We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, this answers a question of S. Watson. Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space with a hereditarily normal square is metrizable. We also solve a problem raised by the second author, proving it consistent with ZFC that every first countable hereditarily normal countable chain condition space is hereditarily separable.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm210-3-4,
author = {Paul B. Larson and Franklin D. Tall},
title = {Locally compact perfectly normal spaces may all be paracompact},
journal = {Fundamenta Mathematicae},
volume = {209},
year = {2010},
pages = {285-300},
zbl = {1211.54034},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm210-3-4}
}
Paul B. Larson; Franklin D. Tall. Locally compact perfectly normal spaces may all be paracompact. Fundamenta Mathematicae, Tome 209 (2010) pp. 285-300. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm210-3-4/