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1、拓扑空间之间的映射对主要拓扑性质的保持性拓扑空间之间的映射对主要拓扑性质的保持性摘摘 要要拓扑学是近代发展起来的一个数学分支，于 19 世纪中期由科学家引入，当时主要研究的是出于数学分析的需要而产生的一些几何问题发展至今，拓扑学主要研究拓扑空间在拓扑变换下的不变性质和不变量 。近年来，拓扑学思想愈来愈渗入到物理学、化学和生物学领域中，愈来愈显示出它的重要地位。在一般拓扑学中，拓扑空间的连通性、可数性、分离性、可度量化、紧性等是整个学科的主要内容，拓扑空间在一些重要映射作用之后是否保持原有性质在拓扑学研究中具有很重要的理论意义和价值。本论文从最基本的拓扑性质出发，讨论各种映射（连续映射，开映射，
2、闭映射，商映射，同胚映射或几种映射复合）对主要拓扑性质是否保持，保持的给出了证明，不保持的给出了反例。首先，论文中所涉及的拓扑性质都是拓扑不变性质；其次，在连续映射下保持的性质有连通性、道路连通性、可分、Lindeloff、紧致、可数紧致、序列紧致，从而它们也都是可商性质，不保持的性质有局部连通性、第一可数性、第二可数性、分离性、可度量化、仿紧致；然后，除了前面所说的可商性质外，还有局部连通性是可商性质，而第一可数性、第二可数性、有关分离性、可度量化、仿紧致都不是可商性质；最后，在开映射和闭映射下，这些拓扑性质都未必能保持，而对于那些在连续映射下也不保持的性质，通过进一步加强映射，发现在连续开
3、映射下保持的有局部连通性、正规和正则，在连续闭映射下保持的有局部连通性、和正1T3.5T4T规。关关键键词词 连通性 可数性 分离性 紧性 映射To Preserve the Topological Properties by Mappings between Two Topographical SpacesABSTRACTTopology, as a branch of mathematics, was introduced by mathematicians in the middle of the nineteenth century and developed in recent ye
4、ars. Its main purpose is to study some geometric problems originated from mathematical analysis. Invariants and invariant properties of a topological space under topological transformations are the main objects of study. In recent years, Topology becomes more and more important, as its ideas gradual
5、ly infiltrated into the field of physics, chemistry and biology. In General Topology, connectivity, countability, Axioms of separation, metrizability, and compactness of a topological space are the main contents of the subject. Whether some important topological properties are preserved under some m
6、ain mappings has theoretical significance and value in the study of Topology. In this paper, based on the basic topological properties, we discuss whether the main topological properties are preserved under various mappings, such as continuous mapping, open mapping, closed mapping, quotient mappings
7、, and homeomorphism mapping. We give the proofs for the affirmative ones and give counterexamples for the negative ones. First of all, all the topological properties involved in this article are topological invariant properties. Secondly, properties preserved under the continuous mappings are connec
8、tivity, path-connectivity, separability, Lindeloff, compactness, countable compact, and sequentially compact. Thus they are preserved under quotient mappings. Properties not preserved under the continuous mappings are the local connectivity, the first countability, the second countability, Axioms of
9、 separation, metrizability, and paracompactness. Then, besides the properties previously mentioned, the local connectivity is also preserved under quotient mappings. While the first countability, the second countability, Axioms of separation, measurability, and paracompactness are not preserved unde
10、r quotient mappings. Finally, under open mappings and closed mapping, these topological properties may not be able to keep. For the properties which are neither preserved under open mappings and closed mapping nor preserved under continuous mappings, we can further consider whether they are preserve
11、d under the strengthened mappings. We find that under continuous open mappings, local connectivity, regularity and normality are preserved; under continuous closed mapping ,the local connectivity, , , and regularity are preserved.1T3.5T4TKEY WORDS connectedness countability Axioms of separation comp
12、actness mapping目目 录录第一章 拓扑基本概念拓扑基本概念.11.1 拓扑，邻域，开集，闭集，点列的极限，基，邻域基.11.2 连续映射，同胚映射，开映射，闭映射.21.3 商拓扑，商映射.3第二章 连通性连通性.42.1 有关连通性.42.2 局部连通空间.52.3 道路连通空间.7第三章 可数性可数性.93.1 第一与第二可数性.93.2 可分空间.103.3 Lindeloff,x yRR x yQx yQ:一一北京邮电大学本科毕业设计（论文）18所得到的商集合为（如 56，）:23:。 / 0 ,2Rx xR自然映射，:/p RR: 0 ,2 ,/xQ xp xxxR Q. 1/,0 ,2URpU:TT它是由两个点组成
