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 ${{\omega }_{1}}$.

We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space that does not include a perfect pre-image of ${{\text{ }\!\!\omega\!\!\text{ }}_{1}}$ is hereditarily paracompact.

Let λ be a regular ordinal with λ≥ω1. Then we prove that (λ+1)×λ is not base-countably metacompact. This implies that base-κ-paracompactness is not an inverse invariant of perfect mappings, which answers a question asked by Yamazaki.

Relations between uniform paracompactness and uniform para-Lindelöfness of a uniform space and its uniform weight are established.

A space has the shrinking property if, for every open cover {Va | a ∈ A}, there is an open cover {Wa | a ∈ A} with for each a ∈ A.lt is strangely difficult to find an example of a normal space without the shrinking property. It is proved here that any ∑-product of metric spaces has the shrinking property.