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# Normality versus paracompactness in locally compact spaces

Published:2017-05-25
Printed: Feb 2018
• Alan Dow,
Department of Mathematics and Statistics, University of North Carolina, Charlotte, North Carolina 28223, USA
• Franklin D. Tall,
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4
 Format: LaTeX MathJax PDF

## Abstract

This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on $\omega_1$, as well as of a strong form of Chang's Conjecture. Together with other improvements, this enables the consistent characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of $\omega_1$.
 Keywords: normal, paracompact, locally compact, countably tight, collectionwise Hausdorff, forcing with a coherent Souslin tree, Martin's Maximum, PFA(S)[S], Axiom R, moving off property
 MSC Classifications: 54A35 - Consistency and independence results [See also 03E35] 54D20 - Noncompact covering properties (paracompact, Lindelof, etc.) 54D45 - Local compactness, $\sigma$-compactness 03E35 - Consistency and independence results 03E50 - Continuum hypothesis and Martin's axiom [See also 03E57] 03E55 - Large cardinals 03E57 - Generic absoluteness and forcing axioms [See also 03E50]

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