semiclosure

Semiclosure is a concept in topology that generalizes closure by using the family of semiclosed sets. Let X be a topological space and A a subset of X. The semiclosure of A, denoted sc(A), is defined as the intersection of all semiclosed sets that contain A. Equivalently, sc(A) is the smallest semiclosed superset of A.

A semiclosed set is a subset S of X satisfying the double inclusion cl(Int(S)) ⊆ S ⊆ cl(S), where cl denotes closure and Int denotes interior. This places semiclosed sets between the interior-driven closure and the full closure, providing an intermediate notion between A and its closure.

Basic properties follow from this definition. For any A ⊆ X, A ⊆ sc(A) ⊆ cl(A). If A is semiclosed, then sc(A) = A; in particular every closed set is semiclosed and hence is its own semiclosure. The concept behaves differently from the strict closure: in some topologies, nontrivial sets may have semiclosures strictly larger than A but still contained in cl(A).

The semiclosure operator is extensive, monotone, and idempotent: A ⊆ sc(A), A ⊆ B implies sc(A) ⊆ sc(B), and sc(sc(A)) = sc(A). It can be viewed as a hull operator relative to the family of semiclosed sets, extracting the smallest semiclosed cover of a given set.

Relation to other notions is as follows: Int(A) ⊆ sc(A) ⊆ cl(A). Semiclosure provides a framework for discussing intermediate notions of closure that lie between a set and its closure, via the intermediate class of semiclosed sets.

See also: semiclosed sets, hull operators.