Table of contents

1 Notation 2 Alternative characterisations 3 Examples 4 Facts about closures

Notation

The closure of S is variously denoted by "Cl(S)" or "". If there is more than one topology on X (say T and T'), then the different topologies may give rise to different closures; this can be indicated in the notation by a subscript, as in "Cl_T(S)". If the topology is itself defined by some other structure, such as a metric d, then "d" can be placed in the subscript instead of "T".

Alternative characterisations

In a metric space X (such as the n-dimensional Euclidean space) the closure Cl(S) is the set {x ∈ X : d(x,S) = 0} of all points in X whose distance from S is 0. Here, d(x,S) is defined as the infimum of the set {d(x,y) : y ∈ S}.

In a first countable space (such as a metric space), Cl(S) 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".

Another characterization of Cl(S) is as follows: an element x of X belongs to Cl(S) if and only if every neighborhood of x contains an element of S. In other words, x ∈ Cl(S) iff x ∈ S or x is a limit point of S.

Examples

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

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 S is closed if and only if Cl(S) = S. In particular, the closure of the empty set is the empty set, and the closure of X itself is X. 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 S is equal to the complement of the interior of the complement of S.

The subset S is dense in X iff Cl(S) = X.