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In the
mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
field of
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
a uniform isomorphism or is a special
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
between
uniform spaces In the mathematical field of topology, a uniform space is a set with additional structure that is used to define '' uniform properties'', such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces a ...
that respects uniform properties. Uniform spaces with uniform maps form a
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
. An
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
between uniform spaces is called a uniform isomorphism.


Definition

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 In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
* f is
uniformly continuous In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
* the
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
f^ is uniformly continuous In other words, a uniform isomorphism is a
uniformly continuous In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
between
uniform spaces In the mathematical field of topology, a uniform space is a set with additional structure that is used to define '' uniform properties'', such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces a ...
whose inverse is also uniformly continuous. If a uniform isomorphism exists between two uniform spaces they are called or . Uniform embeddings A is an injective uniformly continuous map i : X \to Y between uniform spaces whose inverse i^ : i(X) \to X is also uniformly continuous, where the image i(X) has the subspace uniformity inherited from Y.


Examples

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


See also

* — an isomorphism between
topological spaces In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
* — an isomorphism between
metric spaces In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for ...


References

* , pp. 180-4 Homeomorphisms Uniform spaces {{topology-stub