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- Closure (topology)

In mathematics, the **closure** of a subset *S* of points in a topological space consists of all points in *S* together with all limit points of *S*. The closure of *S* may equivalently be defined as the union of *S* and its boundary, and also as the intersection of all closed sets containing *S*. Intuitively, the closure can be thought of as all the points that are either in *S* or "near" *S*. A point which is in the closure of *S* is a point of closure of *S*. 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**\in**S*

*d(x,**s)**<**r*

*x*=*s*

*x*

*S*

*d(x,**S)**:*=inf_{s}*d(x,**s)*=0*.*

This definition generalizes to topological spaces by replacing "open ball" or "ball" with "neighbourhood". Let

*S*

*X.*

*x*

*S*

*x*

*S.*

See main article: Limit point.

The definition of a point of closure is closely related to the definition of a limit point. The difference between the two definitions is subtle but important – namely, in the definition of limit point, every neighbourhood of the point

*x*

*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}*_{X}*S*

*\tau*

*X*

*\tau*

*\operatorname{cl}**S,*

*\overline{S},*

*S**{}*^{-}

*\operatorname{cl}*

*\operatorname{Cl}*

- is the set of all points of closure of
*\operatorname{cl}**S**S.* - is the set
*\operatorname{cl}**S*together with all of its limit points.*S* - is the intersection of all closed sets containing
*\operatorname{cl}**S**S.* - is the smallest closed set containing
*\operatorname{cl}**S**S.* - is the union of
*\operatorname{cl}**S*and its boundary*S**\partial(S).* - is the set of all
*\operatorname{cl}**S*for which there exists a net (valued) in*x**\in**X*that converges to*S*in*x**(X,**\tau).*

The closure of a set has the following properties.

*\operatorname{cl}**S*

*S*

- The set

*S*

*S*=*\operatorname{cl}**S*

- If

*S**\subseteq**T*

*\operatorname{cl}**S*

*\operatorname{cl}**T.*

- If

*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 3 dimensions. 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 be able to distinguish between the interior of 3-ball and the surface, so we distinguish between the open 3-ball, and the closed 3-ball – the closure of the 3-ball. The closure of the open 3-ball is the open 3-ball plus the surface.

- In any space,

*\varnothing*=*\operatorname{cl}**\varnothing.*

- In any space

*X,*

*X*=*\operatorname{cl}**X.*

Giving

R

C

- If

*X*

R

*\operatorname{cl}*_{X}*((*0*,*1*))*=*[*0*,*1*].*

- If

*X*

R

Q

R*.*

Q

R*.*

- If

*X*

C=R^{2,}

*\operatorname{cl}*_{X}*\left(**\{**z**\in*C*:**|**z**|**>*1*\}**\right)*=*\{**z**\in*C*:**|**z**|**\geq*1*\}.*

- If

*S*

*X,*

*\operatorname{cl}*_{X}*S*=*S.*

On the set of real numbers one can put other topologies rather than the standard one.

- If

*X*=R

*\operatorname{cl}*_{X}*((*0*,*1*))*=*[*0*,*1*).*

- If one considers on

*X*=R

*\operatorname{cl}*_{X}*((*0*,*1*))*=*(*0*,*1*).*

- If one considers on

*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.

- In any discrete space, since every set is closed (and also open), every set is equal to its closure.

*X,*

*X*

*A*

*X,*

*\operatorname{cl}*_{X}*A*=*X.*

The closure of a set also depends upon in which space we are taking the closure. For example, if

*X*

R*,*

*S*=*\{**q**\in*Q*:**q*^{2}*>*2*,**q**>*0*\},*

*S*

Q

*S*

*\sqrt*2

*S*

*S*

*\sqrt*2

*S*

Q

*X*

*\sqrt*2

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**\subseteq**X*

*\operatorname{cl}*_{X}*S,*

*\overline{S}*

*S*^{-}

c

*X,*

*S**\subseteq**X*

c*(S)*=*S*

*X*

The closure operator

*\operatorname{cl}*_{X}

*\operatorname{int}*_{X,}

*\operatorname{cl}*_{X}*S*=*X**\setminus**\operatorname{int}*_{X}*(X**\setminus**S),*

and also

*\operatorname{int}*_{X}*S*=*X**\setminus**\operatorname{cl}*_{X}*(X**\setminus**S).*

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}*_{X}*S*=*S.*

- The closure of the empty set is the empty set;
- The closure of

*X*

*X.*

- The closure of an intersection of sets is always a subset of (but need not be equal to) the intersection of the closures of the sets.
- In a union of finitely many sets, the closure of the union and the union of the closures are equal; the union of zero sets is the empty set, and so this statement contains the earlier statement about the closure of the empty set as a special case.
- The closure of the union of infinitely many sets need not equal the union of the closures, but it is always a superset of the union of the closures.

