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The smallest closed set containing a given subset.
The closure of a subset in a topological space is
Equivalently, a point lies in if and only if every neighborhood of intersects .
A set is closed exactly when . Closure is closely tied to limit points and the derived set.
A topological space is an ordered pair where is a set and is a topology on , meaning:
Here denotes the power set of , and the members of are the open sets (whose complements are the closed sets). Many standard constructions—such as the subspace topology, product topology, and quotient topology—produce new topological spaces from existing ones.
A neighborhood of a point in a topological space is a subset such that there exists an open set with .
A closed set in a topological space is a subset such that its complement (the set difference) is a open set.
Closed sets are the natural targets of the closure operation: the closure of is the smallest closed set containing . Closed sets also provide an equivalent formulation of continuity via preimages of closed sets.
A limit point (or accumulation point) of a subset in a topological space is a point such that every neighborhood of satisfies
The set of all limit points of is the derived set of . Limit points also describe closure via the identity .
The derived set of a subset in a topological space is the set
where “limit point” is as in limit point.
The derived set records the accumulation behavior of , and it satisfies the basic relationship with closure: