Document Type

Article

Language

eng

Publication Date

8-2006

Publisher

Elsevier

Source Publication

Topology and its Applications

Source ISSN

0166-8641

Original Item ID

DOI: 10.1016/j.topol.2005.08.009

Abstract

On the surface, the definitions of chainability and Lebesgue covering dimension ⩽1 are quite similar as covering properties. Using the ultracoproduct construction for compact Hausdorff spaces, we explore the assertion that the similarity is only skin deep. In the case of dimension, there is a theorem of E. Hemmingsen that gives us a first-order lattice-theoretic characterization. We show that no such characterization is possible for chainability, by proving that if κ is any infinite cardinal and AA is a lattice base for a nondegenerate continuum, then AA is elementarily equivalent to a lattice base for a continuum Y, of weight κ, such that Y has a 3-set open cover admitting no chain open refinement.

Comments

Accepted version. Topology and its Applications, Vol. 153, No. 14 (August, 2006): 2462-2468. DOI. © 2006 Elsevier. Used with permission.

NOTICE: this is the author’s version of a work that was accepted for publication in Topology and its Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Topology and its Applications, VOL 153, ISSUE 14, (August, 2006) DOI.

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