Closeness is a basic concept in
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 h ...
and related areas in
mathematics. Intuitively we say two sets are close if they are arbitrarily near to each other. The concept can be defined naturally in a
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 set ...
where a notion of distance between elements of the space is defined, but it can be generalized to
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 poi ...
where we have no concrete way to measure distances.
The
closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S
:
Closure operators are d ...
''closes'' a given set by mapping it to a
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
which contains the original set and all points close to it. The concept of closeness is related to
limit point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also conta ...
.
Definition
Given a
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 set ...
a point
is called close or near to a set
if
:
,
where the distance between a point and a set is defined as
:
.
Similarly a set
is called close to a set
if
:
where
:
.
Properties
*if a point
is close to a set
and a set
then
and
are close (the
converse
Converse may refer to:
Mathematics and logic
* Converse (logic), the result of reversing the two parts of a definite or implicational statement
** Converse implication, the converse of a material implication
** Converse nonimplication, a logical ...
is not true!).
*closeness between a point and a set is preserved by
continuous functions
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
*closeness between two sets is preserved by
uniformly continuous functions
Closeness relation between a point and a set
Let
be some set. A relation between the points of
and the subsets of
is a closeness relation if it satisfies the following conditions:
Let
and
be two subsets of
and
a point in
.
[Arkhangel'skii, A. V. General Topology I: Basic Concepts and Constructions Dimension Theory. Encyclopaedia of Mathematical Sciences (Book 17), Springer 1990, p. 9]
*If
then
is close to
.
*if
is close to
then
*if
is close to
and
then
is close to
*if
is close to
then
is close to
or
is close to
*if
is close to
and for every point
,
is close to
, then
is close to
.
Topological spaces have a closeness relationship built into them: defining a point
to be close to a subset
if and only if
is in the closure of
satisfies the above conditions. Likewise, given a set with a closeness relation, defining a point
to be in the closure of a subset
if and only if
is close to
satisfies the
Kuratowski closure axioms In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first form ...
. Thus, defining a closeness relation on a set is exactly equivalent to defining a topology on that set.
Closeness relation between two sets
Let
,
and
be sets.
*if
and
are close then
and
*if
and
are close then
and
are close
*if
and
are close and
then
and
are close
*if
and
are close then either
and
are close or
and
are close
*if
then
and
are close
Generalized definition
The closeness relation between a set and a point can be generalized to any topological space. Given a topological space and a point
,
is called close to a set
if
.
To define a closeness relation between two sets the topological structure is too weak and we have to use a
uniform structure
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 unif ...
. Given a
uniform space
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 unif ...
, sets ''A'' and ''B'' are called close to each other if they intersect all
entourages, that is, for any entourage ''U'', (''A''×''B'')∩''U'' is non-empty.
See also
*
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 po ...
*
Uniform space
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 unif ...
References
{{DEFAULTSORT:Closeness (Mathematics)
General topology