In the mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 field of topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 a uniform isomorphism or uniform homeomorphism is a special isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

 between uniform spaces which respects uniform properties

Uniform property

In the mathematical field of topology a uniform property or uniform invariant is a property of a uniform space which is invariant under uniform isomorphisms....

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Definition

A function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 f between two uniform spaces X and Y is called a uniform isomorphism if it satisfies the following properties

• f is a bijection
  Bijection
  A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
  
  
• f is uniformly continuous
• the inverse function
  Inverse function
  In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
  
   f ^-1 is uniformly continuous

If a uniform isomorphism exists between two uniform spaces they are called uniformly isomorphic or uniformly equivalent.

Examples

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

See also

• homeomorphism
  Homeomorphism
  In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
  
   is an isomorphism between topological spaces
• isometric isomorphism is an isomorphism between metric spaces