If

*S**\subseteq**T**\subseteq**X*

*T*

*X*

*T*

*X*

*\operatorname{cl}*_{T}*S**\subseteq**\operatorname{cl}*_{X}*S*

*S*

*T*

*T*

*S*

*X*

*\operatorname{cl}*_{T}*S**~*=*~**T**\cap**\operatorname{cl}*_{X}*S.*

In particular,

*S*

*T*

*T*

*\operatorname{cl}*_{X}*S.*

If

*S,**T**\subseteq**X*

*S*

*T*

*\operatorname{cl}*_{T}*(S**\cap**T)**~\subseteq~**T**\cap**\operatorname{cl}*_{X}*S*

is guaranteed in general, where this containment could be strict (consider for instance

*X*=*\R*

*T*=*(*-inf*ty,*0*],*

*S*=*(*0*,*inf*ty)*

*T*

*X*

*\operatorname{cl}*_{T}*(S**\cap**T)*=*T**\cap**\operatorname{cl}*_{X}*S*

*S*

*T*

l{U}

*X*

*S**\subseteq**X*

*\operatorname{cl}*_{X}*S*=c*up*_{U}

because

*\operatorname{cl}*_{U}*(S**\cap**U)*=*U**\cap**\operatorname{cl}*_{X}*S*

*U**\in*l{U}

*U**\in*l{U}

*X*

*X*

l{U}

*X*

*S**\subseteq**X*

*X*

*S**\subseteq**X*

*X*

*X*

l{U}

*X*

*S*

*X*

*S**\cap**U*

*U*

*U**\in*l{U}.

One may elegantly define the closure operator in terms of universal arrows, as follows.

The powerset of a set

*X*

*P*

*A**\to**B*

*A*

*B.*

*T*

*X*

*P*

*I**:**T**\to**P.*

*A**\subseteq**X*

*(A**\downarrow**I).*

*\operatorname{cl}**A.*

*A*

*I,*

*A**\to**\operatorname{cl}**A.*

Similarly, since every closed set containing

*X**\setminus**A*

*A*

*(I**\downarrow**X**\setminus**A)*

*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.

- , and use the second property as the definition.
- Because is a closed subset of
*\operatorname{cl}*_{X}*S*the intersection*X,*is a closed subset of*T**\cap**\operatorname{cl}*_{X}*S*(by definition of the subspace topology), which implies that*T*(because*\operatorname{cl}*_{T}*S**\subseteq**T**\cap**\operatorname{cl}*_{X}*S*is the closed subset of*\operatorname{cl}*_{T}*S*containing*T*). Because*S*is a closed subset of*T**\cap**\operatorname{cl}*_{X}*S*from the definition of the subspace topology, there must exist some set*T,*such that*C**\subseteq**X*is closed in*C*and*X*Because*\operatorname{cl}*_{T}*S*=*T**\cap**C.*and*S**\subseteq**\operatorname{cl}*_{T}*S**\subseteq**C*is closed in*C*the minimality of*X,*implies that*\operatorname{cl}*_{X}*S*Intersecting both sides with*\operatorname{cl}*_{X}*S**\subseteq**C.*shows that*T**T**\cap**\operatorname{cl}*_{X}*S**\subseteq**T**\cap**C*=*\operatorname{cl}*_{T}*S.**\blacksquare* - From
*T**:*=*(*-inf*ty,*0 - Let and assume that
*S,**T**\subseteq**X*is open in*T*Let*X.*which is equal to*C**:*=*\operatorname{cl}*_{T}*(T**\cap**S),*(because*T**\cap**\operatorname{cl}*_{X}*(T**\cap**S)*). The complement*T**\cap**S**\subseteq**T**\subseteq**X*is open in*T**\setminus**C*where*T,*being open in*T*now implies that*X*is also open in*T**\setminus**C*Consequently*X.*is a closed subset of*X**\setminus**(T**\setminus**C)*=*(X**\setminus**T)**\cup**C*where*X*contains*(X**\setminus**T)**\cup**C*as a subset (because if*S*is in*s**\in**S*then*T*), which implies that*s**\in**T**\cap**S**\subseteq**\operatorname{cl}*_{T}*(T**\cap**S)*=*C*Intersecting both sides with*\operatorname{cl}*_{X}*S**\subseteq**(X**\setminus**T)**\cup**C.*proves that*T*The reverse inclusion follows from*T**\cap**\operatorname{cl}*_{X}*S**\subseteq**T**\cap**C*=*C.**C**\subseteq**\operatorname{cl}*_{X}*(T**\cap**S)**\subseteq**\operatorname{cl}*_{X}*S.**\blacksquare*