Fuzzy Tight And Rough Set Topology

In this paper, our main purpose is to discuss the properties of different compactness in L - topological spaces and fuzzifying topological spaces, and to study the relationships between rough sets and topology.This paper consists of two parts. They are divided into three chapters.In the first chapter of part one the concepts of S - compactness, S - paracompactness, S - local finite, S - T₂ S - regular and S - Lindel-f property is defined. And some properties of S-compactness and S - paracompactness are proved. For example, the Tychonoff Theorem for S-compactness is true. A weakly induced L - fuzzytopological space is S - compact if and only if ( X ,[δ]) is compact. A S - T₂S-paracompact space is S - regular space. Moreover, the properties of countable S - compactness are studied.In chapter 2, topology is studied by the semantic method of continuous valued logic.Firstly, the concepts of strongly semiopen set and irresolute mapping are defined in fuzz- ifying topological spaces. Moreover, we introduce and study the SS - compactness in fuzzify- ing topological spaces. Many characterizations of SS -compactness are presented.Secondly, the concepts ofβ- open set,α-open set are defined in fuzzifying topo- logical spaces. Moreover we introduce and study theβ- compactness,α-compactness, countableβ- compactness, countableα-compactness andβ- Lindelof property,α- Lindelof property in fuzzifying topological spaces. Many characterizations ofβ- compact- ness,α-compactness, countableβ- compactness, countableα-compactness andβ- Lindel-f property,α-Lindel-f properties are presented.In chapter 3, firstly, the concepts of upper and lower approximation operators of interval -valued fuzzy set are defined. And some properties are studied. Secondly, with the help of the interval-valued fuzzy cut set, we get a parameterize roughness measure of an interval-valued fuzzy rough sets. At last, we proved that the fuzzy interval-valued relation (R|^) can induce a fuzzy topology in fieldU .