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 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 space is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
with a uniform structure. Uniform spaces are
topological space
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 ...
s with additional structure that is used to define
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 ...
such as
completeness,
uniform continuity
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 ...
and
uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
. Uniform spaces generalize
metric space
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 ...
s and
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
s, but the concept is designed to formulate the weakest axioms needed for most proofs in
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
.
In addition to the usual properties of a topological structure, in a uniform space one formalizes the notions of relative closeness and closeness of points. In other words, ideas like "''x'' is closer to ''a'' than ''y'' is to ''b''" make sense in uniform spaces. By comparison, in a general topological space, given sets ''A,B'' it is meaningful to say that a point ''x'' is ''arbitrarily close'' to ''A'' (i.e., in the closure of ''A''), or perhaps that ''A'' is a ''smaller neighborhood'' of ''x'' than ''B'', but notions of closeness of points and relative closeness are not described well by topological structure alone.
Definition
There are three equivalent definitions for a uniform space. They all consist of a space equipped with a uniform structure.
Entourage definition
This definition adapts the presentation of a topological space in terms of
neighborhood system In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbour ...
s. A nonempty collection
of subsets of
is a (or a ) if it satisfies the following axioms:
# If
then
where
is the diagonal on
# If
and
then
# If
and
then
# If
then there is some
such that
, where
denotes the composite of
with itself. The
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials
...
of two subsets
and
of
is defined by
# If
then
where
is the
inverse of
The non-emptiness of
taken together with (2) and (3) states that
is a
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
on
If the last property is omitted we call the space . An element
of
is called a or from the
French word for ''surroundings''.
One usually writes
where
is the vertical cross section of
and
is the canonical projection onto the second coordinate. On a graph, a typical entourage is drawn as a blob surrounding the "
" diagonal; all the different