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 "very 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.
See main article: Adherent point.
For
S
x
S
x
S
x
This definition generalizes to any subset
S
X.
X
d,
x
S
r>0
s\inS
d(x,s)<r
x=s
x
S
d(x,S):=infsd(x,s)=0
inf
This definition generalizes to topological spaces by replacing "open ball" or "ball" with "neighbourhood". Let
S
X.
x
S
x
S
x=s
s\inS
See main article: Limit point of a set.
The definition of a point of closure of a set 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
x
S
x
S
x
x
S
x
x
S
S
S
S
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
x
S
S
x
S
x
For a given set
S
x,
x
S
x
S
x
S
See also: Closure (mathematics).
The of a subset
S
(X,\tau),
\operatorname{cl}(X,S
\operatorname{cl}XS
\tau
X
\tau
\operatorname{cl}S,
\overline{S},
S{}-
\operatorname{cl}
\operatorname{Cl}
\operatorname{cl}S
S.
\operatorname{cl}S
S
S
S
S
S
\operatorname{cl}S
S.
\operatorname{cl}S
S.
\operatorname{cl}S
S
\partial(S).
\operatorname{cl}S
x\inX
S
x
(X,\tau).
The closure of a set has the following properties.
\operatorname{cl}S
S
S
S=\operatorname{cl}S
S\subseteqT
\operatorname{cl}S
\operatorname{cl}T.
A
A
S
A
\operatorname{cl}S.
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).[1]
In a first-countable space (such as a metric space),
\operatorname{cl}S
S.
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.
Consider a sphere 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).
\varnothing=\operatorname{cl}\varnothing
\varnothing
\varnothing
X,
X=\operatorname{cl}X.
Giving
R
C
X
R
\operatorname{cl}X((0,1))=[0,1]
(0,1)
X
[0,1]
X
R
Q
R.
Q
R.
X
C=R2,
\operatorname{cl}X\left(\{z\inC:|z|>1\}\right)=\{z\inC:|z|\geq1\}.
S
X,
\operatorname{cl}XS=S.
On the set of real numbers one can put other topologies rather than the standard one.
X=R
\operatorname{cl}X((0,1))=[0,1).
X=R
\operatorname{cl}X((0,1))=(0,1).
X=R
R
\operatorname{cl}X((0,1))=R.
These examples show that the closure of a set depends upon the topology of the underlying space. The last two examples are special cases of the following.
X,
X
A
X,
\operatorname{cl}XA=X.
The closure of a set also depends upon in which space we are taking the closure. For example, if
X
R,
S=\{q\inQ:q2>2,q>0\},
S
Q
S
\sqrt2
S
S
\sqrt2
S
Q
X
\sqrt2
See also: Closure operator and Kuratowski closure axioms.
A on a set
X
X,
l{P}(X)
(X,\tau)
\operatorname{cl}X:\wp(X)\to\wp(X)
S\subseteqX
\operatorname{cl}XS,
\overline{S}
S-
c
X,
S\subseteqX
c(S)=S
X
The closure operator
\operatorname{cl}X
\operatorname{int}X,
\operatorname{cl}XS=X\setminus\operatorname{int}X(X\setminusS),
and also
\operatorname{int}XS=X\setminus\operatorname{cl}X(X\setminusS).
Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be readily translated into the language of interior operators by replacing sets with their complements in
X.
In general, the closure operator does not commute with intersections. However, in a complete metric space the following result does hold:
A subset
S
X
\operatorname{cl}XS=S.
X
X.
\operatorname{cl}X(S\cupT)=(\operatorname{cl}XS)\cup(\operatorname{cl}XT).
\operatorname{cl}X\left(cupiSi\right) ≠ cupi\operatorname{cl}XSi
I
If
S\subseteqT\subseteqX
T
X
T
X
\operatorname{cl}TS\subseteq\operatorname{cl}XS
S
T
T
S
X
\operatorname{cl}XS
X,
T\cap\operatorname{cl}XS
T
\operatorname{cl}TS\subseteqT\cap\operatorname{cl}XS
\operatorname{cl}TS
T
S
\operatorname{cl}TS
T,
C\subseteqX
C
X
\operatorname{cl}TS=T\capC.
S\subseteq\operatorname{cl}TS\subseteqC
C
X,
\operatorname{cl}XS
\operatorname{cl}XS\subseteqC.
T
T\cap\operatorname{cl}XS\subseteqT\capC=\operatorname{cl}TS.
\blacksquare
It follows that
S\subseteqT
T
T
\operatorname{cl}XS.
\operatorname{cl}TS=T\cap\operatorname{cl}XS
\operatorname{cl}XS;
X=\R,
S=(0,1),
T=(0,infty).
If
S,T\subseteqX
S
T
X=\R
T=(-infty,0],
S=(0,infty)
T
X
\operatorname{cl}T(S\capT)=T\cap\operatorname{cl}XS
S
T
Let
S,T\subseteqX
T
X.
C:=\operatorname{cl}T(T\capS),
T\cap\operatorname{cl}X(T\capS)
T\capS\subseteqT\subseteqX
T\setminusC
T,
T
X
T\setminusC
X.
X\setminus(T\setminusC)=(X\setminusT)\cupC
X
(X\setminusT)\cupC
S
s\inS
T
s\inT\capS\subseteq\operatorname{cl}T(T\capS)=C
\operatorname{cl}XS\subseteq(X\setminusT)\cupC.
T
T\cap\operatorname{cl}XS\subseteqT\capC=C.
C\subseteq\operatorname{cl}X(T\capS)\subseteq\operatorname{cl}XS.
\blacksquare
Consequently, if
l{U}
X
S\subseteqX
\operatorname{cl}U(S\capU)=U\cap\operatorname{cl}XS
U\inl{U}
U\inl{U}
X
X
l{U}
X
S\subseteqX
X
S\subseteqX
X
X
l{U}
X
S
X
S\capU
U
U\inl{U}.
See main article: Continuous function.
A function
f:X\toY
f-1(C)
X
C
Y.
In terms of the closure operator,
f:X\toY
A\subseteqX,
x\inX
A\subseteqX,
f(x)
f(A)
Y.
x
A\subseteqX
x\in\operatorname{cl}XA,
f
A\subseteqX,
f
A
f(A).
f
x\inX
x
A\subseteqX,
f(x)
f(A).
See main article: Open and closed maps.
A function
f:X\toY
C
X
f(C)
Y.
f:X\toY
\operatorname{cl}Yf(A)\subseteqf\left(\operatorname{cl}XA\right)
A\subseteqX.
f:X\toY
\operatorname{cl}Yf(C)\subseteqf(C)
C\subseteqX.
One may define the closure operator in terms of universal arrows, as follows.
The powerset of a set
X
P
A\toB
A
B.
T
X
P
I:T\toP.
A\subseteqX
(A\downarrowI).
\operatorname{cl}A.
A
I,
A\to\operatorname{cl}A.
Similarly, since every closed set containing
X\setminusA
A
(I\downarrowX\setminusA)
A,
\operatorname{int}(A),
A.
All properties of the closure can be derived from this definition and a few properties of the above categories. Moreover, this definition makes precise the analogy between the topological closure and other types of closures (for example algebraic closure), since all are examples of universal arrows.