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).

*‐Topological properties

“…It can be shown that semi-topological properties are a-topological properties as a consequence of Theorem, 2.6 of [4] and Theorem 2 of [3]. We will establish this fact (specifically we will show that tile semi-topological properties are precisely the c-topological properties) as a corollary to the Lifting Theoren proven i, this sectio,.…”

A Note on Semitopological Properties

“…Since all open sets in F(r) are of the form 0 -N where 0 E t and N is nowhere dense in (X, t) [1], we see that if (X, t) is not Urysohn, neither is (X, F(t)), and the proof of Theorem 2 is complete.…”

“…Also, if /: (X, t) -> (Y, o) is a semihomeomorphism, then /: (X, F(t)) -» (Y, F(o)) is a homeomorphism. A new characterization of F(t) as (0 -N\0 E t and N is nowhere dense in (X, t)) was given in [1], and this characterization has simplified the proofs given in this paper and it could be used to simplify the proof given in [3] that the Hausdorff separation axiom is a semitopological property.…”

Semi‐topological properties

“…they proved that for any space (E.), there exists the finest topology F(T) for X having the same family of semi-open sets as (X,T). Crossley [9] PROOF. Suppose P is a semi--topological property.…”