The NZ property and D-spaces

Abstract

We introduce a property which we denote by NZ\({(\kappa)}\), where \({\kappa}\) is a cardinal. We show that chain neighborhood (F) spaces and monotonically metacompact spaces satisfy NZ(1), and that NZ\({(\kappa)}\) implies D if \({\kappa \leq \omega}\). Also, NZ\({(\kappa)}\) is closed under arbitrary subspaces and finite products, and countable products if \({\kappa}\) is infinite. It follows that any countable product of chain neighborhood (F) spaces and monotonically metacompact spaces is hereditarily a D-space. This provides a strong positive answer to a question of X. Yuming. We also prove that spaces satisfying NZ\({(\kappa)}\) are metacompact if \({\kappa}\) is finite, and meta-Lindelof if \({\kappa = \omega}\).