| 1. | In chapter 7 we observe the smallest rectangular cover and triangular cover of any prescribed shape for any triangles of diameter one
在第7章中我们对于直径为1的三角形形成的集族，讨论了其最小的矩形覆盖和最小的三角形覆盖 |
| 2. | Base is one of the most basic and important concepts in topology , but the open property of base restricts the effects of family to be further developed
基是拓扑学中最基本、最重要的概念之一，但是基的“开”性质限制了集族作用的进一步发挥。 |
| 3. | On - weakly hereditarily closure - preserved family and cc - spaces in this paper , we discussed - weakly hereditarily closure - preserved family and cc spaces respectively
本文主要由两个部分组成，分别讨论了-弱遗传闭包保持集族和cc _空间。 |
| 4. | Fuzzy matroids is based on matroids , so they have much comparability . we investigate some conditions that fuzzy subsets make up of fuzzy bases to be similar to the base axiom
摘要模糊拟阵是在一般拟阵的基础上建立起来的，有很多的相似性。文章类似于一般拟阵的基公理讨论模糊集族构成模糊基集所要满足的一些条件。 |
| 5. | In chapter 4 we obtain the helly number for hyperplane transversal to translates of a convex cube in r ~ ( d ) . where we prove that the helly number for such families is 5 when d = 2 , and is greater than or equal to d + 3 when d 3
在第4章中我们探讨了o中超平面横截单位立方体平移形成的集族的heily数，证得碑中此heily数为5 ，在呼中此heily数z民并推广至呼，在胸中此heily数d 3 |
| 6. | In third chapter , we study the relation of the expansion of a space ' s scattered partition and the open refinement of open sets in it . we give the sufficient conditions for a space having the property that its every scattered partition has a open expansion of a given property
第三章详细证明一个空间中的开集族的开加细与空间的散射分解的开膨胀之间的关系，给出了空间的散射分解具有某种性质的开膨胀的充要条件。 |
| 7. | In other words , d . burke and r . engelking and d . lutzer proved that a regular space is metrizable space if and only if it has a - hereditarily closure - preserving base in 1975 , and introduced weakly hereditarily closure - preserving families , which proved that a regular k - space has - weakly hereditarily x closure - preserving bases is metrizable space , too
Burke ， r engelking和d lutzer证明了正则空间是可度量化空间当且仅当它具有遗传闭包保持基，并引入了弱遗传闭包保持集族( weaklyhereditarilyclosure - preservingfamilies ) ，同时证明了具有弱遗传闭包保持基的正则的k空间是可度量化空间。 |
| 8. | In chapter 3 we obtain a new upper bound for " k - l problem " concerning line transversal to unit disks in the plane , where we prove that for a finite disjoint family f of unit disks in the plane , there is a line intersecting all but at most 5 members of f suppose that even " three of the members admit a line transversal . thus we improve the latest upper bound 12
在第3章中我们讨论了直线横截的有关问题，给出了单位圆“ k l ”问题的一个新上界，即证明了对于平面上有限个两两不交的单位圆形成的集族，若其中任三个单位圆可被一直线横截，则存在一条直线至多不与此集族中的5个单位圆相交，从而改进了原有上界12 |
| 9. | On the other hand , locally finite families introduced by p . alexandroff in 1924 have played a fundamental role in the research of metrization problem and paracompact property . research of space with - locally finite networks betters understanding of metric property ' s essence . hereditarily closure - preserving families introduced by n . lasnev in 1966 ties up locally finite families
Alexandroff引入的局部有限集族( locallyfinitefamilies )已在度量化问题及仿紧性的研究中起着不可替代的作用，对于具有局部有限网络空间的探索更加深了人们对度量性的本质了解， 1966年n |
| 10. | And about a given property p which satisfies some conditions , we give a condition for the open sets in the tychnoff product of two spaces which have a open refinement having the property p . the last chapter obtains a result about the limit of the inverse system of a normal mesocompact space and a hereditar ily normal and hereditarily mesocompact space
而且给出了，对满足一定条件的某种性质p ，两个空间的乘积中的每个开集族有p性质的开加细的一个充分条件。第四章详细证明了关于正规中紧空间及遗传正规且遗传中紧空间的逆极限的一个结果。 |
