Uniform isomorphism

Summary

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.

Definition

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A function   between two uniform spaces   and   is called a uniform isomorphism if it satisfies the following properties

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   between uniform spaces whose inverse   is also uniformly continuous, where the image   has the subspace uniformity inherited from  

Examples

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

See also

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References

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