Taras Banakh

Open problems in infinite-dimensional topology

Publisher Summary This chapter discusses open problems in infinite-dimensional topology. The development of infinite-dimensional topology was greatly stimulated by three famous open problem lists—that of Geoghegan, West and Dobrowolski, Mogilski. It is now expected that the future progress will happen on the intersection of infinite-dimensional topology with neighbor areas of mathematics—Dimension Theory, Descriptive Set Theory, Analysis, and Theory of Retracts. This chapter attempts to select the problems whose solution would require creating new methods. It defines a pair as a pair (X, Y) consisting of a space X and a subspace Y ⊂ X. The ω denotes the set of non-negative integers. Many results and objects of infinite-dimensional topology have zero-dimensional counterparts usually considered in Descriptive Set Theory. As a rule, “zero dimensional” results have simpler proofs compared to their higher dimensional counterparts. Some zero-dimensional results are proved by essentially zero-dimensional methods (like those of infinite game theory), and it is an open question to which extent their higher-dimensional analogues are true. The chapter elaborates about the higher dimensional descriptive set theory, concepts of Zn-sets, along with presenting the related questions..

Characterizing the Cantor bi-cube in asymptotic categories

We present the characterization of metric spaces that are micro-, macro- or bi-uniformly equivalent to the extended Cantor set

Topological groups and convex sets homeomorphic to non-separable Hilbert spaces

\{\sum_{i=-n}^\infty\frac{2x_i}{3^i}:n\in\IN ,\;(x_i)_{i\in\IZ}\in\{0,1\}^\IZ\}\subset\IR

Characterization of spaces admitting a homotopy dense embedding into a Hilbert manifold

, which is bi-uniformly equivalent to the Cantor bi-cube

Scattered Subsets of Groups

2^{<\IZ}=\{(x_i)_{i\in\IZ}\in \{0,1\}^\IZ:\exists n\;\forall i\ge n\;x_i=0\}

A 1-dimensional Peano continuum which is not an IFS attractor

endowed with the metric

Algebraically determined topologies on permutation groups

d((x_i),(y_i))=\max_{i\in\IZ}2^i|x_i-y_i|

Packing Index of Subsets in Polish Groups

. Those characterizations imply that any two (uncountable) proper isometrically homogeneous ultrametric spaces are coarsely (and bi-uniformly) equivalent. This implies that any two countable locally finite groups endowed with proper left-invariant metrics are coarsely equivalent. For the proof of these results we develop a technique of towers which can have an independent interest.

WEAKLY P-SMALL NOT P-SMALL SUBSETS IN GROUPS

AbstractLet X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover