Uniform isomorphism

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

is a bijection
is uniformly continuous
the inverse function 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 between uniform spaces whose inverse is also uniformly continuous, where the image has the subspace uniformity inherited from

Examples

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