On Dimensionsgrad, resolutions, and chainable continua

Michael G. Charalambous ¹   ; Jerzy Krzempek ²

¹ Department of Mathematics University of the Aegean 83 200, Karlovassi, Samos, Greece
² Institute of Mathematics Silesian University of Technology Kaszubska 23 44-100 Gliwice, Poland

Fundamenta Mathematicae, Tome 209 (2010) no. 3, pp. 243-265

Résumé

For each natural number $n \geq 1$ and each pair of ordinals $\alpha,\beta$ with $n \leq \alpha \leq \beta \leq \omega({\mathfrak c}^+)$, where $\omega({\mathfrak c}^+)$ is the first ordinal of cardinality ${\mathfrak c}^+$, we construct a continuum $S_{n,\alpha,\beta}$ such that(a) $\dim S_{n,\alpha,\beta}=n$; (b) $\mathop{{\rm trDg}}\nolimits S_{n,\alpha,\beta}=\mathop{{\rm trDgo}}\nolimits S_{n,\alpha,\beta}=\alpha$; (c) $\mathop{\rm trind} S_{n,\alpha,\beta}=\mathop{{\rm trInd}_0}\nolimits S_{n,\alpha,\beta}=\beta$; (d) if $\beta\omega({\mathfrak c}^+)$, then $S_{n,\alpha,\beta}$ is separable and first countable; (e) if $n=1$, then $S_{n,\alpha,\beta}$ can be made chainable or hereditarily decomposable; (f) if $\alpha = \beta\omega({\mathfrak c}^+)$, then $S_{n,\alpha,\beta}$ can be made hereditarily indecomposable; (g) if $n=1$ and $\alpha = \beta\omega({\mathfrak c}^+)$, then $S_{n,\alpha,\beta}$ 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

DOI : 10.4064/fm209-3-3

Keywords: each natural number geq each pair ordinals alpha beta leq alpha leq beta leq omega mathfrak where omega mathfrak first ordinal cardinality mathfrak construct continuum alpha beta dim alpha beta mathop trdg nolimits alpha beta mathop trdgo nolimits alpha beta alpha mathop trind alpha beta mathop trind nolimits alpha beta beta beta omega mathfrak alpha beta separable first countable alpha beta made chainable hereditarily decomposable alpha beta omega mathfrak alpha beta made hereditarily indecomposable alpha beta omega mathfrak alpha beta made chainable hereditarily indecomposable particular answer question raised chatyrko fedorchuk whether every non degenerate chainable space has dimensionsgrad equal moreover establish results enable compute dimensionsgrad number spaces constructed charalambous chatyrko fedorchuk