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Uniform isomorphism

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

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

Examples

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