BREAKING NEWS

## Summary

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

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 uniformly isomorphic or uniformly equivalent.

Uniform embeddings

A uniform embedding 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.