This article is about the relation between two sets. For the description a single set, see
Closedness.
Closeness is a basic concept in
topology 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 where a notion of distance between elements of the space is defined, but it can be generalized to
topological spaces where we have no concrete way to measure distances.
The
closure operator closes a given set by mapping it to a
closed set which contains the original set and all points close to it. The concept of closeness is related to
limit point.
Definition
Given a
metric space
a point
is called close or near to a set
if
,
where the distance between a point and a set is defined as
![{\displaystyle d(p,A):=\inf _{a\in A}d(p,a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c85b81a354af215fe360a79f9b09bf41107170)
where inf stands for
infimum. Similarly a set
is called close to a set
if
![{\displaystyle d(B,A)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d646d39682adaa73a4da0608710918009905245)
where
.
Properties
- if a point
is close to a set
and a set
then
and
are close (the
converse is not true!).
- closeness between a point and a set is preserved by
continuous functions
- 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
.
[1]
- If
then
is close to
.
- if
is close to
then ![{\displaystyle A\neq \emptyset }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0fdcea6d343edeb10e57c35a6a59b5fadec36c4)
- if
is close to
and
then
is close to ![{\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
- if
is close to
then
is close to
or
is close to ![{\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
- 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. 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 ![{\displaystyle B\neq \emptyset }](https://wikimedia.org/api/rest_v1/media/math/render/svg/745f2a85b093b31dac2363089ee19c0dbd35a2ef)
- 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. Given a
uniform space, 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
References
-
^ Arkhangel'skii, A. V.; Pontryagin, L.S. General Topology I: Basic Concepts and Constructions, Dimension Theory. Encyclopaedia of Mathematical Sciences (Book 17), Springer 1990, p. 9.