semiclosed

Semiclosed is a term used in topology to describe a certain class of subsets of a topological space, defined in relation to the interior and closure operators.

Definition: Let X be a topological space, and for a subset A ⊆ X let Int(A) denote its interior and Cl(A) its closure. A subset A is called semiclosed if Cl(Int(A)) ⊆ A. Equivalently, the semi-closure of A, sc(A) = Cl(Int(A)), is contained in A. The property is a way of expressing that A already contains, in a sense, the closure of its interior.

Relations to other notions: Semiclosed sets generalize closed sets, since if A is closed then Cl(Int(A)) ⊆ Cl(A) = A, so A is semiclosed. Open sets are not necessarily semiclosed, but open sets are always semi-open, a related notion defined by A ⊆ Cl(Int(A)). The semiclosed and semi-open concepts form a pair of dual generalized notions between open and closed sets.

Examples: In the real line with the usual topology, the empty set and the whole space are semiclosed. Any closed interval [a,b] is semiclosed because Int([a,b]) = (a,b) and Cl((a,b)) = [a,b]. A non-closed example that is not semiclosed is (0,1], since Int((0,1]) = (0,1) and Cl((0,1)) = [0,1], which is not contained in (0,1].

See also: semi-open sets, interior, closure, topology. Semiclosed sets are used to study generalized closure properties and refinements of topological structure.