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# 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 "". 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 "ClT(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.