Editing Uniform isomorphism (section)

==Definition==

A [[function_(mathematics)|function]] <math>f</math> between two uniform spaces <math>X</math> and <math>Y</math> is called a '''uniform isomorphism''' if it satisfies the following properties

* <math>f</math> is a [[bijection]]
* <math>f</math> is [[uniformly continuous]]
* the [[inverse function]] <math>f^{-1}</math> is uniformly continuous

In other words, a '''uniform isomorphism''' is a [[uniformly continuous]] [[bijection]] between [[uniform spaces]] whose [[Inverse function|inverse]] is also uniformly continuous. 

If a uniform isomorphism exists between two uniform spaces they are called '''{{visible anchor|uniformly isomorphic}}''' or '''{{visible anchor|uniformly equivalent}}'''.

'''Uniform embeddings'''

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>