Semilukkede

Semilukkede is a Danish term used in topology to denote semiclosed sets. If X is a topological space and A is a subset of X, then A is semilukkede when its complement X \ A is semi-open, or equivalently when the closure of the interior of A satisfies cl(Int(A)) ⊆ A. Here Int(A) denotes the interior and cl denotes closure. In this sense, semilukkede sets form a broader class than closed sets: every closed set is semilukkede, but there are semilukkede sets that are not closed.

The concept is part of the broader framework of semi-open and semi-closed sets. A subset U of X is semi-open if U ⊆ cl(Int(U)). A set is semilukkede if and only if its complement is semi-open. These notions generalize the usual open and closed concepts and are used in general topology to study properties of continuity, convergence, and various weaker forms of separation axioms. They often appear in Danish mathematical literature and in discussions that compare different degrees of openness and closedness without requiring full regular open or closed conditions.

Example: In the real line with the standard topology, the half-open interval [a,b) is semilukkede; its interior is (a,b), its closure is [a,b], and [a,b) ⊆ [a,b], so it satisfies the semilukkede condition. Its complement, (-∞,a) ∪ [b,∞), is semi-open, hence semiclosed. The empty set and the whole space are semilukkede as well. In English-language texts, the term semiclosed is typically used for the same concept.