In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
a uniform isomorphism or is a special
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word i ...
between
uniform spaces
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifor ...
that respects
uniform properties. Uniform spaces with uniform maps form a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
. An
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word i ...
between uniform spaces is called a uniform isomorphism.
Definition
A
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
between two uniform spaces
and
is called a uniform isomorphism if it satisfies the following properties
*
is a
bijection
*
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 X ...
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 between
uniform spaces
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifor ...
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
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
In mathematics, a norm is a function (mathematics), function from a real number, real or complex number, complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the Origin (mathematics), origin: it ...
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 the most general settin ...
References
*
John L. Kelley, ''General topology'',
van Nostrand, 1955. P.181.
Homeomorphisms
Uniform spaces
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