Template:Short descriptionIn the mathematical field of topology a uniform isomorphism or Template:Visible anchor is a special isomorphism between uniform spaces that respects uniform properties. Uniform spaces with uniform maps form a category. An isomorphism between uniform spaces is called a uniform isomorphism.
DefinitionEdit
A 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 is also uniformly continuous.
If a uniform isomorphism exists between two uniform spaces they are called Template:Visible anchor or Template:Visible anchor.
Uniform embeddings
A Template:Em 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>
ExamplesEdit
The uniform structures induced by equivalent norms on a vector space are uniformly isomorphic.
See alsoEdit
- Template:Annotated link — an isomorphism between topological spaces
- Template:Annotated link — an isomorphism between metric spaces
ReferencesEdit
- Template:Kelley 1975, pp. 180-4