Topology of hyperspaces, mapping spaces and universal spaces

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共同工作和交流信息。最后，我们取得了许多好的成绩。关于第一个确定各种超空间在什么条件下和在什么空间上是同纯的（即使在不可分的情况下）的对象，通过头部的许多联合工作，我们得到了这样的结果：用Wijsman拓扑确定Banach空间上的超空间；寻找闭集的超空间为ANR的度量空间的条件；证明了赋范空间中闭凸集的超空间是关于Hausdorff均匀性和attach - wets拓扑的ANR，并且对于有限维赋范空间，它是基空间与Hilbert立方的乘积的同胚；指定由各种类型的紧集组成的超空间；证明了无理数空间和Noebeling空间中有界闭集的超空间。对于寻找映射空间为无限维流形并阐明其拓扑结构的第二个目标，Yagasaki将子多面体嵌入空间中包含的连通分量分类为曲面，并将Berlange关于保测度同态群的结果推广到非紧情况。head与Uehara合作阐明了下半连续函数空间的拓扑结构。此外，在与Banakh， Mine和Yagasaki的联合工作中，他证明了具有Whitney拓扑的非紧曲面的同纯群可以作为开集嵌入到Hilbert空间与欧氏空间的直接极限的乘积中。为了丰富不可分无限维通用空间的研究和完成Noebeling空间的表征证明，后者由Nagorko完成，我们没有贡献，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流形的一个立足点。少

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