A Note on Semitopological Classes

Abstract

Abstract.This paper shows that semitopological classes are subsemilattices of the lattice of topologies, and gives a new characterization for the finest topology in the semitopological class.Introduction.In [4] Levine defined a set, A, to be semiopen if there is some open set U so that i/<= A<= c(U), where c( ) denotes closure in the topological space. In [1] it was shown that if (X, r) is a topological space, there is a finest topology [we shall call it F(t)] so that the semiopen sets are the same as for t. If X is a set of points, let T(X) be the lattice of topologies on X. If t e TiX), let [t] denote the equivalence class of all topologies which have the same semiopen sets as r.[t] is called a semitopological class of topologies on X. The object of this note is to show that if t e TiX), [t] is a subsemilattice of TiX) with respect to the usual join operation on topologies, and to give a new characterization for F(t).

On Semi-Open Sets and Semi-Continuity

“…Assume (3). Let E ⊂ X and let y ∈ f (sCl(E)) ⇒ y = f (x) for some x ∈ sCl(E) ⇒ there exists a net {x λ |λ ∈ ∆} in E such that x = s. lim λ∈∆ x λ ⇒ f (x) = lim λ∈∆ f (x λ ) ⇒ f (x) ∈ Cl(f (E)).…”

“…Levine [2] introduced the notion of semi-open sets and semi-continuity. S. Crossley and S. Hildebrand in [1,3] constructed a topology by introducing pre semi-closure and closure operators and proved that this topology is the finest of all topologies for which the semi-open sets are precisely those of (X, T ). We present yet another topology on X which is closely related to the above topology.…”