In
topology, the closure of a subset of points in a
topological space consists of all
points in together with all
limit points of . The closure of may equivalently be defined as the
union of and its
boundary, and also as the
intersection of all
closed sets containing . Intuitively, the closure can be thought of as all the points that are either in or "near" . A point which is in the closure of is a
point of closure of . The notion of closure is in many ways
dual to the notion of
interior.
Definitions
Point of closure
For
as a subset of a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
,
is a point of closure of
if every
open ball centered at
contains a point of
(this point can be
itself).
This definition generalizes to any subset
of a
metric space Fully expressed, for
as a metric space with metric
is a point of closure of
if for every
there exists some
such that the distance
(
is allowed). Another way to express this is to say that
is a point of closure of
if the distance
where
is the
infimum.
This definition generalizes to
topological spaces by replacing "open ball" or "ball" with "
neighbourhood". Let
be a subset of a topological space
Then
is a or of
if every neighbourhood of
contains a point of
(again,
for
is allowed). Note that this definition does not depend upon whether neighbourhoods are required to be open.
Limit point
The definition of a point of closure is closely related to the definition of a
limit point of a set. The difference between the two definitions is subtle but important – namely, in the definition of a limit point
of a set
, every neighbourhood of
must contain a point of
. (Each neighbourhood of
has
but it also must have a point of
that is different from
.) A limit point of
has more strict condition than a point of closure of
in the definitions. The set of all limit points of a set
is called the . A limit point of a set is also called ''cluster point'' or ''accumulation point'' of the set.
Thus, every limit point is a point of closure, but not every point of closure is a limit point. A point of closure which is not a limit point is an
isolated point. In other words, a point
is an isolated point of
if it is an element of
and there is a neighbourhood of
which contains no other points of
than
itself.
For a given set
and point
is a point of closure of
if and only if
is an element of
or
is a limit point of
(or both).
Closure of a set
The of a subset
of a
topological space denoted by
or possibly by
(if
is understood), where if both
and
are clear from context then it may also be denoted by
or
(Moreover,
is sometimes capitalized to
.) can be defined using any of the following equivalent definitions:
- is the set of all points of closure of
- is the set together with all of its limit points.
- is the intersection of all closed sets containing
- is the smallest closed set containing
- is the union of and its boundary
- is the set of all for which there exists a net (valued) in that converges to in
The closure of a set has the following properties.
*
is a
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
superset of
.
* The set
is closed
if and only if .
* If
then
is a subset of
* If
is a closed set, then
contains
if and only if
contains
Sometimes the second or third property above is taken as the of the topological closure, which still make sense when applied to other types of closures (see below).
In a
first-countable space (such as a
metric space),
is the set of all
limits of all convergent
sequences of points in
For a general topological space, this statement remains true if one replaces "sequence" by "
net" or "
filter" (as described in the article on
filters in topology).
Note that these properties are also satisfied if "closure", "superset", "intersection", "contains/containing", "smallest" and "closed" are replaced by "interior", "subset", "union", "contained in", "largest", and "open". For more on this matter, see
closure operator below.
Examples
Consider a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
in a 3 dimensional space. Implicitly there are two regions of interest created by this sphere; the sphere itself and its interior (which is called an open 3-
ball). It is useful to distinguish between the interior and the surface of the sphere, so we distinguish between the open 3-ball (the interior of the sphere), and the closed 3-ball – the closure of the open 3-ball that is the open 3-ball plus the surface (the surface as the sphere itself).
In
topological space:
* In any space,
. In other words, the closure of the empty set
is
itself.
* In any space
Giving
and
the
standard (metric) topology:
* If
is the Euclidean space
of
real numbers, then