On the research of topological dimension of metric spaces

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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