格值Smooth拓扑分子格范畴与L-拓扑空间中若干问题的研究
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(L_1，L_2)-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．
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 pointwise
quasi-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
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(L_1，L_2)-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．
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 pointwise
quasi-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
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42．刘应明，罗懋康，不分明紧化中的预序关系，科学通报，33(1988)，564-466．
43．刘应明，罗懋康，诱导空间与不分明Stone-Cech紧化，中国科学，A辑，4(1987)，359-368．
44. 刘应明，罗懋康, Advances in fuzzy systems—applications and theory Vol.9, Fuzzy topology(World Scientific Publishing Co. Pre. Ltd, Singapore, 1997).
45. Zhongfu Li, Compactness in fuzzy topological spaces, Chinese Kexue Tongbao 6(1983) 321-323.
46．梁基华，完全分配格上的一致结构，科学通报，34(1989)，1049-1051．
47. R. Lowen, Fuzzy uniform spaces, JMAA, 82(1981), 370-385.
48. R. Lowen, Fuzzy topological spaces and fuzzy compactness, JMAA, 56(1976) 621-633.
49. R. Lowen, A comparison of different compactness notions in fuzzy topological spaces, JMAA, 64(1978) 446-454.
50. Meng Guang-Wu. On countably strong fuzzy compact sets in L-fuzzy topological spaces. FSS, 1995(72): 119-123.
51．罗懋康，Fuzzy拓扑空间中的仿紧性与紧性，数学学报，4(1987)，548-552．
52．罗懋康，Fuzzy拓扑空间中的弱仿紧性与紧性，数学年刊，5(1986)，590-596．
53. Maokang Luo, Paracompactness in fuzzy topological spaces, JMAA 130(1988) 55-77.
54. S. Mac Lan, Categories for the Working Mathematician, Springer Verlag, New York, 1971.
55. H. W. Martin, Weakly induced fuzzy topological spaces, JMAA,78(1980) 634-639.
56. Nagata, Modern general topology, Elsevier Science Publishers B.V., 1985.
57．彭育威，格值半连续映射和L-fuzzy Hausdoff N-紧空间，数学年刊，11A(1990)753-760．
58. Pu Paoming and Liu Yingming, Fuzzy topology Ⅰ, Neighborhood structure of a fuzzy point and Moore-Smith convergence, JMAA 76(1980), 571-579.
59. Pu Paoming and Liu Yingming, Fuzzy topology Ⅱ, Product and quotient spaces, JMAA 77(1980), 571-579.
60. A.A.Ramadan, Smooth topological spaces, FSS 48(1992)371-375.
61. S.E.Rodabaugh, A theory of fuzzy uniformities with applications to the fuzzy lines, JMAA, 129(1988), 37-70.
62. S.E.Rodabaugh, E.P.Klement, U.Hohle eds., Applications of Category Theory to Fuzzy Subsets (Kluwer Academic Publishers, 1992).
63. S.E.Rodabaugh, Categorical Foundations of Variable-Basis Fuzzy Topology, in ([23]), 273-388.
64．史福贵，完全分配格上的点式拟一致结构，数学进展，26(1997)，22-28．
65．史福贵，完全分配格上的点式拟一致结构与p.q.度量，数学学报，39(1996)，701-706．
66．史福贵，Fuzzy格上的点式一致结构与度量，科学通报，42(1997)，718-720．
67．史福贵，L_β-集合套和L_α-集合套及其应用，模糊系统与数学，4(1995)63-72．
68．史福贵，L-fuzzy集和素元集合套，数学研究与评论，16(1996)398-402．
69．史福贵等，LF拓扑空间中的F可数紧性．哈尔滨科技大学学报，1992(3)：103-106．
70．史福贵，格上点式一致结构与点式度量理论及其应用，首都师范大学博士论文，2001．
71．史福贵，王国民，L-fuzzy拓扑空间中正则性和正规性的一种新定义，烟台师范学院学报(自然科学版)，3(1995)，20-24．
72．史福贵，郑崇友，L-fuzzy拓扑空间的一种新fuzzy仿紧性，模糊系统与数学，3(1995)40-48．
73．史福贵，模糊子集仿紧性的—个注记，模糊系统与数学，3(1998)3(1998)6-9．
74. Fugui Shi, Chongyou Zheng. Connectivity in L-fuzzy topological molecular lattices. FSS, 1989, 29: 363-370.
75．史福贵，积Fuzzy拓扑分子格的连通性．北京师范学院学报(自然科学版)，1991，2：12-15．
76. A.P.Sostak, On a fuzzy topological structure, Suppl. Rend. Circ. Mat Palermo Ser. Ⅱ 11(1985) 98-103.
77. A.P.Sostak, Two decades of fuzzy topology: basic ideas, notations and results, Russian Math. Surveys, 44 (1989) 125-186.
