Frink-Tukey度量化引理

一致空间的度量化问题是一致空间的基本问题之一,其主要工具是Tukey度量化引理.证明在拓扑空间的度量化问题中起主要工具之一的Frink引理与Tukey度量化引理如出一辙,可将它们称之为Frink-Tukey度量化引理.     

度量空间的k－映射像

Ｅ．Ｈａｌｆａｒ（［６］）引入了ｋ－映射概念，本文给出了度量空间ｋ－映像的一个内在刻划，并由此得到一些度量化定理。     

关于伪欧氏空间

本文提出了一些新概念，ｐ—伪转置阵，ｐ—伪正交阵，ｐ—伪对称阵，从而能以这些矩阵为工具研究伪欧氏空间的性质以及空间中两个特殊线性变换，伪正交变换和伪对称变换．本文得到的主要结果是定理２，定理３，定理４及定理６．本文还指出了北大编《高等代数》第二版中的伪正交变换习题的一个错误．     

度量空间的k—映射像

Ｅ．Ｈａｌｆａｒ（［６］引入了ｋ－映射概念，本文给出了度量空间ｋ－映像的一个内在刻划，并由此得到一些度量化定理。     

关于CWC映射与度量化定理

本文讨论CWC映射在度量化定理中的应用, 通过Nagata条件和弱$\gamma$空间条件等给出拓扑空间一些新的度量化定理, 推广了R.E. Hodel和J.Nagata等获得的度量化定理.     

关于k－半分层空间的度量化

本文将给出ｋ－半分层空间的若干度量化定理以及分层空间和σ－空间的度量化定理．     

广义H-空间的理论及其应用*

本文给出了广义H-空间的完备性特征性质和紧性特征性质,同时也研究了这一空间的度量化定理.作为这些理论的应用.我们得到了Menger概率度量空间的完备性特征和紧性特征.给出了该空间的度量化函数的具体形式.     

关于仿紧性与拓扑空间的可度量性

引入σ-相对垫状开加细等概念并用它们刻划了仿紧性.引入线性遗传闭包保持集族的概念;证明了下列度量化定理:正则T_1空间是可度量的当且仅当它具有可数伪特征且有σ-线性遗传闭包保持基.     

完全分配格上的点式p.q度量及诱导映射族

本文在完全分配格上引入一种点式p,q度量,并以由它诱导出的一类保交映射族为工具,讨论p,q 度量分子格的Moor-Smith式收敛,闭包,连续等拓补性质,给出拟一致分子格p,q度量化定理一种构造性证明。     

完全分配格上的点式拟一致结构与p．q．度量

在完全分配格上建立了点式拟一致结构理论．讨论了诱导拓扑分子格中闭包，局部基，连续等性质．证明了每个拓扑分子格皆可点式拟一致化．另外借助纯距离函数与真正的远域映射族给出了［８］中ｐ．ｑ．度量的等价定义与刻画，得到了点式拟一致分子格的ｐ．ｑ．度量化定理．     

On Antosik's Lemma and the Antosik-Mikusinski Basic Matrix Theorem

That Antosik's Lemma is not a special case of the Antosik-Mikusinski Basic Matrix Theorem will be shown and, an equivalent form of the Antosik-Mikusinski Basic Matrix Theorem will also be presented in this paper.

     

On the structure of linear semigroups

Green's Lemma [1, Lemma 2.2] is one of the most important theorems in the theory of semigroups. The main purpose of this note is to establish a generalized Green's Lemma and a generalized Clifford and Miller's Theorem [1, p. 59] in linear semigroups. A generalized Green's Lemma describes the behavior of certain mappings between two distinct D-classes.     

Bishop's Lemma

Bishop's Lemma is a centrepiece in the development of constructive analysis. We show that

• 1. its proof requires some form of the axiom of choice; and that
• 2. the completeness requirement in Bishop's Lemma can be weakened and that there is a vast class of non‐complete spaces that Bishop's Lemma applies to.

     

A note on kernels and Sperner’s Lemma

The kernel-solvability of perfect graphs was first proved by Boros and Gurvich, and later Aharoni and Holzman gave a shorter proof. Both proofs were based on Scarf’s Lemma. In this note we show that a very simple proof can be given using a polyhedral version of Sperner’s Lemma. In addition, we extend the Boros–Gurvich theorem to h-perfect graphs and to a more general setting.     

The origin of “Zorn's Lemma”

The paper investigates the claim that “Zorn's Lemma” is not named after its first discoverer, by carefully tracing the origins of several related maximal principles and of the name “Zorn's Lemma.” Previously unpublished information supplied by Zorn is included.     

A note on extremal disconnectedness in bitopological spaces

Pairwise extremally disconnected bitopological spaces exhibit properties similar to those of pairwise normal bitopological spaces. Due to this fact, we get results which resemble Uryshon’s Lemma and Tietze’s Extension Theorem for the pairwise extremally disconnected bitopological spaces.     

Upper and lower bounds of Borel–Cantelli Lemma in a general measure space

It is known that the Borel–Cantelli Lemma plays an important role in probability theory. Many attempts were made to generalize its second part. In this article, we investigate the upper and lower bounds of Borel–Cantelli Lemma for the nonnegative functions in a general measure space. Our results extend the corresponding results obtained in a probability space. Some examples including dependent random variables are illustrated to our results.     

A generalization of the Mitchell Lemma: The Ulmer Theorem and the Gabriel–Popescu Theorem revisited

We prove a generalization of the Mitchell Lemma, and we show that it is a key lemma that can be used in order to deduce in a unified easier way several important results. Thus, the Ulmer Theorem, the generalized Gabriel–Popescu Theorem and the generalized Takeuchi Lemma are all consequences of the generalized Mitchell Lemma.     

Some Applications of ABHYANKAR's Lemma

ABHYANKAR 's Lemma provides a simple method of producing unramified extensions of number fields. By application of the lemma we simultaneously simplify and strengthen some recent results of ISHIDA concerning unramified abelian extensions and class numbers.     

Asymmetry in the Converse of Schur's Lemma

We show that the converse of Schur's Lemma can hold in the category of right modules, but not the category of left modules, over an appropriate ring. We exhibit classes of rings over which this left-right asymmetry does not occur, and provide new constructions of rings over whose module categories the converse of Schur's Lemma holds. We propose various open problems and avenues for further research concomitant to our work.