  Generalized topological spaces in the sense of Cs\'{a}sz\'{a}r have two main
features which distinguish them from typical topologies. First, these families
of subsets are not closed under intersections. Second, we allow for the
possibility that the whole space is not open. Hence, some points of the
universe may be beyond any open set. In this paper we assume that such points
are associated with certain open neighbourhoods by means of a special function
F. We study various properties of the structures obtained in this way. We
introduce the notions of F-interior and F-closure and we discuss issues of
convergence and continuity in this new setting.
