Some results on property (a)

A space X has property (a) if for every open cover U of X and for each dense D⫅X there is a closed discrete F⫅D such that St(F,U)=X. In this paper, the relationship between property (a) and normality is investigated. A consistent example of a normal space without property (a) is constructed. A weakening of property (a) is defined and studied. Examples of nonmetrizable Moore spaces with property (a) are given. The question for which spaces X with property (a) the product X×(ω+1) also has property (a) is studied.