Revisiting Cauty's proof of the Schauder conjecture.
On Convex Sets with Convex-Hereditary CEP
CEP stands for the compact extension property. We characterize nonlocally convex complete metric linear spaces with convex-hereditary CEP.
A Polish AR-Space with no Nontrivial Isotopy
The Polish space Y constructed in [vM1] admits no nontrivial isotopy. Yet, there exists a Polish group that acts transitively on Y.
The space of ANR’s in
The hyperspaces and in consisting respectively of all compact absolute neighborhood retracts and all compact absolute retracts are studied. It is shown that both have the Borel type of absolute -spaces and that, indeed, they are not -spaces. The main result is that is an absorber for the class of all absolute -spaces and is therefore homeomorphic to the standard model space of this class.
Classification of function spaces with the pointwise topology determined by a countable dense set
Failure of the Factor Theorem for Borel pre-Hilbert spaces
In every infinite-dimensional Fréchet space X, we construct a linear subspace E such that E is an -subset of X and contains a retract R so that is not homeomorphic to . This shows that Toruńczyk’s Factor Theorem fails in the Borel case.
Applications of some results of infinite-dimensional topology to the topological classification of operator images
This volume consists of three relatively independent articles devoted to the topological study of the so-called operator images and weak unit balls of Banach spaces. These articles are: “The topological classification of weak unit balls of Banach spaces” by T. Banakh, “The topological and Borel classification of operator images” by T. Banakh, T. Dobrowolski and A. Plichko, and “Operator images homeomorphic to ” by T. Banakh. The articles summarize investigations that has been done by these authors...
Topological type of weakly closed subgroups in Banach spaces
The main result says that nondiscrete, weakly closed, containing no nontrivial linear subspaces, additive subgroups in separable reflexive Banach spaces are homeomorphic to the complete Erdős space. Two examples of such subgroups in which are interesting from the Banach space theory point of view are discussed.
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 .
Closed subgroups in Banach spaces
We show that zero-dimensional nondiscrete closed subgroups do exist in Banach spaces E. This happens exactly when E contains an isomorphic copy of . Other results on subgroups of linear spaces are obtained.
Selections and near-selections in metric linear spaces without local convexity
We characterize the AR property in convex subsets of metric linear spaces in terms of certain near-selections.
Les hyperespaces des rétractes absolus et des rétractes absolus de voisinage du plan
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