Selection theorems for partitions of Polish spaces
Selections that characterize topological completeness
We show that the assertions of some fundamental selection theorems for lower-semicontinuous maps with completely metrizable range and metrizable domain actually characterize topological completeness of the target space. We also show that certain natural restrictions on the class of the domains change this situation. The results provide in particular answers to questions asked by Engelking, Heath and Michael [3] and Gutev, Nedev, Pelant and Valov [5].
Selections using orderings (non-separable case)
Sequentially complete spaces
Set-valued mappings on metric spaces
Several fixed point theorems concerning -distance.
Some extensions on the Sadovskii functor.
Some fixed point theorems
Some fixed point theorems for weak contraction conditions of integral type.
Some observations on local uniform boundedness principles
Some problems on Suslin spaces and topological algebras
Some properties of the Hausdorff distance in metric spaces.
Some properties of the Hausdorff distance in complete metric spaces are discussed. Results obtained in this paper explain ideas used in the theory of measures of noncompactness.
Some results on Ciric's quasi-contraction mappings.
Some results on metric trees
Using isometric embedding of metric trees into Banach spaces, this paper will investigate barycenters, type and cotype, and various measures of compactness of metric trees. A metric tree (T, d) is a metric space such that between any two of its points there is a unique arc that is isometric to an interval in ℝ. We begin our investigation by examining isometric embeddings of metric trees into Banach spaces. We then investigate the possible images x₀ = π((x₁ + ... + xₙ)/n), where π is a contractive...
Spaces of ANR's
Spaces of ANR's. II
Stability of convergent continuous descent methods.
Stabilizers of closed sets in the Urysohn space
Building on earlier work of Katětov, Uspenskij proved in [8] that the group of isometries of Urysohn's universal metric space 𝕌, endowed with the pointwise convergence topology, is a universal Polish group (i.e. it contains an isomorphic copy of any Polish group). Answering a question of Gao and Kechris, we prove here the following, more precise result: for any Polish group G, there exists a closed subset F of 𝕌 such that G is topologically isomorphic to the group of isometries of 𝕌 which map...