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，首席研究员一直在与研究人员合作进行研究，并邀请W。来自波兰的Kubis和T.来自乌克兰的Banakh进行联合工作和交流信息。最后，我们得到了许多很好的结果。关于第一个对象指定在什么条件下和什么空间的每一个不同的超空间是同胚（即使在不可分的情况下），通过他的许多合作工作，我们得到了如下结果：用Wijsman拓扑指定Banach空间上的超空间，找到了度量空间的闭集超空间是ANR的条件;证明了赋范空间中的闭凸集超空间是关于Hausdorff一致性和Attouch-Wets拓扑的ANR，并且对于有限维赋范空间，它同胚于基空间和Hilbert立方体的乘积;指定由各种类型的超空间组成的超空间;证明 ...更多信息 在无理数空间和Noebeling空间中的有界闭集的超空间。关于第二个目标，找到映射空间是无限维流形，并澄清其拓扑结构，Yagasaki分类的连通分量的包含在空间中的嵌入的一个子多面体到一个表面和推广Berlange的结果组的测度保持同态到非紧的情况。头合作上原澄清拓扑结构的空间下半连续函数。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。关于充实不可分无限维泛空间的研究和完成Noebeling空间的特征证明的目的，后者是Nagorko完成的，我们不能给出任何贡献，但Head和Mine可以得到LF-空间中开集的分类定理，这可以成为研究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