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On cohesive almost zero-dimensional spaces

Part of: Peculiar spaces Special properties Fairly general properties

Published online by Cambridge University Press:  15 July 2020

Jan J. Dijkstra  and
David S. Lipham
Jan J. Dijkstra
Affiliation:
PO Box 1180, Crested Butte, CO 81224, USA e-mail: jan.dijkstra1@gmail.com
David S. Lipham*
Affiliation:
Department of Mathematics, Auburn University at Montgomery, Montgomery, AL 36117, USA e-mail: dlipham@aum.edu
*
e-mail: dsl0003@auburn.edu
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Abstract

We investigate C-sets in almost zero-dimensional spaces, showing that closed $\sigma $C-sets are C-sets. As corollaries, we prove that every rim-$\sigma $-compact almost zero-dimensional space is zero-dimensional and that each cohesive almost zero-dimensional space is nowhere rational. To show that these results are sharp, we construct a rim-discrete connected set with an explosion point. We also show that every cohesive almost zero-dimensional subspace of $($Cantor set$)\!\times \mathbb R$ is nowhere dense.

Keywords

Almost zero-dimensionalcohesiverationalLelek fanexplosion pointErdős space

MSC classification

Primary: 54F45: Dimension theory 54F50: Spaces of dimension $leq 1$; curves, dendrites 54D35: Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54G20: Counterexamples

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 2 , June 2021 , pp. 429 - 441
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Copyright
© Canadian Mathematical Society 2020

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