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Wikipedia / Closure (topology)

In topology, the closure of a subset of points in a topological space consists of all pointss in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection of all closed sets containing . Intuitively, the closure can be thought of as all the points that are either in or "very near" . A point which is in the closure of is a point of closure of . The notion of closure is in many ways dual to the notion of interior.

Definitions

Point of closure

as a subset of a Euclidean space if every open ball itself).

of a metric space is the infimum.

This definition generalizes to topological spaces by replacing "open ball" or "ball" with "neighbourhood is a or is allowed). Note that this definition does not depend upon whether neighbourhoods are required to be open.

Limit point

The definition of a point of closure of a set is closely related to the definition of a limit point of a set is called the . 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 itself.

(or both).

Closure of a set

The of a topological space.) can be defined using any of the following equivalent definitions:

  1. is the set of all points of closure

  2. together with all of its limit points.)

  3. is the intersection of all closed sets

  4. and its boundary

  5. for which there exists a net

The closure of a set has the following properties.

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).

In a first-countable space (such as a metric space is the set of all limits of all convergent sequences 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:

  • itself.

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.

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

  • In any indiscrete space In other words, every non-empty subset of an indiscrete space is dense.

is the set of rational numbers, with the usual relative topology is both closed and open 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 greater than .

Closure operator

A is a mapping of the power set, into itself which satisfies the Kuratowski closure axioms. Given a topological space then a topological space is obtained by defining the closed sets of these subsets form the open sets of the topology).

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

Properties

is closed In particular:

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

  • 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 finitelyly 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.

  • is infinite.

is a subspace is endowed with the subspace topology (by definition of the subspace topology is the

if and only if

).

is any open cover is endowed with the subspace topology is a manifold are domains of coordinate charts if and only if it is "locally closed is any open cover

Functions and closure

Continuity

between topological spaces is continuous if and only if the preimage

is then this terminology allows for a plain English

Closed maps

is a (strongly) closed map

Categorical interpretation

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

The powerset may be realized as a partial order category in which the objects are subsets and the morphisms are inclusion maps is a subcategory can be identified with the comma category

with terminal object the interior

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

References

Bibliography