Net Spaces in Categorical Topology

In the paper we introduce the notion of S-nets, S a non-empty construct , as a generalization of the usual nets. S-net spaces and continuous maps between them are then deened in a natural way. For S-net spaces we introduce a number of axioms and study the categories obtained. We nd some topological and cartesian closed categories of S-net spaces and describe their relations to certain known topological categories. 0. Introduction Cartesian closed topological categories belong to the most important objects studied in categorical topology. The usefulness of these categories for applications to many classical branches of mathematics is well known. And the interest in cartesian closed categories became even stronger after discovering the important role that they play in computer science. When looking for new cartesian closed topological categories categorical topologists usually start from the category of topological spaces and generalize or modify the considered axioms of topology. These axioms depend on the chosen access to topology (closure operations, neighborhood structures, systems of open sets, etc.). In this connection the most often used access is that one which views topologies as convergence structures and uses lters for describing the convergence. This lter access gave arise to a rich hierarchy of useful categories and an intensive study is being performed in the area at present. In this paper we will also view topologies as convergence structures, but we shall use nets instead of lters for expressing the convergence. This access is closely related to that one using lters and some connections between certain types of categories of convergence lter spaces and those of convergence net spaces are described in 14]. Of course, from the classical point of view lters are more suitable for investigations in topology than nets which have, essentially, an order-theoretic nature. However, in our considerations we will not deal with the usual nets, but