A continuous version of Liapunov's convexity theorem
A contribution to the topological classification of the spaces Ср(X)
We prove that for each countably infinite, regular space X such that is a -space, the topology of is determined by the class of spaces embeddable onto closed subsets of . We show that , whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set for the multiplicative Borel class if . For each ordinal α ≥ 2, we provide an example such that is homeomorphic to .
A converse of the Arsenin–Kunugui theorem on Borel sets with σ-compact sections
Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel metric) space L onto a metric space M such that f(F) is a Borel subset of M if F is closed in L. We show that then is a set for all except countably many y ∈ M, that M is also Luzin, and that the Borel classes of the sets f(F), F closed in L, are bounded by a fixed countable ordinal. This gives a converse of the classical theorem of Arsenin and Kunugui. As a particular case we get Taĭmanov’s theorem saying that the image of...
A counterexample concerning products in the shape category
We exhibit a metric continuum X and a polyhedron P such that the Cartesian product X × P fails to be the product of X and P in the shape category of topological spaces.
A counter-example concerning quasi-homeomorphisms of compacta
A decomposition theorem for a class of continua for which the set function T is continuous
We prove a decomposition theorem for a class of continua for which F. B.. Jones's set function 𝓣 is continuous. This gives a partial answer to a question of D. Bellamy.
A degree for a class of acyclic-valued vector fields in Banach spaces
A degree theory for almost continuous functions
A dense subsemigroup of S(R) generated by two elements
A dimension raising hereditary shape equivalence
We construct a hereditary shape equivalence that raises transfinite inductive dimension from ω to ω+1. This shows that ind and Ind do not admit a geometric characterisation in the spirit of Alexandroff's Essential Mapping Theorem, answering a question asked by R. Pol.
A discontinuous function with a connected closed graph
A dual of the compression-expansion fixed point theorems.
A factorization theorem and its application to extremally disconnected resolutions
A few remarks on the set of finite-to-one maps of the Cantor set
A fixed point conjecture for Borsuk continuous set-valued mappings
The main result of this paper is that for n = 3,4,5 and k = n-2, every Borsuk continuous set-valued map of the closed ball in the n-dimensional Euclidean space with values which are one-point sets or sets homeomorphic to the k-sphere has a fixed point. Our approach fails for (k,n) = (1,4). A relevant counterexample (for the homological method, not for the fixed point conjecture) is indicated.
A fixed point index for bimaps
A fixed point principle for locally expansive multifunctions
A fixed point result of Seghal-Smithson type
A fixed point theorem for multivalued maps in symmetric spaces.
A Fixed Point Theorem in Reflexive Banach Spaces