The Study on Recent Generalization in Continuous and Homeomorphism Maps Using Bi Topology and Fuzzy Topological Spaces

Abstract

In the year 1965 L.A.Zadeh and his team [88] investigated and presented the speculation on quotFuzzy Subsetquot. This speculation consequently helped researchers to find many applications in the area of science and technology. Subsequently in the year 1968 fuzzy topological spaces were presented by C.L.Chang [16]. After C.L.Chang s contribution, number of other reserchers K.K.Azad [1 and 2], M.Ferraro and D.H.Foster [23], A.Haydar Es [230], R.Lowen [38], A.S.Mashhour et.al [44-47], M.N.Mukherjee and B.Ghosh [48], M.N.Mukherjee and S.8.Sinha [49], P.Sundaram [64-65], R.H.Warren [83-84], C.K.Wong [85-87], and many more researchers put their efforts for the evolution of fuzzy topological spaces. In particular from 2000 to 2004 reserachers in this field such as M.K.R.S.Veerakumar [70], M.S.Jayasheela Reddy [31], R.Devi and M.Muthtamil Selvan [19] references demonstrates the idea behind gand#8727;and#8722;and#119888;and#119897;and#119900;and#119904;and#119890;and#119889;and#119904;and#119890;and#119905;and#119904;and and#119879;1/2and#8727;,*and#119879;1/2and#8722;and#119891;and#119906;and#119911;and#119911;and#119910;and#119905;and#119900;and#119901;and#119900;and#119897;and#119900;and#119892;and#119894;and#119888;and#119886;and#119897;and#119904;and#119890;and#119905;and#119904;. A new class of fuzzy sets, gand#8727;and#8722;and#119888;and#119897;and#119900;and#119904;and#119890;and#119889;and#119891;and#119906;and#119911;and#119911;and#119910;and#119904;and#119890;and#119905; and its application which helps in developing two new topological spacesand#8722;fuzzyand#8722;Tand#8727;and#8727;and fuzzyand#8722;Tand#8727;and#8727;. After analysis it is witnessed that any gand#8727;and#8727;and#8722;and#119888;and#119897;and#119900;and#119904;and#119890;and#119889;,and#119904;and#119890;and#119898;and#119894;and#119888;and#119897;and#119900;and#119904;and#119890;and#119889;and#119886;and#119899;and#119889;, semi closed, and and#61537;-closed fuzzy set respectively is a gand#8727;and#8727;and#8722;and#119888;and#119897;and#119900;and#119904;and#119890;and#119889;and#119891;and#119906;and#119911;and#119911;and#119910;and#119904;and#119890;and#119905; however not vice versa. Similarly, all gand#8727;and#8727;and#8722;and#119888;and#119897;and#119900;and#119904;and#119890;and#119889;and#119894;and#119904; gand#8722;and#119888;and#119897;and#119900;and#119904;and#119890;and#119889;is and gand#8727;and#8722;and#119888;and#119897;and#119900;and#119904;and#119890;and#119889;and#119891;and#119906;and#119911;and#119911;and#119910;and#119904;and#119890;and#119905; respectively however not vice versa. After detailed analysis on gand#8727;and#8727;and#8722;and#119888;and#119897;and#119900;and#119904;and#119890;and#119889;and#119891;and#119906;and#119911;and#119911;and#119910;and#119904;and#119890;and#119905;, it is witnessed that all and#119879;1/2and#8722;and#119904;and#119901;and#119886;and#119888;and#119890;and#119904; is Tand#8727;and#8727;and#119886;and#119899;and#119889;and#8727;and#8727;and#119879; fuzzy spaces correspondingly. Also the analysis of fgand#8727;and#8727;and#8722;and#119888;and#119900;and#119899;and#119905;and#119894;and#119899;and#119906;and#119900;and#119906;and#119904; and fgand#8727;and#8727;and#8722;and#119888;and#119897;and#119900;and#119904;and#119890;and#119889;and#119898;and#119886;and#119901;and#119904; and concluded as every and#119891;and#8722;and#119888;and#119900;and#119899;and#119905;and#119894;and#119899;and#119906;and#119900;and#119906;and#119904;,semi continuous and and#8722;and#61537;and#8722;continuous, functions correspondingly are fgand#8727;and#8727;and#8722;and#119888;and#119900;and#119899;and#119905;and#119894;and#119899;and#119906;and#119900;and#119906;and#119904; however vice versa is not true. Moreover, all fgand#8727;and#8727;and#8722;and#119888;and#119900;and#119899;and#119905;and#119894;and#119899;and#119906;and#119900;and#119906;and#119904; is a and#119891;gsand#8722;and#119888;and#119900;and#119899;and#119905;and#119894;and#119899;and#119906;and#119900;and#119906;and#119904; and is a and#119891;gspand#8722;and#119888;and#119900;and#119899;and#119905;and#119894;and#119899;and#119906;and#119900;and#119906;and#119904; function, respectively. A characterization of fgand#8727;and#8727;and#8722;and#119888;and#119900;and#119899;and#119905;and#119894;and#119899;and#119906;and#119900;and#119906;and#119904; is carried out and r