The hyperspaces $ANR(ℝ^n)$ and $AR(ℝ^n)$ in $2^{ℝ^n} (n ≥ 3)$ 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 $G_{δσ δ}$-spaces and that, indeed, they are not $F_{σ δσ }$-spaces. The main result is that $ANR(ℝ^n)$ is an absorber for the class of all absolute $G_{δσ δ}$-spaces and is therefore homeomorphic to the standard model space $Ω_3$ of this class.
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In every infinite-dimensional Fréchet space X, we construct a linear subspace E such that E is an $F_{σδσ}$-subset of X and contains a retract R so that $R × E^{ω}$ is not homeomorphic to $E^{ω}$. This shows that Toruńczyk's Factor Theorem fails in the Borel case.
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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 for the past 10 years. All that started in the late 80s with the following question by T. Dobrowolski: Is the topological type of an operator image fully determined by its Borel type? Let us recall that an operator image is a space of the form TX, where T:X → Y is a continuous linear operator between Fréchet spaces (operator images often appear in analysis and topology, for example, the space $C_{p}*(X)$ of bounded continuous functions on a countable space X with the topology of pointwise convergence can be considered as an operator image of the Banach space C(βX)). In the early 90s T. Dobrowolski obtained a positive answer to the above question for operator images of low Borel complexity. Namely, he showed that every infinite-dimensional separable σ-complete operator image is homeomorphic to one of the spaces: l², Σ, or Σ × l², where Σ is the linear hull of the standard Hilbert cube in the Hilbert space l² (a space is σ-complete if it is a countable union of closed completely-metrizable subspaces). The same result was proved independently by T. Banakh. The topological structure of operator images of higher Borel complexity remained unclear. However, it was known that the topological type of $C_{p}*(X)$ is determined by its Borel type for the case of the second multiplicative Borel class. More precisely, each absolute $F_{σδ}$-space $C_{p}*(X)$ over a nondiscrete space X is homeomorphic to $Σ^{ω}$. Thus the conjecture appeared: The spaces l², Σ, Σ × l² and $Σ^{ω}$ exhaust all possible topological types of infinite-dimensional operator images that are absolute $F_{σδ}$-spaces. Numerous attempts to confirm this conjecture were unsuccessful (though many of those attempts lead to very fruitful developments in infinite-dimensional topology). Finally, in 1998, T. Banakh found a counterexample to the above conjecture. The counterexample came from the study of the weak topology of the closed unit balls of Banach spaces. It turned out that the topological type of an operator image TX depends much on the geometric properties of the Fréchet space X as well as on the properties of the weak topology of X. This topic, which is of independent interest, is considered in detail in the first article of this volume, that is, "The topological classification of weak unit balls of Banach spaces". An example of a pathological Banach space constructed in the last section of this article is applied in the remaining two articles, strictly devoted to studying operator images. The first of them, "The topological and Borel classification of operator images", deals with some general questions in the area, and also with operator images of high Borel complexity, while the second one is restricted to the study of operator images homeomorphic to $Σ^{ω}$. We refer the reader to Introductions that the three articles start with for more detailed information on their contents.
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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 $Ω_α$.
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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.
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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.
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