A study on some aspects of bitopological spaces

Abstract

J. C. Kelly initiated the systematic study of bitopological spaces in 1963. Since newlinethen many contributions to bitopological settings were done by several authors. newlineThe aim of the present research project is to develop a suitable notion of a kind newlineof semi open sets, separation axioms, generalized and#923; and#8722; sets, semi continuous functions, newlinesemi irresolute functions, connectedness and compactness in bitopological spaces and newlineto study some related notions. newlineWe study and introduce the concepts of and#119894;and#119895; and#8722; and#120575; semi open sets, and#119894;and#119895; and#8722; and#120575;and#119904; newlinecontinuous functions, and#119894;and#119895; and#8722; and#120575;and#119904; irresolute functions in bitopological spaces and newlineinvestigate some of their characterizations and relationships with other sets. Also newlinewe introduce and investigate the following separation axioms in bitopological spaces: newlineand#119894;and#119895; and#120575;and#119904; and#8722; and#119879;and#119894; and and#119894;and#119895; and#120575;and#119904; and#8722; and#119877;and#119894; newlinefor and#119894; = 0, 1, 2, and investigate some of their properties newlineand relationships with other spaces. newlineWe introduced and#119894;and#119895; and#8722; and#948;s connected, and#119894;and#119895; and#8722; and#120575;and#119904; extremally disconnected and and#119894;and#119895; and#8722; and#948;s newlinecompact bitopological spaces and study some of their characterizations and properties. newlineWe introduce the notions of and#119894;and#119895; and#8722; and#923;and#948; newlines newlinesets and and#119892; and#8722; and#119894;and#119895; and#8722; and#923;and#948; newlines newlinesets and study some newlineof their fundamental properties. Also, we introduce the new notions of and#119894;and#119895; and#8722; and#119879; newlineVand#948; newlines newlinespace, newlineand#119894;and#119895; and#8722; and#923;and#948; newlines newlinecontinuous functions, and#119894;and#119895; and#8722; and#923;and#948; newlines newlineirresolute functions, and#119894;and#119895; and#8722; and#923;and#948; newlines newlineopen functions and newlineand#119894;and#119895; and#8722; and#923;and#948; newlines newlinehomeomorphism functions and investigate some of their properties and newlinerelationships with other functions and spaces. Also we introduced the notions of newlinepairwise and#923;and#948; newlines and#8722; Tand#119894; and pairwise and#923;and#948; newlines and#8722; Rand#119894; bitopological spaces for and#119894; = 0,1,2 and newlineinvestigate their properties and study the relationships with other spaces. We introduced the notion of a closure operator, called and#119894;and#119895; and#8722; semi (and#120575;, and#120579;)-closure newlineoperator and we introduce a new class of a sets, called and#119894;and#119895; and#8722; semi (and#120575;, and#120579;)-open sets and newlinestudy some of their properties. Using these sets, we shall also define the notion of newlinegeneralized and#119894;and#119895; and#8722; and#119904;and#8896;and#120575; newlineand#120579; newlinesets. The major properties of this new concept will be studied. newlineFinally