78. R.Srivastava, On separation axioms in a newly defined fuzzy topology, FSS 62(1994)341-346.
79．孙叔豪，完全分配格上的拟一致结构，陕西师范大学学报，3(1985)，9-15．
80．王戈平，不分明集的一个分解定理及其在不分明拓扑学中的应用，科学通报，26(1981)，259-262．
81．王戈平，完全分配格上的弱辅助序与广义序同态，数学季刊，4(1988)，76-83．
82．王戈平，田晓明，一类完备格的直积分解与Fuzzy格的构造，数学学报，1(1993)，45-52．
83．王国俊，完全分配格上的点式拓扑，陕西师范大学学报，1，2(1985)，1-17、1-15．
84．王国俊，Theory of topological molecular lattices, FSS, 47(1992), 351-376.
85．王国俊，L-fuzzy拓扑空间论，西安：陕西师范大学出版社，1988．
86．王国俊，广义序同态理论，东北数学，2(1985)，141-152．
87．王国俊，拓扑分子格理论，西安：陕西师范大学出版社，1988．
88．王国俊，广义拓扑分子格，中国科学，12(1983)，1063-1072．
89．王国俊，王灏，拟一致结构与拓扑，数学学报，2(1993)，207-216．
90．王国俊，徐罗山，内蕴拓扑与Hutton单位区间的细致化，中国科学(A辑)，7(1992)，705-712．
91. Guojun Wang, A new fuzzy compactness defined by fuzzy nets, JMAA, 94(1983) 1-23.
92. C. K. Wong, Fuzzy points and local properties of fuzzy topology, JMAA, 43(1974), 316-328.
93. Wu Congxin, Wu Jianrong, Fuzzy quasi-uniformities and fuzzy bitopological spaces, FSS, 46(1992), 133-137.
94．徐晓泉，完全分配格范畴中乘积与上积的结构，科学通报，9(1990)，643-646．
95．徐罗山，格与拓扑研究—范畴论方法与拓扑分解方法及应用，四川大学博士论文，1992．
96．宣立新．强序列紧性，可数强紧性与强列紧性．南京师范大学学报(自然科学版)，1989(2)：14-19．
97．杨忠强，分子格范畴的Cartesian闭性，四川大学博士论文，1990．
98．易耘，不分明拓扑空间的层次结构，数学学报，6(1991)，729-736．
99. M. S. Ying, A new approach for fuzzy topology (Ⅰ);(Ⅱ);(Ⅲ), FSS, 39 (1991) 303-321; 47(1992), 221-232; 55 (1993), 193-207.
