格值Smooth拓扑分子格范畴与L-拓扑空间中若干问题的研究

【作者】 张杰；

【导师】 郑崇友；

【作者基本信息】 首都师范大学 ， 基础数学， 2002， 博士

【摘要】 本文的主要目的是在拓扑分子格理论的基础之上建立格值Smooth拓扑分子格理论的基本框架，研究格值smooth拓扑分子格范畴的相关性质，并对L-拓扑空间中Fuzzy仿紧性和连通性等问题作深入地研究。 本文由两部分组成，第一部分是关于格值Smooth拓扑分子格及其范畴的研究。第二部分是关于L-拓扑空间中若干拓扑性质的研究。全文共分为四章。 第一部分的主要内容如下： 第一章在完全分配格上给出格值Smooth拓扑分子格的概念，它以一般拓扑空间、L-拓扑空间和拓扑分子格概念为特例。借助文[67，68]中提出的分子集合套和素元集合套理论，结合文[83]并以截集芦F_[a]和F^[a]为工具刻画L上smooth余拓扑F，从而给出了L-smooth余拓扑的刻画定理、分解定理、表现定理和由余拓扑族构造L-smooth余拓扑的充要条件．引入完全分配格上L-smooth闭包算子的定义，证明了这一概念的的合理性，给出了L-smooth闭包算子的若干等价条件，从而揭示了它与分子格上的闭包算子的内在联系。定义L-smooth连续广义序同态，L-smooth闭广义序同态等概念，研究了它们的一些性质与特征，得到了它们与连续广义序同态和闭广义序同态之间的关系，并给出了L-smooth拓扑分子格的一种乘积。 第二章 研究 L-smooth拓扑分子格与L-smooth连续广义序同态所构成的范畴STML（L）的相关性质．证明了范畴STML（L）是拓扑范畴，拓扑分子格和连续广义序同态所构成的范畴TML是范畴STML（L）的双反射满子范畴，以及两个函子的右伴随的存在性定理。以文[106]的相关概念为基础，研究了范畴STML（L）中的极限与逆极限问题，从而给出了范畴STML（L）的另一种乘积，且这两种乘积在范畴意义下是同构的。构造性地给出一对函子，并以此证明了范畴STML（L）的一个表示定理，即范畴STML（L）与范畴FSTS（L）（由L-fuzzifying scott拓扑空间与保定向并和way-below关系的L-fuzzifying连续映射所构成的范畴）等价。研究了范畴STML（C）的一些性质，其中C为完全分配格范畴的一个子范畴，L-smooth余拓扑的值域L∈C，态射为（L₁，L₂）-smooth连续的广义序同态。 第三章 以文[70]的点式拟一致结构为背景，在完全分配格上（或Fuzzy格上）建立一种格值smooth点式拟一致结构概念．它以文[70]的点式拟一致结构为特款。获得了它的刻画定理，分解定理，表现定理及构造条件，揭示了它与点式拟一致结构之间的关系。研究了它与L-smooth拓扑的关系．证明了每个L-smooth拓扑分子格都可以层点式拟一致化，在一定条件下可L-smooth点式拟一致化，每个L-smooth点式拟一致结构在一定条件下可以诱导出一个L-smooth余拓扑。给出了L-smooth点式拟一致连续广义序同态的定义，得到了它的等价刻画以及它和点式拟一致连续广义序同态的关系。定义L-smooth点式拟一致结构的L-smooth基的概念，从而给出了F格上的L-smooth点式一致结构概念，获得了若干L-smooth一致结构的等价刻画。 第二部分的主要内容如下： 第四章首先，在占—拓扑空间中利用文[72]中fuzzy集的α-局部有限族概念定义一种新的强fuzzy仿紧性。它是强fuzzy紧概念的直接一般化，具有层次特色，并且与某些fuzzy分离公理具有良好的相容性。给出了强可数紧集是强fuzzy仿紧的，以及弱诱导h 首都师范大学博士学位论文2002年空间为强fsZy仿紧的充要条件．证明了每个强fUZZy紧集和fUZZy单位区间I厂)都是强fumy仿紧的；强fuZZy紧集和强fuZZy仿紧集的乘积是强fuZZy仿紧集；强TZ的强fUZZy仿紧空间是强S“一正则的；强TZ的强fUZ。＊仿紧空间是强S”一正规的；强S”一正则的强 Lindeloef空间是强 fUzzy仿紧的等结果． 其次，借助文 I691中 a一。聚点与。一聚点概念给出了可数强 F紧集的两个刻画定理：（1）Iazy集 A是可数强F紧的当且仅当 Va e M瞩；A中每个 a可数无限集皆有高为口的a一。聚点；（2）T的L一拓扑空间中的fuZzy集A是可数强F紧的当且仅当 Va 6 MK)；A中每个 a可数无限集皆有高为 a的 a一聚点、证明了强F紧集和可数强F紧集的乘积是可数强F紧集，从而得良紧集与可数强F紧集的乘积是可数强F紧集，可数强F紧集在L值Zadeh型函数下的逆不变性． 最后，基于文［叫的思想，借助开集定义一种具有fuZZy特色的 O厂连通性，当 8二 0时，它以文 ［109］中的 O－连通性为特例，并较好地保持了连通性的良好性质，证明了它是L－好的推广，给出了若干等价刻画． 本文部分结果已发表《Journal of Fuzzy Mathematics ))、《数学研究与评论》、《模糊系统与数学》等杂志上，详见附录．更多还原

