Topological Space Decomposition And Continuous Mapping Of Several Approximate Compactness

In this paper, two kinds of approximate compactness are introduced and studied in [0,1]-fuzzy topological spaces and L-fuzzy topological spaces. We define some approximate open sets, and discuss many decompositions of generalized continuity and its weak forms.The theory of compactness is one of the most important theories in general topological spaces. Compactness and its strong and weak forms play an important role in general topology. Based on Zadeh’s theory of fuzzy sets, Chang introduced the concepts of [0,1]-fuzzy topolog-ical space and its compactness. Lowen introduced the concepts of fuzzy compactness, strong fuzzy compactness and ultra-fuzzy compactness in [0,1]-fuzzy topological spaces. G.J.Wang introduced the concept of nice compactness in [0,1]-fuzzy topological spaces. It has various good properties, and is valued and affirmed by international scholars. F.G.Shi introduced a new kind of fuzzy compactness in L-topological spaces. This new concept does not depend on the structure of basis lattice L which does not require any distributivity. S.Z.Bai introduced and studied various approximate compactness in view of the above various methods. In diis paper, by means of fuzzy strongly preopen sets and fuzzy strongly preclosed sets, fuzzy strong-precompactness and fuzzy strong-preclosed spaces are introduced in [0,1]-fuzzy topological spaces. Fuzzy filterbases are used to characterize these concepts. We introduce a new notion of strong-precompactness in L-fuzzy topological spaces by means of strongly preopen L-sets and their inequality. We investigate some fundamental properties of strong-precompactness and present its some equivalent characterizations.In2002, A.Csaszar introduced a new concept of generalized topological space. With the development of ten years, the basic theories and the basic framework of generalized topol-ogy have been established. In this paper, we study some new properties and characterizations of (π, g’)-continuity in generalized topological spaces. We also discuss several new charac-terizations of weak (g, g’)-continuity and almost (g, g’)-continuity. The interrelationships of (π, g’)-continuity, weak (g, g’)-continuity and almost (g, g’)-continuity in generalized topologi-cal spaces are presented. We defineωμ-open sets and so on in generalized topological spaces, and introduce a new set D(w1,W2)={A:iW1A=iW2A], where w1,w2are weak structures on a nonempty set X. Many decompositions of generalized continuity and its weak forms are presented.