D-14 - Countable Paracompactness, Countable Metacompactness, and Related Concepts

Abstract

A space is countably paracompact (respectively countably metacompact) if every countable open cover has a locally finite (respectively, point-finite) open refinement. Countable paracompactness generally goes hand in hand with normality—so much so that spaces that are normal but not countably paracompact are singled out by the term “Dowker spaces” while spaces that are countably paracompact but not normal are widely termed anti-Dowker. Dowker space is defined either as a normal space that is not countably paracompact, or as a space that is not countably metacompact. On the other hand, so few regular spaces were known not to be countably metacompact that Brian M. Scott, in his time, referred to the few then-known examples as “almost Dowker spaces”. Morita P-spaces are an important special class of countably metacompact spaces. These spaces are often classed as “generalized metric spaces” because the normal ones are precisely those normal spaces whose product with every metric space is normal.