k-映射的指数定律

Exponential Law of k-Mapping

摘要: 本文介绍了指数定律和k-空间即紧生成空间相关的研究历史，并给出了k-映射的指数定律的一个证明。本文用赋予终拓扑的方法定义了拓扑空间的k化，接下来依次引入k-空间、k-映射、k-映射空间的定义。本文利用一系列定理说明k化了的k-映射空间中可以存在指数定律，证明了指数定律还是某种k-空间之间的同胚。在k-空间的基础上，本文介绍了弱Hausdorff的性质以及它对k-空间性质的影响，并利用对角集对弱Hausdorff的紧生成空间进行判定。最后，本文以数学分析中的含参积分为例说明弱Hausdorff的紧生成空间以及对角集在 中的应用。为此本文还介绍了滤子基，展示了对角集在一致连续定义中的作用，并证明了如何通过滤子基将极限的换序推广到一般情形。本文的主要结果是给出k-映射的指数定律的一个证明，并举出了一个k-空间的具体例子，展示了对角集在k-空间判定和含参积分中的应用。

Abstract: This paper introduces the research history of exponential law and k-space, which is also called compactly generated space, and gives a proof of exponential law of k-mapping. In this paper, the k-ficaton of topological space is defined by giving the final topology, and then the definitions of k-space, k-mapping and k-mapping space are introduced in turn. In this paper, a series of theorems are used to illustrate that there can be an exponential law in k-mapping spaces with k-ficaton, and it is proved that the exponential law is also a homeomorphism between some k-spaces. On the basis of k-space, this paper introduces the properties of weakly Hausdorff and its influence on the proper-ties of k-space, and uses diagonal set to determine whether a compactly generated space is weakly Hausdorff. Finally, this paper takes the parametric integral in mathematical analysis as an example to illustrate the application of compactly generated weakly Hausdorff spaces and diagonal set in . Therefore, this paper also introduces the filter base, explaining the role of diagonal set in the definition of uniform continuity, and proves how to extend the order change of limit to the general case through filter base. The main results of this paper are giving a proof of the exponential law of k-mapping, and giving a specific example of k-spaces, showing the application of diagonal set in k-space determination and parametric integral.