Research of topological properties of metric spaces and dimensions

本研究计画主要探讨度量空间的拓扑性质，以及与度量空间相关的维数。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）度量空间及其维数（C）度量空间中的紧化及其维数（D）黎曼流形中的切球丛、超曲面及其维数（A）、（B）、（C）、（D）的研究主要由Y.Tanaka、T.后藤、T.Kimura和K.Morishita等人完成。关于（A），田中义雄（Y.Tanaka）考虑了下列经典问题（1）和（2），问题（3）（1）利用度量空间的某些良商空间刻画某些拓扑空间（2）对于k-空间X，Y，什么是乘积空间X x Y是k-空间的充分必要条件？（3）对于具有某种k-网或关于某种覆盖具有弱拓扑的空间X，研究空间X的良好拓扑结构对于上述问题，Y.Tanaka在国际著名期刊Topology and its Applications，Houston J.Math.，关于（B）-（D），研究者们在Topology and its Appl.，基金数学，或其他国际期刊等详细信息，包括相关论文和科学报告等，在我们的研究报告中给出（在科学研究补助金（14540066）下），2005年。《三月》（第1 -340页）。

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).