Research of topological properties of metric spaces and dimensions

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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).