Closure (topology)

Helping orphans the way you would do it

In topology and mathematical analysis, the closure of a subset of a topological space is the smallest closed subset of which contains . This can be constructed by intersecting all closed supersets of in .

Table of contents

1 Notation
2 Alternative characterisations
3 Examples
4 Facts about closures

Notation

The closure of is written as or . If there is more than one topology on (say and ), then the different topologies may give rise to different closures; this can be indicated in the notation by a subscript, as in "". If the topology is itself defined by some other structure, such as a metric , then "" can be placed in the subscript instead of "".

Alternative characterisations

In a metric space (such as the -dimensional Euclidean space) the closure is the set of all points in whose distance from is 0. Here, is defined as the infimum of the set .

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".

Another characterization of is as follows: an element of belongs to if and only if every neighborhood of contains an element of . In other words, iff or is a limit point of .

Examples

The closure of the open interval (0,1) in the real numbers is the closed interval [0,1]. If denotes the set of all rational numbers greater than the square root of 2, then the closure of in the rational numbers is ; the closure of in the real numbers is the set of all real numbers greater than or equal to .

In the trivial topology, the closure of any non-empty set is the whole space. In the discrete topology, the closure of any set is that set itself.

Facts about closures

The set is closed if and only if . In particular, the closure of the empty set is the empty set, and the closure of itself is . 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 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.

The closure operation can be characterized by the Kuratowski closure axioms; in particular, this operation is an example of a closure operator.

The closure of the set is equal to the complement of the interior of the complement of .

The subset is dense in iff .

If is a subspace of containing , then the closure of computed in is equal to the intersection of and the closure of computed in : . In particular, is dense in iff is a subset of .