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

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.

See also

• Homeomorphism – Mapping which preserves all topological properties of a given space — an isomorphism between topological spaces
• Isometric isomorphism – Distance-preserving mathematical transformationPages displaying short descriptions of redirect targets — an isomorphism between metric spaces

References

• John L. Kelley, General topology, van Nostrand, 1955. P.181.