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 between two uniform spaces
and
is called a uniform isomorphism if it satisfies the following properties
is a bijection
is uniformly continuous
- the inverse function
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 between uniform spaces whose inverse
is also uniformly continuous, where the image
has the subspace uniformity inherited from
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
- Isometric isomorphism – Distance-preserving mathematical transformation — an isomorphism between metric spaces
References
- John L. Kelley, General topology, van Nostrand, 1955. P.181.