Factorising usco mappings

AbstractWe deal with factorisations through metrizable spaces of compact-valued u.s.c. mappings. In case the domain has some higher separation axioms, we found some natural relationship with the graph of such mappings. For an arbitrary domain, we related such factorisations to compact-valued continuous expansions

Coarse infinite-dimensionality of hyperspaces of finite subsets

We consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property

On a selection theorem of Blum and Swaminathan

summary:Blum and Swaminathan [Pacific J. Math. 93 (1981), 251--260] introduced the notion of \Cal B-fixedness for set-valued mappings, and characterized realcompactness by means of continuous selections for Tychonoff spaces of non-measurable cardinal. Using their method, we obtain another characterization of realcompactness, but without any cardinal assumption. We also characterize Dieudonné completeness and Lindelöf property in similar formulations