On expandable discrete collections

Expandability-type properties, which are more general than both normality and countable paracompactness, are considered. We give a common generalization of theorems related to closed mappings of normal or countably paracompact spaces onto a q-space. We also consider these properties in product spaces and prove theorems which are parallel to Katĕtovʹs theorem on hereditary normality in product spaces. An application to a space of closed subsets is given too.