Normality versus paracompactness in locally compact spaces

TL;DR

This paper corrects and strengthens a proof showing that in certain models, locally compact normal spaces are collectionwise Hausdorff, using advanced set-theoretic tools like saturation of ideals and Chang's Conjecture.

Contribution

It provides a correct proof of a key result linking local compactness, normality, and collectionwise Hausdorff property under forcing models, with new set-theoretic techniques.

Findings

Locally compact normal spaces are collectionwise Hausdorff in certain models.

Characterization of locally compact hereditarily paracompact spaces.

Use of saturation of non-stationary ideals and Chang's Conjecture in the proof.

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 characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of \omega_1.