Research of topological properties of metric spaces and dimensions

本研究项目是关于度量空间的拓扑性质以及与度量空间相关的维度的研究。主要研究内容如下：(A)度量空间的商空间及其乘积空间的拓扑性质或结构(B)度量空间与维度(C)度量空间中的紧化及其维度(D)黎曼流形中的切球丛、超曲面及其维度(A)；(B)；(C)；(D)主要由田中、后藤、木村、森下；和M.Sekizawa，Y.Tanaka分别考虑了下列经典问题(1)和(2)，以及问题(3)(1)利用度量空间的某些良商空间来刻画某些拓扑空间(2)对于k-空间X，Y，乘积空间X x Y是k-空间的充要条件是什么？(3)对于具有某一k-网的空间X，或关于某一覆盖具有弱拓扑的空间X，研究空间X的良拓扑结构。田中在国际著名期刊、《拓扑学及其应用》、《休斯顿数学》、《拓扑学学报》等(联合)论文中得到了一些好的答案或结果。(B)-(D)，研究人员在《拓扑学及其应用》、《基金数学》等国际期刊上的(联合)论文中也获得了一些好的结果。这些论文的详细内容，包括相关论文和科学报告等，载于我们2005年的研究报告(本科学研究助学金(14540066)项下)。3月(pp.1-340)。

This research project is the investigations of topological properties around metric spaces, and dimensions related to metric spaces. The main objects of the research are the following :(A)Topological properties or structures of quotient spaces of metric spaces, and their product spaces(B)Metric spaces and dimensions(C)Compactifications in metric spaces, and their dimensions(D)Tangent sphere bundles, hyper surfaces in Riemannian manifold, and their dimensionsThe research on (A);(B);(C); and (D) has been done by Y.Tanaka mainly ; T.Goto ; T.Kimura and K.Morishita ; and M.Sekizawa, respectivelyConcerning (A), Y.Tanaka considered the following classic questions (1) & (2), and a question (3)(1)Characterize some topological spaces by means of certain nice quotient spaces of metric spaces(2)For k-spaces X, Y, what is a necessary and sufficient conditions for the product space X x Y to be a k-space?(3)For a space X having a certain k-network, or having a weak topology with respect to certain covering, investigate nice topological structures of the space XFor the above questions, Y.Tanaka obtained some nice answers or results in his (joint) papers in famous international journals, Topology and its Applications, Houston J.Math., or Topology Proceedings, and so on.Concerning (B)-(D), some nice results also obtained by the investigators in their (joint) papers in Topology and its Appl., Fund Math., or other international journals, etcThe details for these, containing related papers and science reports, etc., are given in our Research-Report (under this grant-in-aid for scientific research (14540066)), 2005. March (pp.1-340).