Closure (topology)
In topology, 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.
1 Definitions
1.1 Point of closure
Main article: Adherent point
For {\displaystyle S}S as a subset of a Euclidean space, {\displaystyle x}x is a point of closure of {\displaystyle S}S if every open ball centered at {\displaystyle x}x contains a point of {\displaystyle S}S (this point can be {\displaystyle x}x itself).
This definition generalizes to any subset {\displaystyle S}S of a metric space {\displaystyle X.}X. Fully expressed, for {\displaystyle X}X as a metric space with metric {\displaystyle d,}d, {\displaystyle x}x is a point of closure of {\displaystyle S}S if for every {\displaystyle r>0}r > 0 there exists some {\displaystyle s\in S}s\in S such that the distance {\displaystyle d(x,s)<r}{\displaystyle d(x,s)<r} ({\displaystyle x=s}{\displaystyle x=s} is allowed). Another way to express this is to say that {\displaystyle x}x is a point of closure of {\displaystyle S}S if the distance {\displaystyle d(x,S):=\inf _{s\in S}d(x,s)=0}{\displaystyle d(x,S):=\inf _{s\in S}d(x,s)=0} where {\displaystyle \inf }\inf is the infimum.
This definition generalizes to topological spaces by replacing “open ball” or “ball” with “neighbourhood”. Let {\displaystyle S}S be a subset of a topological space {\displaystyle X.}X. Then {\displaystyle x}x is a point of closure or adherent point of {\displaystyle S}S if every neighbourhood of {\displaystyle x}x contains a point of {\displaystyle S}S (again, {\displaystyle x=s}{\displaystyle x=s} for {\displaystyle s\in S}s\in S is allowed).[1] Note that this definition does not depend upon whether neighbourhoods are required to be open.
1.2 Limit point
Main article: Limit point of a set
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 {\displaystyle x}x of a set {\displaystyle S}S, every neighbourhood of {\displaystyle x}x in question must contain a point of {\displaystyle S}S other than {\displaystyle x}x itself. (Each neighbourhood of {\displaystyle x}x can have {\displaystyle x}x but it must have a point of {\displaystyle S}S that is different from {\displaystyle x}x.) The set of all limit points of a set {\displaystyle S}S is called the derived set of {\displaystyle S.}S. 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 {\displaystyle x}x is an isolated point of {\displaystyle S}S if it is an element of {\displaystyle S}S and there is a neighbourhood of {\displaystyle x}x which contains no other points of {\displaystyle S}S than {\displaystyle x}x itself.[2]
For a given set {\displaystyle S}S and point {\displaystyle x,}x, {\displaystyle x}x is a point of closure of {\displaystyle S}S if and only if {\displaystyle x}x is an element of {\displaystyle S}S or {\displaystyle x}x is a limit point of {\displaystyle S}S (or both).
1.3 Closure of a set
See also: Closure (mathematics)
The closure of a subset {\displaystyle S}S of a topological space {\displaystyle (X,\tau ),}{\displaystyle (X,\tau ),} denoted by {\displaystyle \operatorname {cl} _{(X,\tau )}S}{\displaystyle \operatorname {cl} _{(X,\tau )}S} or possibly by {\displaystyle \operatorname {cl} _{X}S}{\displaystyle \operatorname {cl} _{X}S} (if {\displaystyle \tau }\tau is understood), where if both {\displaystyle X}X and {\displaystyle \tau }\tau are clear from context then it may also be denoted by {\displaystyle \operatorname {cl} S,}{\displaystyle \operatorname {cl} S,} {\displaystyle {\overline {S}},}{\displaystyle {\overline {S}},} or {\displaystyle S{}^{-}}{\displaystyle S{}^{-}} (Moreover, {\displaystyle \operatorname {cl} }{\displaystyle \operatorname {cl} } is sometimes capitalized to {\displaystyle \operatorname {Cl} }{\displaystyle \operatorname {Cl} }.) can be defined using any of the following equivalent definitions:
{\displaystyle \operatorname {cl} S}{\displaystyle \operatorname {cl} S} is the set of all points of closure of {\displaystyle S.}S.
{\displaystyle \operatorname {cl} S}{\displaystyle \operatorname {cl} S} is the set {\displaystyle S}S together with all of its limit points.[3]
{\displaystyle \operatorname {cl} S}{\displaystyle \operatorname {cl} S} is the intersection of all closed sets containing {\displaystyle S.}S.
{\displaystyle \operatorname {cl} S}{\displaystyle \operatorname {cl} S} is the smallest closed set containing {\displaystyle S.}S.
{\displaystyle \operatorname {cl} S}{\displaystyle \operatorname {cl} S} is the union of {\displaystyle S}S and its boundary {\displaystyle \partial (S).}{\displaystyle \partial (S).}
{\displaystyle \operatorname {cl} S}{\displaystyle \operatorname {cl} S} is the set of all {\displaystyle x\in X}x\in X for which there exists a net (valued) in {\displaystyle S}S that converges to {\displaystyle x}x in {\displaystyle (X,\tau ).}{\displaystyle (X,\tau ).}
The closure of a set has the following properties.[4]
{\displaystyle \operatorname {cl} S}{\displaystyle \operatorname {cl} S} is a closed superset of {\displaystyle S}S.
The set {\displaystyle S}S is closed if and only if {\displaystyle S=\operatorname {cl} S}{\displaystyle S=\operatorname {cl} S}.
If {\displaystyle S\subseteq T}S\subseteq T then {\displaystyle \operatorname {cl} S}{\displaystyle \operatorname {cl} S} is a subset of {\displaystyle \operatorname {cl} T.}{\displaystyle \operatorname {cl} T.}
If {\displaystyle A}A is a closed set, then {\displaystyle A}A contains {\displaystyle S}S if and only if {\displaystyle A}A contains {\displaystyle \operatorname {cl} S.}{\displaystyle \operatorname {cl} S.}
Sometimes the second or third property above is taken as the definition of the topological closure, which still make sense when applied to other types of closures (see below).[5]
In a first-countable space (such as a metric space), {\displaystyle \operatorname {cl} S}{\displaystyle \operatorname {cl} S} is the set of all limits of all convergent sequences of points in {\displaystyle S.}S. 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.