Closure (topology)

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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.

Contents

Definitions

Point of closure

For a subset of a Euclidean space, is a point of closure of if every open ball centered at contains a point of (this point may be itself).

This definition generalizes to any subset of a metric space Fully expressed, for a metric space with metric is a point of closure of if for every there exists some such that the distance (again, is allowed). Another way to express this is to say that is a point of closure of if the distance

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 [1] 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 in question must contain a point of the set other than itself. The set of all limit points of a set is called the derived set of

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 if there is a neighbourhood of which contains no other points of other 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

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".

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 (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 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

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 1]

In particular, is dense in if and only if is a subset of

If but is not necessarily a subset of then only

is guaranteed in general, where this containment could be strict (consider for instance with the usual topology, and [proof 2] ) although if is an open subset of then the equality will hold [proof 3] (no matter the relationship between and ). 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

Categorical interpretation

One may elegantly 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. 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
    2. From and it follows that and which implies
    3. 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

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    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