【作者】 鲁莉；

【导师】 陈东立；

【作者基本信息】 西安建筑科技大学 ， 应用数学， 2012， 硕士

【摘要】 拓扑学对于分析学的发展起了极大的推动作用,它的概念和方法在多个学科中都有着直接广泛的应用。而紧性是拓扑学中非常重要的概念之一,研究有关几类紧性的各种性质是非常必要的。非标准分析理论在近半个世纪以来得到飞速发展,并将之运用到其他经典数学及物理等理论中,尤其是关于无限和微观的方面。本文便是将此理论应用于拓扑学中。本文首先概述非标准分析的相关理论,即非标准全域和标准全域的构造过程,非标准分析中的形式语言,以及非标准模型的构造及其性质.其次通过单子重新定义了拓扑,并且给出拓扑中相关概念的定义,以及有关定理的证明.最后分别给出紧性、相对紧、局部紧的非标准刻画,并证明这种刻画与标准分析中的定义是相符的.在此刻画的基础上应用转换原理、内性定理等讨论有关定理,最终得到有关三种紧性空间乘积的性质.非标准分析的方法使得原有标准分析中的定义更能突出其本质意义,使得原来的证明方法更为简化。这种貌似离散的非标准定义与一般拓扑的定义是致的。它为拓扑学的研究提供一种新的方法,也有助于非标准分析理论在拓扑学中的发展,有一定的参考价值和实际意义。更多还原

【Abstract】 Topology has promoted the progress of analytics greatly. The concept and tech-nology of the topology have been widely applied to many subjects. It is necessary tostudy the propery of compact, which is the key concept in topology.Nonstandard analysis theory has developed rapidly recently. It has been appliedto classical mathematics and phisics theory, especially in unlimited and micro aspect.Nonstandard analysis theory is introduced to Topology in this paper.Firstly, the related theory of nonstandard analysis is described, including thestructure in nonstandard global domain and standard global domain, formal language,and the configuration of nonstandard model with its nature.Secondly, the topology is redefined by monad. The related defination and certifi-cation is proposed.Finally, the nonstandard characterization is provided in compact, locally compact,relatively compact situation. It proves that the nonstandard characterization agreeswell with the definition of nonstandard analysis. The natrue of three compact spacepruduct is obtained by intruducing the conversion principle and internal theorem.Nonstandard analysis can not only makes the definition original standard analysisclear, but also simplifies the certification. It has the same consistency with the normaltopology. Also, it can improve the progress the nonstandard analysis theory in topology.It provides a novel method in the study of Tomology, and has some reference value andpractical significance.更多还原