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.
Definitions
Point of closure
For S a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S.This definition generalizes to any subset S of a metric space X. Fully expressed, for X a metric space with metric d, x is a point of closure of S if for every r > 0, there is a y in S such that the distance d < r. Another way to express this is to say that x is a point of closure of S if the distance d := inf = 0.
This definition generalizes to topological spaces by replacing "open ball" or "ball" with "neighbourhood". Let S be a subset of a topological space X. Then x is a point of closure of S if every neighbourhood of x contains a point of S. 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. The difference between the two definitions is subtle but important — namely, in the definition of limit point, every neighbourhood of the point x in question must contain a point of the set other than x itself.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 is an isolated point of S if it is an element of S and if there is a neighbourhood of x which contains no other points of S other than x itself.
For a given set S and point x, x is a point of closure of S if and only if x is an element of S or x is a limit point of S.
Closure of a set
The closure of a set S is the set of all points of closure of S, that is, the set S together with all of its limit points. The closure of S is denoted cl, Cl, or. The closure of a set has the following properties.- cl is a closed superset of S.
- cl is the intersection of all closed sets containing S.
- cl is the smallest closed set containing S.
- cl is the union of S and its boundary ∂.
- Set S is closed if and only if S = cl.
- If S is a subset of T, then cl is a subset of cl.
- If A is a closed set, then A contains S if and only if A contains cl.
In a first-countable space, cl is the set of all limits of all convergent sequences of points in S. For a general topological space, this statement remains true if one replaces "sequence" by "net" or "filter".
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 in 3 dimensions. Implicitly there are two regions of interest created by this sphere; the sphere itself and its interior. 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 topological space:
- In any space,.
- In any space X, X = cl.
- If X is the Euclidean space R of real numbers, then cl) = .
- If X is the Euclidean space R, then the closure of the set Q of rational numbers is the whole space R. We say that Q is dense in R.
- If X is the complex plane C = R2, then cl =.
- If S is a finite subset of a Euclidean space, then cl = S.
- If X = R, where R has the lower limit topology, then cl) =
0, 1). - If one considers on R the [discrete topology in which every set is closed, then cl) =.
- If one considers on R the trivial topology in which the only closed sets are the empty set and R itself, then cl) = R.
- In any discrete space, since every set is closed, every set is equal to its closure.
- In any indiscrete space X, since the only closed sets are the empty set and X itself, we have that the closure of the empty set is the empty set, and for every non-empty subset A of X, cl = X. In other words, every non-empty subset of an indiscrete space is dense.
Closure operator
A closure operator on a set X is a mapping of the power set of X,, into itself which satisfies the Kuratowski closure axioms.Given a topological space, the mapping − : S → S− for all is a closure operator on X. Conversely, if c is a closure operator on a set X, a topological space is obtained by defining the sets S with c = S as closed sets.
The closure operator − is dual to the interior operator o, in the sense that
and also
where X denotes the underlying set of the topological space containing S, and the backslash refers to the set-theoretic difference.
Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be easily translated into the language of interior operators, by replacing sets with their complements.
In general, the closure operator does not commute with intersections. However, in a complete metric space the following result does hold:
Theorem :
Let X be a complete metric space and let be a sequence of subsets of X.
- If each is closed in X then.
- If each is open in X then.
Facts about closures
- The closure of the empty set is the empty set;
- The closure of itself is.
- The closure of an intersection of sets is always a subset of 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.
Categorical interpretation
One may elegantly define the closure operator in terms of universal arrows, as follows.The powerset of a set X may be realized as a partial order category P in which the objects are subsets and the morphisms are inclusions whenever A is a subset of B. Furthermore, a topology T on X is a subcategory of P with inclusion functor. The set of closed subsets containing a fixed subset can be identified with the comma category. This category — also a partial order — then has initial object Cl. Thus there is a universal arrow from A to I, given by the inclusion.
Similarly, since every closed set containing X \ A corresponds with an open set contained in A we can interpret the category as the set of open subsets contained in A, with terminal object, the interior of 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, since all are examples of universal arrows.