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On the research of topological dimension of metric spaces

论度量空间的拓扑维数研究

基本信息

批准号：
09640108
负责人：
HATTORI Yasunao
金额：
$ 1.92万
依托单位：
Shimane University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640108/
关键词：
metric spaces topological dimension topological groups finitistic spaces K-approximation ordered spaces selection complex projective plane 連続写係収束

项目摘要

In the present research project, we investigated some dimensiontheoretic properties of metrizable spaces and its applications. Hattori considered with Yamada on two transfinite dimensions - the large transfinite dimension Ind and the order dimension O-dim - of metric spaces. They proved that for a metric space X, X has hid if and only if X has O-dim. Furthermore, the inequality Ind X < O-dim X holds. This answers the question of F.G.Arenas affirmatively. Hattori considered a convergence structure of metric groups. He proved that there is a metric group topology on the real line such that it is weaker than the usual one and the sequence {2n : n = 1, 2, ...} converges to 0. He also proved that there is no such group topology preserving the linearity. This answers the Fric's question. Yamada investigated the group structures of free groups of metrizable spaces and proved that the structures of a free group are determined by its neighborhood systems. Shakhmatov researched the topological p … More roperties of compactly generated groups and free abelian groups from dimension theoretic points of view. Hattori investigated an extension property on ordered spaces and proved that every perfect ordered space with a special condition has the Dugundji extension property. He furthered this research with G.Gruenhage and H.Ohta.Shey determined closed subspace of ordered spaces which has the Dugundji extension property. This solves the problems of Heath-Lutzer and van Douwen. Miwa considered the perfect normality and certain covering properties of ordered spaces. Nogura and Shakhmatov gave some theorems about the existence of continuous selections on the hyperspaces of metric spaces under some dimensionality conditions. Kikkawa and Maeda considered the circles in the complex projective plane. Hattori investigated metric finitistic spaces and gave a universal space theorem and a Pasynkov's type of factorization theorem for finitistic spaces. Aikawa researched the harmanic functions and measure from dimension theoretic points of view. Less

在本研究项目中，我们研究了可度量空间的一些维数理论性质及其应用。 Hattori 与 Yamada 一起考虑了度量空间的两个超限维度——大超限维度 Ind 和阶维 O-dim。他们证明，对于度量空间 X，当且仅当 X 具有 O-dim 时，X 才隐藏。此外，不等式 Ind X < O-dim X 成立。这肯定地回答了F.G.阿里纳斯的问题。 Hattori 考虑了度量组的收敛结构。他证明了实线上存在一个度量群拓扑，其强度比通常的群拓扑弱，并且序列 {2n : n = 1, 2, ...} 收敛于 0。他还证明了不存在这样的保持线性的群拓扑。这回答了弗里克的问题。山田研究了可度量空间的自由群的群结构，并证明了自由群的结构是由其邻域系统决定的。沙赫玛托夫从维数论的角度研究了紧生成群和自由交换群的拓扑性质。 Hattori 研究了有序空间的扩展性质，并证明了每个具有特殊条件的完美有序空间都具有 Dugundji 扩展性质。他与 G.Gruenhage 和 H.Ohta.Shey 进一步开展了这项研究，确定了具有 Dugundji 扩展性质的有序空间的闭合子空间。这解决了 Heath-Lutzer 和 van Douwen 的问题。 Miwa 考虑了有序空间的完美常态和某些覆盖属性。野仓和沙赫马托夫给出了在一定维数条件下度量空间超空间上连续选择存在性的定理。吉川和前田考虑了复射影平面中的圆。 Hattori 研究了度量有限空间并给出了有限空间的通用空间定理和 Pasynkov 类型的因式分解定理。相川从量纲理论的角度研究了调和函数及其测度。较少的