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.

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

If a uniform isomorphism exists between two uniform spaces they are called uniformly isomorphic or uniformly equivalent.

Examples[]

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

See also[]

• Homeomorphism—an isomorphism between topological spaces
• Isometric isomorphism—an isomorphism between metric spaces

References[]

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

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