Research On Generalized Open Sets In Topological Spaces

In this thesis, the main purpose is to discuss the generalized open sets and study its properties in general topological spaces, L-topological spaces and fuzzy bitopological spaces in the further way. The primary studies in this paper are the following:(1)Research on generalized open sets is a very important field in general topological spaces. In 1963,Levine introduced the concept of semi-open sets. In 1965, Njastad introduced the notion ofα-open sets. In 1982, the concept of preopen sets are studied by Mashour. In 1983, the notion ofβ-open sets are studied by Abd El-Monsef. In 1997, the concept ofγ-open sets are introuduced by A.Csaszar, etc. In this thesis, first, strong semi-open sets, one type of generalized open sets, is introduced in a new way, and its properties are researched, and then the relations of the relative concepts are made clear by taking some anti-examples.(2) The properties of semi-subset in general topological spaces are studied by Li Hou-yuan in 1991. In this thesis, a series of concepts of strong semi-subset are introduced. Then,by comparing to subset and semi-subset in general topological spaces, some properties of strong semi-subset in general topological spaces are discussed and researched ,and the further research and generalization on Kuratowski's 14 Sets Theorem in general topological spaces made the two important theorems introduced in general topological spaces .(3) Separation axioms are very important in general topological spaces and L-topological spaces. In 1984,the concepts of S i-separation are introduced in general topological spaces by Hu Qin-ping. In 2005, Ai Wei-hong introduced the notion of strong S i-separation in general topological spaces. In this thesis, according to the new generalized open sets—strong semi-open set in general topological spaces, a series of concepts, including the stong semi-open set, strong semi-R-neighbourhood and so on, are introduced in L-topological spaces, and the concepts of strong S i-separation are introduced in L-topological spaces and the relative properties are researched.(4) Kandil introduced the concept of fuzzy bitopological spaces, the concept of fuzzy semi-open sets and fuzzy semi-continuity in fuzzy bitopological spaces was studied by S.S.Thakur in Kandils'means. In this thesis, the concepts of fuzzy preopen sets and fuzzy precontinuous mappings in fuzzy topological spaces due to Bin Shahana have been generalized to fuzzy bitopological spaces, and some of their properties are studied.