[Disjunction] and [Conjunction]-Sets of Generalized Topologies (Report‪)‬

1. Introduction In 199[gamma], Professor A. Csaszar [1] nicely presented the open sets and all weak forms of open sets in a topological space X in terms of monotonic functions defined on [??](X), the collection of all subsets of X. For each such function [gamma], he defined a collection [mu] of subsets of X, called the collection of [gamma]-open sets. A is said to be [gamma]-open if A [subset] [gamma](A). B is said to be [gamma]-closed if its complement is [gamma]-open. With respect to this collection of subsets of X, for A [subset] X, the [gamma]-interior of A, denoted by [i.sub.[gamma]](A), is defined as the largest [gamma]-open set contained in A and the [gamma]-closure of A, denoted by [i.sub.[gamma]](A), is the smallest [gamma]-closed set containing A. It is established that [mu] is a generalized topology [3]. In [5], [gamma]-semiopen sets are defined and discussed. In [7], [gamma][alpha]-open sets, [gamma]-preopen sets and [gamma]-open sets are defined and discussed. [gamma][beta]-open sets are defined in [6]. If [alpha] is the family of [gamma][alpha]-open sets, is the family of all [gamma]-semiopen sets, is the family of all [gamma]-preopen sets, [pi] is the family of all [gamma][beta]-open sets and [beta] is the family of all [gamma]-open sets, then each collection is a generalized topology. Since every topological space is a generalized topological space, we prove that some of the results established for topological spaces are also true for the generalized topologies [Omega] = {[mu], [alpha], [sigma], [pi], b, [beta]}. In section 2, we list all the required definitions and results. In section 3, we define the [[conjunction].sub.k] and [[disjunction].sub.k], operators for each k [member of] [Omega] and discuss its properties. Then, we define [[conjunction].sub.k]-sets, [[disjunction].sub.k]-sets, g. [[conjunction].sub.k]-sets and g. [[disjunction].sub.k]-sets and characterize these sets. In section 4, for each k [member of] [Omega], we define and characterize the separation axioms k - [T.sub.i], i = 0, 1, 2 and k - [R.sub.i], i = 0, 1.