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.

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

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

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

• 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

• Kelley, John L. (1975) [1955]. General Topology. Graduate Texts in Mathematics. Vol. 27 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90125-1. OCLC 1365153., pp. 180-4

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