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Neighborhood spaces, pretopological spaces, and closure spaces are topological space generalizations which can be characterized by means of their associated interior (or closure) 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 generalp-stacks.

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A basic theory for probabilistic convergence spaces based on filter convergence is introduced. As in Florescu's previous theory of probabilistic convergence structures based on nets, one is able to assign a probability that a given filter converges to a given point. Various concepts and theorems pertaining to convergence spaces are extended to the realm of probabilistic convergence spaces, and illustrated by means of examples based on convergence in probability and convergence almost everywhere. Diagonal axioms due to Kowalsky and Fischer are also studied, first for convergence spaces and then in the setting of probabilistic convergence spaces.1991 Mathematics subject classification (Amer. Math. Soc): 54A20, 54E70, 60A05.