ON SOME NEW MAXIMAL AND MINIMAL SETS VIA θ-OPEN SETS

Abstract

Abstract. Nakaoka and Oda ([1] and [2]) introduced the notion of max-imal open sets and minimal closed sets in topological spaces. In thispaper, we introduce new classes of sets called maximal θ -open sets, min-imal θ -closed sets, θ -semi maximal open and θ -semi minimal closed andinvestigate some of their fundamental properties. 1. Introduction and preliminariesGeneralized open sets play a very important role in General Topology andthey are now the research topics of many topologists worldwide. Indeed asignificant theme in General Topology and Real Analysis concerns the variouslymodified forms of continuity, separation axioms etc by utilizing generalizedopen sets. One of the most well-known notions and also an inspiration sourceis the notion of θ -open sets introduced by N. V. Veli˘cko [3] in 1968. Since thecollection of θ -open sets in a topological space ( X,τ ) forms a topology τ θ on X then the union of two θ -open sets is of course θ -open. Moreover τ = τ θ if andonly if ( X,τ ) is regular.F. Nakaoka and N. Oda in [1] and [2] introduced the notion of maximal opensets and minimal closed sets. The purpose of the present paper is to introducethe concept of a new class of open sets called maximal