100. M. S. Ying, On the method of neighborhood systems in fuzzy topology, FSS, 68(1994) 227-238.
101．张德学，刘应明，完备Heyting代数的L-fuzzy谱，四川大学学报(自)，4(1993)，553-555．
102. Dexue Zhang, Subcategories of Fuzzy Convergence Spaces, Journal of Fuzzy Mathematics, Vol.8, No.3, 2000, 769-787.
103．张杰，郑崇友，序列充分多的拓扑分子格的乘积与直积的若干性质，东北数学，3(1989)，338-342．
104．张杰，(α，γ)-Hausdorff分离公理，辽宁工学院学报，1990，Vol．9，No．1，53-56．
105．赵彬，诱导空间的权、特征与浓度，科学通报，35(1990)，569-571．
106．赵彬，分子格范畴中的极限及其应用，四川大学博士论文，1993．
107．赵东升，Fuzz上的拟一致连续序同态，数学进展，3(1987)，305-308．
108. Dongsheng Zhao, N-compactness in L-fuzzy topological spaces, JMAA,128(1987) 64-79.
109．郑崇友，Fuzzy拓扑空间的连通性．模糊数学，1982，3：59-66．
110. Chongyou Zheng. L-fuzzy unit interval and fuzzy connectedness. FSS, 1988, 27: 73-76.
111．郑崇友，樊磊，崔宏斌，Frame与连续格，首都师范大学出版社，2000．
2. G. Artico and R.Moresco, Fuzzy proximities and totally bounded fuzzy uniformities, JMAA, 99(1984), 320-337.
3. C. L. Chang, Fuzzy topological spaces, JMAA, 24(1968), 182-190.
4. Yixiang Chen, Papracompactness on fuzzy topological spaces, FSS, 53(1993), 335-342.
5. K.C.Chattopadyay, R.N.Hazra and S.K.Samanta, Gradation of openness: fuzzy topology, FSS 49(1992) 237-242.
6. Yixiang Chen, Paracompactness in L-fuzzy topological spaces, FSS 53(1993) 335-342.
7. H. B. Cui, C. Y. Zheng, Stratification structure on a kind of completely distributive lattices and their applications in the theory of topological molecular lattices, FSS, 106 (1999), 449-454.
8. R. Engelking, General Topology (Polish Scientific Publishers,Warszawa,1977).
9. T. E. Gantner et al., Compactness in fuzzy topological spaces, JMAA, 62(1978) 547-562.
10. M. K. El Gayyar, E.E. Kerre and A.A. Ramadan, Almost compactness and near compactness in smooth topological spaces, FSS 62(1994) 193-202 .
11．樊磊，崔宏斌，郑崇友，分子格的直积分解与广义序同态的构造，科学通报，21(1987)，1611-1614．
12．樊太和，分子格范畴分子格范畴中的积运算，科学通报，32(1987)，505-510．
13．樊太和，拓扑分子格范畴，四川大学博士论文，1990．
14．方锦暄，单值模糊映射，数学物理学报，3(1984)，311-317．
15. G. Gierz et al., A compendium of continuous lattices (Springer Verlag, Berlin, 1980).
16. R. N. Hazra, S.K.Samanta and K.C.Chattopadyay, Fuzzy topology redefined, FSS 45(1992) 79-80.
17．何明，L-fuzzy集上的双诱导映射，科学通报，6(1986)475．
18. H. Herrilich and G. E. Strecher, Category Theory, Helderman-Verlag, Berlin, 1979.
19. H. Herrilich, Categorical topology, General topology Appl. ,1971, 1: 1-15.
20. U. Hohle, Uppersemicontinuous fuzzy sets and applications, JMAA, 78 (1980) 659-673.
21. U. Hohle, Manuscripfa Math., 38(1982),289-323.
22. U. Hohle, A. P. Sostak, general theory of fuzzy topological spaces, FSS, 73 (1995) 131-149.
23. U. Hohle and Rodabaugh, eds., Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, Volume 3(1999), Kluwer Academic Publishers (Dordrecht).
24. U. Hohle, A. P. Sostak, Axiomatic Foundations of Fixed-Basis Fuzzy Topology, in ([23]), 123-272.
25. B. Hutton, Uniformities on fuzzy topological spaces, JMAA 58(1977), 559-571.
26. B. Hutton, Normality in fuzzy topological spaces, JMAA, 50(1975), 74-79.
27. B. Hutton, Product of fuzzy topological spaces, Topology and its Applications, 11(1980), 59-67.
29．金晨辉，郑崇友，分子格范畴中乘积的既约性，模糊系统与数学，1(1987)，45-50．
28. B. Hutton, I. Reilly, Separation axioms in fuzzy topological spaces, FSS, 3(1980), 93-104.
30. T. Kubiak, On Fuzzy Topology, Ph.D. Thesis, A. Mickiewicz, Poznan (1985)
31. S. R. T. Kudri, Compactness in fuzzy topological spaces, FSS 67(1994) 329-336.
32. S. G. Li, Inverse limits in category LTop (I), FSS, Vol. 108, No. 2, 1999, 235-241.
33．李生刚，关于格上拓扑学中某些问题的研究，四川联合大学博士论文，1998．
34．李永明，Locale的函数空间与逻辑语义，四川联合大学博士论文，1998．
35．李永明，王国俊，拓扑空间范畴，拓扑Fuzzy范畴与拓扑分子格范畴间的反射与余反射，数学学报，41(1998)，4，731-736．
36．刘应明，Fuzzy集论中邻属关系的分析，数学年刊，A辑，4(1984)，461-466．
37．刘应明，不分明完全正则性的点式刻画与嵌入定理，中国科学，A辑，26(1983)，138-147．
38. 刘应明，Intersection operation on union-preserving mappings in completely distributive lattices, JMAA, 84(1984), 249-255.
39. Yingming Liu, Compactness and Tychnoff Theorem in fuzzy topological spaces, Acta Mathematica Sinica 24(1981) 260-268.
