively. Let δ and ϵ be the proximities generated by ( 𝒰 and 𝒱 respectively. It is known that X and Y are proximity spaces with proximities δ and ϵ respectively. Furthermore, we have the following:

Theorem 1.

f:XY is proximity continuous.


Let A,B be any subsets of X with AδB. We want to show that f(A)ϵf(B), or equivalently,


for any V𝒱. Pick any V𝒱. Since f is uniformly continuous, there is U𝒰 such that


for any xX. As a result,


which implies that


Similarly f(U[B])V[f(B)]. Now, AδB is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to U[A]U[B], so we can pick




and therefore


This shows that f is proximity continuous. ∎

Title proof that uniformly continuous is proximity continuous
Canonical name ProofThatUniformlyContinuousIsProximityContinuous
Date of creation 2013-03-22 18:07:50
Last modified on 2013-03-22 18:07:50
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Proof
Classification msc 54E15
Classification msc 54C08
Classification msc 54C05
Classification msc 54E17
Classification msc 54E05