Locally compact perfectly normal spaces may all be paracompact
Volume 210 / 2010
Fundamenta Mathematicae 210 (2010), 285-300
MSC: Primary 54D15, 54D20, 54D45, 03E35, 54A35; Secondary 03E50, 03E65, 03E75, 54A20, 54A25.
DOI: 10.4064/fm210-3-4
Abstract
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 At a crucial step we use some still unpublished results announced by Todorcevic.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 Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space with a hereditarily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.
