Some Results On L-topological Spaces

The differences of L-topological space and general topological space are derived from the former's layer structures . Comprehensive research of this kind of structures is the base of L-topological space theory . As we know that both separation and paracompactness have always played essential roles in topology and topological space theory have been enriched by the introduction of the notions of relative topological space and relative topological properties . In this paper , we firstly discuss the structures of moleculely generated lattices , then with this kind of lattice as the value-fields of fuzzy sets , we define a new type of fuzzy topological space , introduce the notions of layer separation axioms and layer paracompactness and study their properties . At last relative separation and relative paracompactness of L-fuzzy topology are defined and some properties of them are investigated . the main content of this paper is as follows .Chapter one , further properties of moleculely generated lattice are studied . We discuss the relationship between this kind of lattice and topological space , give the equivalent characterizations of distributive moleculely generated lattice , prove that every distributive moleculely generated lattice is isomorphic to the lattice of closed subsets of some topological space , and inversely , the lattice of closed subset of every topological space must be distributive and moleculely generated . At last , we give the sufficient and necessary conditions of a moleculely generated lattice being a completely distributive lattice , further prove that every distributive moleculely generated lattice with an order-reversing involution is a fuzzy lattice .Chapter two , in L-topological space ,we define a new set of separation axioms, say layer separation axioms , give their equivalent characterizations , study their various elementary properties . Base on that , we give the comparisons between this new set of separation axioms and another defined by professor Wang in Theory of L-fuzzy Topological Spaces . The results show that the former is weaker than the later and the two are harmonious . As an example of application of the layer separations , we show that H( X) unit interval satisfy layer T?separation but Hutton unit interval does not satisfy even layer T0 separation . At the end of this chapter , we weaken the layer separations , introduce the ultra-separation axioms.In the chapter three , we define a new type of fuzzy paracompactness-layer paracompactnessand obtain a series relevant properties . We show that, if a fuzzy topological space is I-paracompact, then it must be layer paracompact; if a L-fuzzy topological space is II-paracompact, then it must be layer paracompact. But generally , layer paracompactness can imply neither I-paracompactness nor II-paracompactness , I-paracompactness can not imply layer paracompactness .In chapter four , some results on relative compactness and relative paracompactness in general topology are generalized to the case of L-fuzzy topology . The notions of relative nature compactness and relative fuzzy paracompactness are introduced and a series relevant properties are investigated in detail .