Closure (topology)

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

The closure of S is variously denoted by "Cl(S)" or "S" with a horizontal line over it. 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.

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; for infinitely many sets, this need not be the case. However in any case, the closure of a union of sets is always a superset of the union of the closures of the sets. Since zero is a finite number and the union of zero sets is the empty set, this is another way to see that the empty set is its own closure; that is, the empty set is closed.

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.

If A is a subspace of X containing S, then the closure of S computed in A is equal to the intersection of A and the closure of S computed in X: Cl_A(S) = A ∩ Cl_X(S). In particular, S is dense in A iff A is a subset of Cl_X(S).