On Dimensionsgrad, resolutions, and chainable continua

Michael G. Charalambous ; Jerzy Krzempek

Fundamenta Mathematicae, Tome 209 (2010), p. 243-265 / Harvested from The Polish Digital Mathematics Library

Résumé

For each natural number n ≥ 1 and each pair of ordinals α,β with n ≤ α ≤ β ≤ ω(⁺), where ω(⁺) is the first ordinal of cardinality ⁺, we construct a continuum $S_{n,α,β}$ such that (a) $dim S_{n,α,β} = n$; (b) $trDg S_{n,α,β} = trDgo S_{n,α,β} = α$; (c) $trind S_{n,α,β} = trInd₀S_{n,α,β} = β$; (d) if β < ω(⁺), then $S_{n,α,β}$ is separable and first countable; (e) if n = 1, then $S_{n,α,β}$ can be made chainable or hereditarily decomposable; (f) if α = β < ω(⁺), then $S_{n,α,β}$ can be made hereditarily indecomposable; (g) if n = 1 and α = β < ω(⁺), then $S_{n,α,β}$ can be made chainable and hereditarily indecomposable. In particular, we answer the question raised by Chatyrko and Fedorchuk whether every non-degenerate chainable space has Dimensionsgrad equal to 1. Moreover, we establish results that enable us to compute the Dimensionsgrad of a number of spaces constructed by Charalambous, Chatyrko, and Fedorchuk.

Détail

Publié le : 2010-01-01

Zbl 1213.54048

EUDML-ID : urn:eudml:doc:282600