【Abstract】 In this paper,our main purpose is to build a concept of lattice valued smooth topological molecular lattice,to research related properties of its category,and to study the fuzzy paracom-pactness and connectness in the L-topological spaces.This paper is consist of two part. There are four chapters. In the first part,the concept of lattice valued smooth topological molecular lattice and its category are researched. In the second part,some topological properties in L-topological spaces are deliberated.In the first chapter,the concepts of -smooth topological molecular lattices,-smooth closure operator,L-smooth continuous GOH and -smooth closed GOH are defined,the concept of -smooth topological molecular lattices is the generalization of general topological spaces and topological molecular lattices. Based on the theory of molecular set nests and prime elements set nested in [67,68],with the cuted set F and JM,we have obtain the equivalent theorem,the decomposition theorems,the representation theorems about L-smooth co-topology,the equivalent conditions of constructing a -smooth co-topology by a family co-topologies and the equivalent theorem of closure operator. We have proved that -smooth closure operator is reasonable,discussed the relations between -smooth closure operator and closure operator,-smooth continuous GOH and continuous GOH,and given a kind of product of -smooth topological molecular lattices.In the second Chapter,some properties of the category STML which is consisted of the -smooth topological molecular lattices and -smooth continuous GOH. We have proved that STML is a topological category and the category TML is a bireflective full subcategory of STML,the existence of some functors’ right adjunction. Based on [106],we have researched the limits and inverse limits in the category STML,given the other one kind of product of -smooth topological molecular lattices. By constructing two functor,we have proved a representation theorem of the category STML that the category STML is equivalent with the category FSTS,where FSTS is consisted of -fuzzifying scott topological spaces and the mappings which are preserving directed-join and way-below relation and continuous. Besides,the category STML(C) has been discussed,where C is a subcategory of the category of the completely distributive lattices and GOHs, C,morphisms are (1,2)-smooth continuous GOH.In the third chapter,Based on [70],we introduce a concept of the lattice valued smooth pointwise quasi-uniformity on completely distributive lattices,and pointwise quasi-uniformity in [70] is a special case of it. The equivalent theorem,the decomposition theorems,the representation theorems about -smooth pointwise quasi-uniformity have been obtained. These theorems show the inherent relation between -smooth pointwise quasi-uniformity and point-wise quasi-uniformity. It is proved that every -smooth topological molecular lattice is stratum pointwise quasi-uniformizable,and -smooth pointwise quasi-uniformizable if it satisfies some conditions,every -smooth pointwise quasi-uniformity can induce an -smooth co-topology if it satisfies some conditions. Meanwhile we introduce the concepts of -smooth pointwisequasi-uniform continuous GOH and L-smooth pointwise uniformity on fuzzy lattices,discussed some properties,characteristics and equivalent depictions.In the fourth chapter,the first,with the concepts of a- locally finite family of fuzzy sets in [72],we defined a new kind of strong fuzzy paracompactness. It is a direct generalization of the concept of strong fuzzy compactness,has layer characteristics,and has a good compatibility with some kind of fuzzy separation axioms. It is proved that strong countable compact set is a strong fuzzy paracompact if and only if it is a strong fuzzy compact,strong fuzzy compact set and fuzzy unit interval are strong fuzzy paracompact,the product of a strong fuzzy compact set and a strong paracompact set is strong fuzzy paracompact,a strong T2Strong fuzzy paracompact space is strong 5-regular and strong S-normal,a strong更多还原