On infinite-dimensional Cantor manifolds

Continuum many Fréchet types of hereditarily strongly infinite-dimensional Cantor manifolds

“…However, in Section 4 we shall describe an example, obtained jointly with Roman Pol, of a compactum without 1-dimensional subcontinua, still containing a 1-dimensional subset (we did not find any examples to this effect in the literature). It cannot be excluded that this phenomenon occurs among the Cantor manifolds considered in [13]. Our present construction combines some ideas from [5] and [13], and it involves the method of "condensation of singularities" which we discuss in Section 2.…”

“…It cannot be excluded that this phenomenon occurs among the Cantor manifolds considered in [13]. Our present construction combines some ideas from [5] and [13], and it involves the method of "condensation of singularities" which we discuss in Section 2. This technique yields in fact a stronger version of incomparability, explained in Remark 3.3.…”

“…A collection of continuum many pairwise incomparable Cantor manifolds without any non-trivial finite-dimensional subcontinua was indicated in [13]. However, in Section 4 we shall describe an example, obtained jointly with Roman Pol, of a compactum without 1-dimensional subcontinua, still containing a 1-dimensional subset (we did not find any examples to this effect in the literature).…”

On hereditarily indecomposable continua, Henderson compacta and a question of Yohe

“…Let A = {A ∈ K(C) : M (A) embeds in K}. Standard arguments show that A is analytic (see [19], proof of Lemma 6.3). Since A α ∈ A for uncountably many α, one concludes that there exists A 0 ∈ A with t ∈ A 0 \ Q.…”

“…Let us notice that in [19] we have constructed a family of continuum many Cantor manifolds without non-trivial finite-dimensional subcontinua that are not embeddable into each other, and Chatyrko and the author constructed in [6] such a family of hereditarily strongly infinite-dimensional Cantor manifolds. However, these continua are not hereditarily indecomposable.…”