In
mathematics, 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 ...
of
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 ...
s with
domain is called a and is said to (or just ) if for any two distinct elements
and
of
there exists a function
such that
[.]
Separating sets can be used to formulate a version of the
Stone–Weierstrass theorem for real-valued functions on a
compact Hausdorff space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
with the topology of
uniform convergence. It states that any subalgebra of this space of functions is dense if and only if it separates points. This is the version of the theorem originally proved by
Marshall H. Stone.
Examples
* The
singleton set consisting of the
identity function on
separates the points of
* If
is a
T1 normal topological space, then
Urysohn's lemma states that the set
of
continuous functions on
with
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
(or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
) values separates points on
* If
is a
locally convex Hausdorff topological vector space over
or
then the
Hahn–Banach separation theorem implies that
continuous linear functional In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear ...
s on
separate points.
See also
*
References
{{mathlogic-stub
Set theory