Editing Uniform space (section)

==Uniform continuity==
{{Main|Uniform continuity}}

Similar to [[continuous function]]s between [[topological space]]s, which preserve [[topological properties]], are the [[uniformly continuous function]]s between uniform spaces, which preserve uniform properties. 

A uniformly continuous function is defined as one where inverse images of entourages are again entourages, or equivalently, one where the inverse images of uniform covers are again uniform covers. Explicitly, a function <math>f : X \to Y</math> between uniform spaces is called '''{{em|{{visible anchor|uniformly continuous}}}}''' if for every entourage <math>V</math> in <math>Y</math> there exists an entourage <math>U</math> in <math>X</math> such that if <math>\left(x_1, x_2\right) \in U</math> then <math>\left(f\left(x_1\right), f\left(x_2\right)\right) \in V;</math> or in other words, whenever <math>V</math> is an entourage in <math>Y</math> then <math>(f \times f)^{-1}(V)</math> is an entourage in <math>X</math>, where <math>f \times f : X \times X \to Y \times Y</math> is defined by <math>(f \times f)\left(x_1, x_2\right) = \left(f\left(x_1\right), f\left(x_2\right)\right).</math>

All uniformly continuous functions are continuous with respect to the induced topologies. 

Uniform spaces with uniform maps form a [[Category (mathematics)|category]]. An [[isomorphism]] between uniform spaces is called a {{visible anchor|uniform isomorphism|text=[[uniform isomorphism]]}}; explicitly, it is a [[#uniformly continuous|uniformly continuous]] [[bijection]] whose [[Inverse function|inverse]] is also uniformly continuous. 
A '''{{em|{{visible anchor|uniform embedding}}}}''' is an injective uniformly continuous map <math>i : X \to Y</math> between uniform spaces whose inverse <math>i^{-1} : i(X) \to X</math> is also uniformly continuous, where the image <math>i(X)</math> has the subspace uniformity inherited from <math>Y.</math>