Nearly metacompact spaces

A space X is called nearly metacompact (meta-Lindelof) provided that if U is an open cover of X then there is a dense set D subset of or equal to X and an open refinement V of ZA that is point-finite (point-countable) on D. We show that countably compact, nearly meta-Lindelof T-3-spaces are compact. That nearly metacompact radial spaces are meta-Lindelof. Every space can be embedded as a closed subspace of a nearly metacompact space. We give an example of a countably compact, nearly meta-Lindelof T-2-space that is not compact and a nearly metacompact T-2-space which is not irreducible. (C) 1999 Elsevier Science B.V. All rights reserved.