semiopen

Semiopen is a term used in topology to describe a certain kind of subset of a topological space. Let X be a topological space and A ⊆ X. A is called semiopen if A ⊆ cl(int(A)), where int(A) is the interior of A and cl denotes closure. Equivalently, since int(A) is an open set contained in A, A is semiopen if there exists an open set U with U ⊆ A ⊆ cl(U) (taking U = int(A)).

Basic properties and examples

- Open sets are semiopen, because if A is open then int(A) = A and A ⊆ cl(A).

- The empty set and the whole space X are semiopen, since int(∅) = ∅ and cl(∅) = ∅, while int(X) = X and cl(X) = X.

- In the real line with the standard topology, every half-open interval [a,b) is semiopen, since int([a,b)) = (a,b) and cl((a,b)) = [a,b].

- The Cantor set, having empty interior, is not semiopen because it is not contained in the closure of its interior (which is empty).

Persistence and limitations

- The union of any family of semiopen sets is semiopen: if A_i ⊆ cl(int(A_i)) for all i, then ∪A_i ⊆ cl(∪int(A_i)) ⊆ cl(int(∪A_i)).

- Finite intersections need not be semiopen. For example, in R, [0,1] and [1,2] are semiopen, but their intersection {1} is not semiopen.

Relation to other notions

- A semiopen set sits between an open set and the closure of that open set. It is related to, but distinct from, regular open and regular closed sets, which satisfy equalities involving interior and closure.

This concept is used in the study of generalized separation axioms and in fuzzy topology, where semiopen sets help describe intermediate openness-like behavior.