Is a perfectly normal space always a paracompact space?
A topological space $X$ is called a perfectly normal space if $X$ is a normal space and every closed subset of $X$ is a $G_\delta$-set.
Thanks for your help.
2026-04-14 01:37:48.1776130668
Is a perfectly normal space always a paracompact space?
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Some counterexamples:
http://matwbn.icm.edu.pl/ksiazki/fm/fm97/fm9714.pdf
http://dantopology.wordpress.com/2012/12/02/a-subspace-of-bings-example-g/
http://matwbn.icm.edu.pl/ksiazki/fm/fm78/fm78129.pdf (consistent example)
http://www.ams.org/journals/proc/1993-118-03/S0002-9939-1993-1137225-2/S0002-9939-1993-1137225-2.pdf; Ostaszewski spaces are all very beautiful examples, as they are hereditarily separable, perfectly normal, locally compact but not Lindelöf and a paracompact separable space is Lindelöf. Of course we need $\diamondsuit$ but we do need some axiom for such a nice space:
An interesting consistency result: it is consistent that all locally compact perfectly normal spaces are paracompact: http://www.math.toronto.edu/tall/publications/LT.pdf