40．刘应明，梁基华，格值诱导空间的完全正则性，四川大学学报(自)，1(1993)，23-28．
41．刘应明，梁基华，不分明拓扑学—层次结构和点式处理，数学进展，4(1994)，304-321．
42．刘应明，罗懋康，不分明紧化中的预序关系，科学通报，33(1988)，564-466．
43．刘应明，罗懋康，诱导空间与不分明Stone-Cech紧化，中国科学，A辑，4(1987)，359-368．
44. 刘应明，罗懋康, Advances in fuzzy systems—applications and theory Vol.9, Fuzzy topology(World Scientific Publishing Co. Pre. Ltd, Singapore, 1997).
45. Zhongfu Li, Compactness in fuzzy topological spaces, Chinese Kexue Tongbao 6(1983) 321-323.
46．梁基华，完全分配格上的一致结构，科学通报，34(1989)，1049-1051．
47. R. Lowen, Fuzzy uniform spaces, JMAA, 82(1981), 370-385.
48. R. Lowen, Fuzzy topological spaces and fuzzy compactness, JMAA, 56(1976) 621-633.
49. R. Lowen, A comparison of different compactness notions in fuzzy topological spaces, JMAA, 64(1978) 446-454.
50. Meng Guang-Wu. On countably strong fuzzy compact sets in L-fuzzy topological spaces. FSS, 1995(72): 119-123.
51．罗懋康，Fuzzy拓扑空间中的仿紧性与紧性，数学学报，4(1987)，548-552．
52．罗懋康，Fuzzy拓扑空间中的弱仿紧性与紧性，数学年刊，5(1986)，590-596．
53. Maokang Luo, Paracompactness in fuzzy topological spaces, JMAA 130(1988) 55-77.
54. S. Mac Lan, Categories for the Working Mathematician, Springer Verlag, New York, 1971.
55. H. W. Martin, Weakly induced fuzzy topological spaces, JMAA,78(1980) 634-639.
56. Nagata, Modern general topology, Elsevier Science Publishers B.V., 1985.
57．彭育威，格值半连续映射和L-fuzzy Hausdoff N-紧空间，数学年刊，11A(1990)753-760．
58. Pu Paoming and Liu Yingming, Fuzzy topology Ⅰ, Neighborhood structure of a fuzzy point and Moore-Smith convergence, JMAA 76(1980), 571-579.
59. Pu Paoming and Liu Yingming, Fuzzy topology Ⅱ, Product and quotient spaces, JMAA 77(1980), 571-579.
60. A.A.Ramadan, Smooth topological spaces, FSS 48(1992)371-375.
61. S.E.Rodabaugh, A theory of fuzzy uniformities with applications to the fuzzy lines, JMAA, 129(1988), 37-70.
62. S.E.Rodabaugh, E.P.Klement, U.Hohle eds., Applications of Category Theory to Fuzzy Subsets (Kluwer Academic Publishers, 1992).
63. S.E.Rodabaugh, Categorical Foundations of Variable-Basis Fuzzy Topology, in ([23]), 273-388.
64．史福贵，完全分配格上的点式拟一致结构，数学进展，26(1997)，22-28．
65．史福贵，完全分配格上的点式拟一致结构与p.q.度量，数学学报，39(1996)，701-706．
66．史福贵，Fuzzy格上的点式一致结构与度量，科学通报，42(1997)，718-720．
67．史福贵，L_β-集合套和L_α-集合套及其应用，模糊系统与数学，4(1995)63-72．
68．史福贵，L-fuzzy集和素元集合套，数学研究与评论，16(1996)398-402．
69．史福贵等，LF拓扑空间中的F可数紧性．哈尔滨科技大学学报，1992(3)：103-106．
70．史福贵，格上点式一致结构与点式度量理论及其应用，首都师范大学博士论文，2001．
71．史福贵，王国民，L-fuzzy拓扑空间中正则性和正规性的一种新定义，烟台师范学院学报(自然科学版)，3(1995)，20-24．
72．史福贵，郑崇友，L-fuzzy拓扑空间的一种新fuzzy仿紧性，模糊系统与数学，3(1995)40-48．
73．史福贵，模糊子集仿紧性的—个注记，模糊系统与数学，3(1998)3(1998)6-9．
74. Fugui Shi, Chongyou Zheng. Connectivity in L-fuzzy topological molecular lattices. FSS, 1989, 29: 363-370.
75．史福贵，积Fuzzy拓扑分子格的连通性．北京师范学院学报(自然科学版)，1991，2：12-15．
76. A.P.Sostak, On a fuzzy topological structure, Suppl. Rend. Circ. Mat Palermo Ser. Ⅱ 11(1985) 98-103.
77. A.P.Sostak, Two decades of fuzzy topology: basic ideas, notations and results, Russian Math. Surveys, 44 (1989) 125-186.
