### Revisiting Cauty's proof of the Schauder conjecture.

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CEP stands for the compact extension property. We characterize nonlocally convex complete metric linear spaces with convex-hereditary CEP.

The Polish space Y constructed in [vM1] admits no nontrivial isotopy. Yet, there exists a Polish group that acts transitively on Y.

The hyperspaces $ANR\left({\mathbb{R}}^{n}\right)$ and $AR\left({\mathbb{R}}^{n}\right)$ in ${2}^{{\mathbb{R}}^{n}}\left(n\ge 3\right)$ 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 ${G}_{\delta \sigma \delta}$-spaces and that, indeed, they are not ${F}_{\sigma \delta \sigma}$-spaces. The main result is that $ANR\left({\mathbb{R}}^{n}\right)$ is an absorber for the class of all absolute ${G}_{\delta \sigma \delta}$-spaces and is therefore homeomorphic to the standard model space ${\Omega}_{3}$ of this class.

In every infinite-dimensional Fréchet space X, we construct a linear subspace E such that E is an ${F}_{\sigma \delta \sigma}$-subset of X and contains a retract R so that $R\times {E}^{\omega}$ is not homeomorphic to ${E}^{\omega}$. This shows that Toruńczyk’s Factor Theorem fails in the Borel case.

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 ${\Sigma}^{\omega}$” by T. Banakh. The articles summarize investigations that has been done by these authors...

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

We prove that for each countably infinite, regular space X such that ${C}_{p}\left(X\right)$ is a ${Z}_{\sigma}$-space, the topology of ${C}_{p}\left(X\right)$ is determined by the class ${F}_{0}\left({C}_{p}\left(X\right)\right)$ of spaces embeddable onto closed subsets of ${C}_{p}\left(X\right)$. We show that ${C}_{p}\left(X\right)$, whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set ${\Omega}_{\alpha}$ for the multiplicative Borel class ${M}_{\alpha}$ if ${F}_{0}\left({C}_{p}\left(X\right)\right)={M}_{\alpha}$. For each ordinal α ≥ 2, we provide an example ${X}_{\alpha}$ such that ${C}_{p}\left({X}_{\alpha}\right)$ is homeomorphic to ${\Omega}_{\alpha}$.

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.

We characterize the AR property in convex subsets of metric linear spaces in terms of certain near-selections.

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