Bericht über einige neue Ergebnisse und Probleme auf dem Gebiet der anschaulichen Topologie.
Best proximity pairs for upper semicontinuous set-valued maps in hyperconvex metric spaces.
Best proximity pairs theorems for continuous set-valued maps.
Best proximity sets and equilibrium pairs for a finite family of multimaps.
Biequivalences and topology in the alternative set theory
Bifurcations in One Dimension. I. The Nonwandering Set.
Bijections of scattered spaces onto compact Hausdorff
Bijections onto compact spaces
Bi-Lipschitz embeddings into low-dimensional Euclidean spaces
Bilipschitz extensions from smooth manifolds.
We prove that every compact C1-submanifold of Rn, with or without boundary, has the bilipschitz extension property in Rn.
Bimeasurable maps
Bimorphisms in pro-homotopy and proper homotopy
A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. The category is balanced if every bimorphism is an isomorphism. In the paper properties of bimorphisms of several categories are discussed (pro-homotopy, shape, proper homotopy) and the question of those categories being balanced is raised. Our most interesting result is that a bimorphism f:X → Y of is an isomorphism if Y is movable. Recall that is the full subcategory of consisting of...
Bi-quotient images of ordered spaces.
Blumberg property versus almost continuity.
Booleanization
Booleanization of uniform frames
Booleanization of frames or uniform frames, which is not functorial under the basic choice of morphisms, becomes functorial in the categories with weakly open homomorphisms or weakly open uniform homomorphisms. Then, the construction becomes a reflection. In the uniform case, moreover, it also has a left adjoint. In connection with this, certain dual equivalences concerning uniform spaces and uniform frames arise.
Bootstrapping Kirszbraun's extension theorem
We show how Kirszbraun's theorem on extending Lipschitz mappings in Hilbert space implies its own generalization. There is a continuous extension operator preserving the Lipschitz constant of every mapping.
Borel classes of uniformizations of sets with large sections
We give several refinements of known theorems on Borel uniformizations of sets with “large sections”. In particular, we show that a set B ⊂ [0,1] × [0,1] which belongs to , α ≥ 2, and which has all “vertical” sections of positive Lebesgue measure, has a uniformization which is the graph of a -measurable mapping. We get a similar result for sets with nonmeager sections. As a corollary we derive an improvement of Srivastava’s theorem on uniformizations for Borel sets with sections.
Borel sets with σ-compact sections for nonseparable spaces
We prove that every (extended) Borel subset E of X × Y, where X is complete metric and Y is Polish, can be covered by countably many extended Borel sets with compact sections if the sections , x ∈ X, are σ-compact. This is a nonseparable version of a theorem of Saint Raymond. As a by-product, we get a proof of Saint Raymond’s result which does not use transfinite induction.
Borsuk's quasi-equivalence is not transitive
Borsuk's quasi-equivalence relation on the class of all compacta is considered. The open problem concerning transitivity of this relation is solved in the negative. Namely, three continua X, Y and Z lying in ℝ³ are constructed such that X is quasi-equivalent to Y and Y is quasi-equivalent to Z, while X is not quasi-equivalent to Z.