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 $$f$$ between two uniform spaces $$X$$ and $$Y$$ is called a uniform isomorphism if it satisfies the following properties


 * $$f$$ is a bijection
 * $$f$$ is uniformly continuous
 * the inverse function $$f^{-1}$$ 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 $$i : X \to Y$$ between uniform spaces whose inverse $$i^{-1} : i(X) \to X$$ is also uniformly continuous, where the image $$i(X)$$ has the subspace uniformity inherited from $$Y.$$

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