Dually discrete spaces

A neighbourhood assignment in a space X is a family O = { O x : x ∈ X } of open subsets of X such that x ∈ O x for any x ∈ X . A set Y ⊆ X is a kernel of O if O ( Y ) = ⋃ { O x : x ∈ Y } = X . We obtain some new results concerning dually discrete spaces, being those spaces for which every neighbourhood assignment has a discrete kernel. This is a strictly larger class than the class of D-spaces of [E.K. van Douwen, W.F. Pfeffer, Some properties of the Sorgenfrey line and related spaces, Pacific J. Math. 81 (2) (1979) 371–377].