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.

Definitions

Point of closure

Main article: Adherent point

For as a subset of a Euclidean space, 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 point of closure or adherent point of if every neighbourhood of contains a point of (again, for is allowed).[1] Note that this definition does not depend upon whether neighbourhoods are required to be open.

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 of a set , every neighbourhood of in question must contain a point of other than itself. (Each neighbourhood of can have but it must have a point of that is different from .) The set of all limit points of a set is called the derived set of 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.[2]

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

See also: Closure (mathematics)

The closure 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:

  1. is the set of all points of closure of
  2. is the set together with all of its limit points.[3]
  3. is the intersection of all closed sets containing
  4. is the smallest closed set containing
  5. is the union of and its boundary
  6. 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.[4]

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), 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 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:

Giving and the standard (metric) topology:

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

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.

The closure of a set also depends upon in which space we are taking the closure. For example, if is the set of rational numbers, with the usual relative topology induced by the Euclidean space and if then is both closed and open in because neither nor its complement can contain , which would be the lower bound of , but cannot be in because is irrational. So, has no well defined closure due to boundary elements not being in . However, if we instead define to be the set of real numbers and define the interval in the same way then the closure of that interval is well defined and would be the set of all real numbers greater than or equal to .

Closure operator

See also: Closure operator and Kuratowski closure axioms

A closure operator on a set is a mapping of the power set of , into itself which satisfies the Kuratowski closure axioms. Given a topological space , the topological closure induces a function that is defined by sending a subset to where the notation or may be used instead. Conversely, if is a closure operator on a set then a topological space is obtained by defining the closed sets as being exactly those subsets that satisfy (so complements in of these subsets form the open sets of the topology).[6]

The closure operator is dual to the interior operator, which is denoted by in the sense that

and also

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

In general, the closure operator does not commute with intersections. However, in a complete metric space the following result does hold:

Theorem[7] (C. Ursescu) — Let be a sequence of subsets of a complete metric space

Facts about closures

A subset is closed in if and only if In particular:

If and if is a subspace of (meaning that is endowed with the subspace topology that induces on it), then and the closure of computed in is equal to the intersection of and the closure of computed in :

Proof

Because is a closed subset of the intersection is a closed subset of (by definition of the subspace topology), which implies that (because is the smallest closed subset of containing ). Because is a closed subset of from the definition of the subspace topology, there must exist some set such that is closed in and Because and is closed in the minimality of implies that Intersecting both sides with shows that

It follows that is a dense subset of if and only if is a subset of It is possible for to be a proper subset of for example, take and

If but is not necessarily a subset of then only

is always guaranteed, where this containment could be strict (consider for instance with the usual topology, and [proof 1]), although if happens to an open subset of then the equality will hold (no matter the relationship between and ).

Proof

Let and assume that is open in Let which is equal to (because ). The complement is open in where being open in now implies that is also open in Consequently is a closed subset of where contains as a subset (because if is in then ), which implies that Intersecting both sides with proves that The reverse inclusion follows from

Consequently, if is any open cover of and if is any subset then:

because for every (where every is endowed with the subspace topology induced on it by ). This equality is particularly useful when is a manifold and the sets in the open cover are domains of coordinate charts. In words, this result shows that the closure in of any subset can be computed "locally" in the sets of any open cover of and then unioned together. In this way, this result can be viewed as the analogue of the well-known fact that a subset is closed in if and only if it is "locally closed in ", meaning that if is any open cover of then is closed in if and only if is closed in for every

Functions and closure

Continuity

Main article: Continuous function

A function between topological spaces is continuous if and only if the preimage of every closed subset of the codomain is closed in the domain; explicitly, this means: is closed in whenever is a closed subset of

In terms of the closure operator, is continuous if and only if for every subset

That is to say, given any element that belongs to the closure of a subset necessarily belongs to the closure of in If we declare that a point is close to a subset if then this terminology allows for a plain English description of continuity: is continuous if and only if for every subset maps points that are close to to points that are close to Thus continuous functions are exactly those functions that preserve (in the forward direction) the "closeness" relationship between points and sets: a function is continuous if and only if whenever a point is close to a set then the image of that point is close to the image of that set. Similarly, is continuous at a fixed given point if and only if whenever is close to a subset then is close to

Closed maps

Main article: Open and closed maps

A function is a (strongly) closed map if and only if whenever is a closed subset of then is a closed subset of In terms of the closure operator, is a (strongly) closed map if and only if for every subset Equivalently, is a (strongly) closed map if and only if for every closed subset

Categorical interpretation

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

The powerset of a set may be realized as a partial order category in which the objects are subsets and the morphisms are inclusion maps whenever is a subset of Furthermore, a topology on is a subcategory of 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 Thus there is a universal arrow from to given by the inclusion

Similarly, since every closed set containing corresponds with an open set contained in we can interpret the category as the set of open subsets contained in with terminal object the interior of

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.

See also

Notes

  1. ^ From and it follows that and which implies

References

  1. ^ Schubert 1968, p. 20
  2. ^ Kuratowski 1966, p. 75
  3. ^ Hocking & Young 1988, p. 4
  4. ^ Croom 1989, p. 104
  5. ^ Gemignani 1990, p. 55, Pervin 1965, p. 40 and Baker 1991, p. 38 use the second property as the definition.
  6. ^ Pervin 1965, p. 41
  7. ^ Zălinescu 2002, p. 33.

Bibliography