© Hindawi Publishing Corp. NEIGHBORHOOD SPACES

Neighborhood spaces, pretopological spaces, and closure spaces are topological space generalizations which can be characterized by means of their associated interior (or clo-sure) operators. The category NBD of neighborhood spaces and continuous maps contains PRTOP as a bicoreflective subcategory and CLS as a bireflective subcategory, whereas TOP is bireflectively embedded in PRTOP and bicoreflectively embedded in CLS. Initial and final structures are described in these categories, and it is shown that the Tychonov theorem holds in all of them. In order to describe a successful convergence theory in NBD, it is necessary to replace filters by more general p-stacks. 2000 Mathematics Subject Classification: 54A05, 54A10. 1. Introduction. In his original definition of topology, Hausdorff [7] assigned to each point x in a set X a system of neighborhoods subject to certain axioms. We will define a neighborhood structure ν on X in the same way, using axioms weaker than those of Hausdorff, but just strong enough to include both pretopological and closure spaces as special cases of neighborhood spaces. Of particular significance i