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.

## Contents

* 1 Definition
* 2 Examples
* 3 See also
* 4 References

## 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->Y between uniform spaces whose inverse i−1:i(X)->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.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Uniform isomorphism. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Uniform_isomorphism&oldid=292042"