Dually discrete spaces

A neighbourhood assignment in a space X is a family of open subsets of X such that x∈Ox for any x∈X. A set Y⊆X is a kernel of O if . 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].