Uniform isomorphism

Short description: Uniformly continuous homeomorphism

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.

See also

• Homeomorphism – Mapping which preserves all topological properties of a given space — an isomorphism between topological spaces

References

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