In the mathematical field of topology a uniform isomorphism or 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.

Definition

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 or .

Uniform embeddings

A 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>

Examples

The uniform structures induced by equivalent norms on a vector space are uniformly isomorphic.

See also

  • — an isomorphism between topological spaces
  • — an isomorphism between metric spaces

References

  • , pp. 180-4