Uniformly Isomorphic

In the mathematical field of topology a uniform isomorphism or 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 $f^$ 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 or .

Uniform embeddings

A is an injective uniformly continuous map $i : X \to Y$ between uniform spaces whose inverse $i^ : i(X) \to X$ 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

* — an isomorphism between topological spaces
* — an isomorphism between metric spaces

References

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

Homeomorphisms
Uniform spaces

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