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 $f$ 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

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.