On the research of topological dimension of metric spaces

在本研究项目中，我们研究了可度量化空间的一些维度理论性质及其应用。Hattori和Yamada在度量空间的两个超维--大的超维Ind和序维O-dim上进行了讨论。他们证明了对于度量空间X，X有隐当且仅当X有O维。此外，不等式Ind X&lt；O-dim X成立。这肯定地回答了F·G·阿里纳斯的问题。Hattori考虑了度量群的收敛结构。他证明了实直线上存在一个度量群拓扑，使得它比通常的度量群拓扑弱，且序列{2n：n=1，2，…}收敛到0。他还证明了不存在这种保持线性的群拓扑。这回答了金砖四国的问题。Yamada研究了可度量空间的自由群的群结构，证明了自由群的结构是由其邻域系统决定的。沙赫马托夫研究了拓扑p-…从量纲理论的观点看紧生成群和自由阿贝尔群的更多性质。Hattori研究了序空间上的一个扩张性质，证明了每一个具有特殊条件的完美序空间都具有Dugundji扩张性质。他与G.Gruenage和H.Ohta.Shey进一步研究了序空间中具有Dugundji扩张性质的闭子空间。这解决了Heath-Lutzer和Van Douwen的问题。Miwa考虑了序空间的完全正规性和某些覆盖性质。Nogura和Shakhmatov给出了度量空间的超空间在一定的维度条件下存在连续选择的一些定理。Kikkawa和Maeda考虑了复射影平面上的圆。Hattori研究了度量有限空间，给出了有限空间的普适空间定理和Pasynkov型因式分解定理。爱川从量纲理论的角度对哈曼函数和度量进行了研究。较少

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