Collections of higher-dimensional hereditarily indecomposable continua, with incomparable Fréchet types

Abstract

We construct a family {Y s : s ∈ S} of cardinality 2 ℵ 0 of hereditarily indecomposable continua which are: (a) n-dimensional Cantor manifolds, for any given natural number n, or (b) hereditarily strongly infinite-dimensional Cantor manifolds, or else (c) countable-dimensional continua of every given transfinite inductive dimension, small or large, such that if h : Y s → Y s is an embedding then s = s and h is the identity.

Fully closed maps and non-metrizable higher-dimensional Anderson–Choquet continua

“…A metric Cook continuum must have dimension ≤ 2 (Maćkowiak [38]), and if it is hereditarily indecomposable, then it must be one-dimensional (Krzempek [32]). On the other hand, several authors investigated rigidity properties of higher-dimensional continua (J.J. Charatonik [8], M. Reńska [45], E. Pol [40][41][42][43]34], see [32] for more references). In [32] the present author constructed a metric, n-dimensional (arbitrary n > 1), hereditarily indecomposable continuum no two of whose disjoint n-dimensional subcontinua are homeomorphic, but he was not able to ensure that the continuum be Anderson-Choquet.…”