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. Uniform spaces with uniform maps form a category. An isomorphism between uniform spaces is called a uniform isomorphism.

Definition

A function [math]\displaystyle{ f }[/math] between two uniform spaces [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] is called a uniform isomorphism if it satisfies the following properties

• [math]\displaystyle{ f }[/math] is a bijection
• [math]\displaystyle{ f }[/math] is uniformly continuous
• the inverse function [math]\displaystyle{ f^{-1} }[/math] 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 [math]\displaystyle{ i : X \to Y }[/math] between uniform spaces whose inverse [math]\displaystyle{ i^{-1} : i(X) \to X }[/math] is also uniformly continuous, where the image [math]\displaystyle{ i(X) }[/math] has the subspace uniformity inherited from [math]\displaystyle{ Y. }[/math]

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.

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