In the
mathematical field of
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
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 is ...
between
uniform spaces that respects
uniform properties 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.
Since uniform spaces come as topological spaces and uniform isomorphisms are homeomorphism ...
. Uniform spaces with uniform maps form a
category. 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 is ...
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
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
*
is
uniformly continuous
* the
inverse function is uniformly continuous
In other words, a uniform isomorphism is a
uniformly continuous bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
between
uniform spaces whose
inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when ad ...
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 points ...
* — an isomorphism between
metric spaces
References
*
John L. Kelley
John L. Kelley (December 6, 1916, Kansas – November 26, 1999, Berkeley, California) was an American mathematician at the University of California, Berkeley, who worked in general topology and functional analysis.
Kelley's 1955 text, ''General ...
, ''General topology'',
van Nostrand, 1955. P.181.
Homeomorphisms
Uniform spaces
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