Uniform isomorphism


In the mathematical field of topology a uniform isomorphism or uniform homeomorphism 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 edit

A function   between two uniform spaces   and   is called a uniform isomorphism if it satisfies the following properties

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 uniformly isomorphic or uniformly equivalent.

Uniform embeddings

A uniform embedding is an injective uniformly continuous map   between uniform spaces whose inverse   is also uniformly continuous, where the image   has the subspace uniformity inherited from  

Examples edit

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

See also edit

References edit