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Topology of hyperspaces, mapping spaces and universal spaces

超空间、映射空间和通用空间的拓扑

基本信息

批准号：
17540061
负责人：
SAKAI Katsuro
金额：
$ 2.3万
依托单位：
University of Tsukuba
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

项目摘要

To achieve three objects mentioned at the beginning of this project, the head investigator have been making researches in cooperation with investigators and invited W. Kubis from Poland and T. Banakh from Ukraine to do joint works and to exchange information. Finally, we obtained many good results. Concerning the first object to specify under what conditions and to what spaces each of various hyperspaces is homeomorphic (even in non-separable case), by many joint works conducted by the head, we had such results as: specifying hyperspaces on Banach spaces with the Wijsman topology; finding conditions of metric spaces whose hyperspaces of closed sets are ANR's; proving that the hyperspace of closed convex sets in a normed space is an ANR with respect to Hausdorff uniformity and Attouch-Wets topology and for a finite-dimensional normed space it is homeomorphic to the product of the base space and the Hilbert cube; specifying hyperspaces consisting of compacta of various types; proving tha … More t the hyperspaces of bounded closed sets in the space of irrationals and the Noebeling spaces. Concerning the second object to find mapping spaces being infinite-dimensional manifold and to clarify their topological structure, Yagasaki classified the connected component of the inclusion in the space of embeddings of a subpolyhedron into a surface and generalized Berlange's result on the group of measure-preserving homeo-morphisms to non-compact case. The head collaborated Uehara on clarifying topological structure of the space of lower semi-continuous functions. Moreover, in the joint work with Banakh, Mine and Yagasaki, he proved that the homeomorphism group of non-compact surfaces with the Whitney topology can be embedded in the product of the Hilbert space and the direct limit of Euclidean spaces as open sets. Concerning the object to enrich studies on non-separable infinite-dimensional universal spaces and to complete the proof of characterization of Noebeling spaces, the latter was done by Nagorko and we could not give any contribution but the head and Mine could obtain the classification theorem on open sets in LF-spaces, which can be a foothold on studying LF-manifolds. Less

为了实现本项目开始时提到的三个目标，首席调查员一直在与调查人员合作进行研究，并邀请波兰的W.Kubis和乌克兰的T.Banakh共同开展工作和交流信息。最后，我们取得了很多不错的结果。关于第一个关于各种超空间在什么条件下以及在什么空间上是同胚的(即使在不可分的情况下)的问题，通过许多负责人的联合工作，我们得到了如下结果：用Wijsman拓扑指定Banach空间上的超空间；找到闭集的超空间是ANR的条件；证明了赋范空间中的闭凸集的超空间是关于Hausdorff一致和Atouch-Wets拓扑的ANR；对于有限维赋范空间，它是基空间和Hilbert立方体的乘积的同胚超空间；指定了由各种类型的紧集组成的超空间；证明了…更多的是无理空间和Noebeling空间中的有界闭集超空间。关于第二个目标，即找到无限维流形上的映射空间并阐明其拓扑结构，Yagaki将次多面体嵌入到曲面空间中的包含的连通分支分类，并将Berlange关于保测同态群的结果推广到非紧的情形。这位负责人与上原合作，澄清了下半连续函数空间的拓扑结构。此外，在与Banakh、Mine和Yagaki的共同工作中，他证明了具有Whitney拓扑的非紧曲面的同胚群可以嵌入到Hilbert空间和欧氏空间的直接极限的乘积中作为开集。关于丰富不可分无限维泛在空间的研究和完成Noebeling空间刻画的证明，后者是Nagorko所做的，我们不能给出任何贡献，但Head和Mine得到了LF-空间中开集的分类定理，这可以作为研究LF-流形的立足点。较少