78. R.Srivastava, On separation axioms in a newly defined fuzzy topology, FSS 62(1994)341-346.
79．孙叔豪，完全分配格上的拟一致结构，陕西师范大学学报，3(1985)，9-15．
80．王戈平，不分明集的一个分解定理及其在不分明拓扑学中的应用，科学通报，26(1981)，259-262．
81．王戈平，完全分配格上的弱辅助序与广义序同态，数学季刊，4(1988)，76-83．
82．王戈平，田晓明，一类完备格的直积分解与Fuzzy格的构造，数学学报，1(1993)，45-52．
83．王国俊，完全分配格上的点式拓扑，陕西师范大学学报，1，2(1985)，1-17、1-15．
84．王国俊，Theory of topological molecular lattices, FSS, 47(1992), 351-376.
85．王国俊，L-fuzzy拓扑空间论，西安：陕西师范大学出版社，1988．
86．王国俊，广义序同态理论，东北数学，2(1985)，141-152．
87．王国俊，拓扑分子格理论，西安：陕西师范大学出版社，1988．
88．王国俊，广义拓扑分子格，中国科学，12(1983)，1063-1072．
89．王国俊，王灏，拟一致结构与拓扑，数学学报，2(1993)，207-216．
90．王国俊，徐罗山，内蕴拓扑与Hutton单位区间的细致化，中国科学(A辑)，7(1992)，705-712．
91. Guojun Wang, A new fuzzy compactness defined by fuzzy nets, JMAA, 94(1983) 1-23.
92. C. K. Wong, Fuzzy points and local properties of fuzzy topology, JMAA, 43(1974), 316-328.
93. Wu Congxin, Wu Jianrong, Fuzzy quasi-uniformities and fuzzy bitopological spaces, FSS, 46(1992), 133-137.
94．徐晓泉，完全分配格范畴中乘积与上积的结构，科学通报，9(1990)，643-646．
95．徐罗山，格与拓扑研究—范畴论方法与拓扑分解方法及应用，四川大学博士论文，1992．
96．宣立新．强序列紧性，可数强紧性与强列紧性．南京师范大学学报(自然科学版)，1989(2)：14-19．
97．杨忠强，分子格范畴的Cartesian闭性，四川大学博士论文，1990．
98．易耘，不分明拓扑空间的层次结构，数学学报，6(1991)，729-736．
99. M. S. Ying, A new approach for fuzzy topology (Ⅰ);(Ⅱ);(Ⅲ), FSS, 39 (1991) 303-321; 47(1992), 221-232; 55 (1993), 193-207.
100. M. S. Ying, On the method of neighborhood systems in fuzzy topology, FSS, 68(1994) 227-238.
101．张德学，刘应明，完备Heyting代数的L-fuzzy谱，四川大学学报(自)，4(1993)，553-555．
102. Dexue Zhang, Subcategories of Fuzzy Convergence Spaces, Journal of Fuzzy Mathematics, Vol.8, No.3, 2000, 769-787.
103．张杰，郑崇友，序列充分多的拓扑分子格的乘积与直积的若干性质，东北数学，3(1989)，338-342．
104．张杰，(α，γ)-Hausdorff分离公理，辽宁工学院学报，1990，Vol．9，No．1，53-56．
105．赵彬，诱导空间的权、特征与浓度，科学通报，35(1990)，569-571．
106．赵彬，分子格范畴中的极限及其应用，四川大学博士论文，1993．
107．赵东升，Fuzz上的拟一致连续序同态，数学进展，3(1987)，305-308．
108. Dongsheng Zhao, N-compactness in L-fuzzy topological spaces, JMAA,128(1987) 64-79.
109．郑崇友，Fuzzy拓扑空间的连通性．模糊数学，1982，3：59-66．
110. Chongyou Zheng. L-fuzzy unit interval and fuzzy connectedness. FSS, 1988, 27: 73-76.
111．郑崇友，樊磊，崔宏斌，Frame与连续格，首都师范大学出版社，2000．