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On Convex Sets with Convex-Hereditary CEP

Tadeusz Dobrowolski — 2011

Bulletin of the Polish Academy of Sciences. Mathematics

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

Tadeusz Dobrowolski — 2008

Bulletin of the Polish Academy of Sciences. Mathematics

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 $ℝ^{n}$

Tadeusz Dobrowolski; Leonard Rubin — 1994

Fundamenta Mathematicae

The hyperspaces $A N R (ℝ^{n})$ and $A R (ℝ^{n})$ in $2^{ℝ^{n}} (n \geq 3)$ 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 $G_{δ σ δ}$ -spaces and that, indeed, they are not $F_{σ δ σ}$ -spaces. The main result is that $A N R (ℝ^{n})$ is an absorber for the class of all absolute $G_{δ σ δ}$ -spaces and is therefore homeomorphic to the standard model space $Ω_{3}$ of this class.

Classification of function spaces with the pointwise topology determined by a countable dense set

Tadeusz Dobrowolski; Witold Marciszewski — 1995

Fundamenta Mathematicae

Failure of the Factor Theorem for Borel pre-Hilbert spaces

Tadeusz Dobrowolski; Witold Marciszewski — 2002

Fundamenta Mathematicae

In every infinite-dimensional Fréchet space X, we construct a linear subspace E such that E is an $F_{σ δ σ}$ -subset of X and contains a retract R so that $R \times E^{ω}$ is not homeomorphic to $E^{ω}$ . 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

Taras Banakh; Tadeusz Dobrowolski; Anatoliĭ Plichko — 2000

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 $Σ^{ω}$ ” by T. Banakh. The articles summarize investigations that has been done by these authors...

Topological type of weakly closed subgroups in Banach spaces

Tadeusz Dobrowolski; Janusz Grabowski; Kazuhiro Kawamura — 1996

Studia Mathematica

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 $ℓ^{1}$ which are interesting from the Banach space theory point of view are discussed.

A contribution to the topological classification of the spaces Ср(X)

Robert Cauty; Tadeusz Dobrowolski; Witold Marciszewski — 1993

Fundamenta Mathematicae

We prove that for each countably infinite, regular space X such that $C_{p} (X)$ is a $Z_{σ}$ -space, the topology of $C_{p} (X)$ is determined by the class $F_{0} (C_{p} (X))$ of spaces embeddable onto closed subsets of $C_{p} (X)$ . We show that $C_{p} (X)$ , whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set $Ω_{α}$ for the multiplicative Borel class $M_{α}$ if $F_{0} (C_{p} (X)) = M_{α}$ . For each ordinal α ≥ 2, we provide an example $X_{α}$ such that $C_{p} (X_{α})$ is homeomorphic to $Ω_{α}$ .

Closed subgroups in Banach spaces

Fredric Ancel; Tadeusz Dobrowolski; Janusz Grabowski — 1994

Studia Mathematica

We show that zero-dimensional nondiscrete closed subgroups do exist in Banach spaces E. This happens exactly when E contains an isomorphic copy of $c_{0}$ . Other results on subgroups of linear spaces are obtained.