Uniform isomorphism

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 {{visible anchor|uniformly isomorphic}} or {{visible anchor|uniformly equivalent}}.

Uniform embeddings

A {{em|{{visible anchor|uniform embedding}}}} is an injective uniformly continuous map $i\; :\; X\; \backslash to\; Y$ between uniform spaces whose inverse $i^\{-1\}\; :\; i(X)\; \backslash to\; X$ is also uniformly continuous, where the image $i(X)$ has the subspace uniformity inherited from $Y.$

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

• {{annotated link|Homeomorphism}} — an isomorphism between topological spaces
• {{annotated link|Isometric isomorphism}} — an isomorphism between metric spaces

• {{Kelley 1975}}, pp. 180-4

{{Metric spaces}}

Category:Homeomorphisms

Category:Uniform spaces

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