Nearly metacompact spaces

A space X is called nearly metacompact (meta-Lindelöf) provided that if U is an open cover of X then there is a dense set D⫅X and an open refinement V of U that is point-finite (point-countable) on D. We show that countably compact, nearly meta-Lindelöf T3-spaces are compact. That nearly metacompact radial spaces are meta-Lindelöf. Every space can be embedded as a closed subspace of a nearly metacompact space. We give an example of a countably compact, nearly meta-Lindelöf T2-space that is not compact and a nearly metacompact T2-space which is not irreducible.