A Study Of The Accumulation Point, Derived Set And Derived Operator In L-fuzzy Topological Spaces

The topological structure of the L-fuzzy topological space(L - fts for short) which have stratifications is more complicated than that of the general topological space, so various accumulation points and derived sets of a LF-set in L - fts were defined (see [1]-[11]) since the two basic concepts of accumulation point and derived set of a F-set were introduced for the first time by B. Pu and Y. ,Liu in their original paper [4] concerning " pointlike " fuzzy topology in 1977. In this dissertation, all kinds of accumulation points and derived sets (defined in [1]-[11]) together with some basic results are introduced in Chapter 1.Chapter 2 is devoted to the intrinsic relation between the l-accumulation point and all kinds of accumulation points defined in [2] and [4]-[ll]. In ?.1, we firstly investigate these kinds of accumulation points defined in [4]-[10], and reach a conclusion that the above each accumulation point is not only one kind of w-accumulation point but also that of l-accumulation point. Then we farther study the intrinsic relation between /-accumulation point and w-accumulation point in ?.2, and respectively provide some sufficient conditions under which the following two formulae of will hold. Furthermore, we construct several Fuzzy lattices to give the counterexamples that dissatisfy above two formulae. Based on this work, in ?.3 we prove the " Classification Theorem for /-accumulation point " using the " Classification Theorem for w-accumulation point ".Chapter is mainly devoted to some important properties of l-derived set and N-derived set. Firstly the " Decomposition Theorem for /-derived set "is proved in ?.1. As its application, the decomposition formula of a LF-set or molecule is given. In ?.2, the concept of derived set preserving is introduced, and we prove that every locally finite family of subsets is /-derived set preserving, moreover, the conclusion that every locally finite family of subsets is iV-derived set preserving is generalized. In ?.3, the C.T.Yang's theorem of the molecular form (" molecular form" for short) concerning /-derived set and iV-derived set are discussed, and we obtained two sufficient conditions under which the " molecular form" concerning N-derived set will hold: is finite. Furthermore, we raise a new question for further study of the Open Question whether the " molecular form" concerning l-derived set will hold. Besides, we prove that the positive solution to the new question will answer not only the Open Question but also the long-standing question whether the " molecular form" concerning iV-derived set hold!The main aim of Chapter 4 is to study the l-derived operator and TV-derived operator on Lx. In ?.1, the concept of l-derived operator is introduced, and we proved the one to one correspondence between the family of all the l-derived operator and that of all the L-fuzzy topology. In ?.2, the relations between ST-1 separation and other four separations are systematically researched, and several examples are given. Subsequently, the characterization theorem for induced spaces by iV-derived operator and some other results in [24]-[25] are briefly introduced in ?.3. Using these results, we draw a comparison of the properties between /-derived set and iV-derived set, and detailedly analyze the advantage and drawback of respective properties of two derived sets in ?.4. Our analysis shows that /-derived operator can't characterize induced spaces, and that the reason that cause the above drawback is essential and inevitably. Furthermore, an interesting question is naturally raised according to the above analysis.