Perfect pre-images of collectionwise normal spaces

Abstract:

We show that the following are true for perfect maps: (1) collectionwise normality with respect to compact (Lindelöf) sets is preserved inversely; (2) collectionwise normality with respect to countably compact (${\omega _1}$-compact) sets is preserved inversely if and only if the domain space is normal with respect to countably compact (${\omega _1}$-compact) sets; and (3) if $P$ is any property such that (i) $P$ is preserved by perfect maps, (ii) the free union of spaces satisfying $P$ also satisfies $P$, (iii) $P$ is closed hereditary, and (iv) $P$ plus collectionwise normality implies countable metacompactness, then collectionwise normality with respect to closed $P$-sets is preserved inversely if the domain space is normal with respect to closed $P$-sets. Examples of such a property $P$ are paracompactness, submetacompactness, stratifiability and countable metacompactness.