Net Spaces in Categorical Topology
by
Josef Šlapal
Technical University of Brno

Let S be a non-empty concrete category over the category of non-empty sets. By an S-net in a set X we understand any pair (S, f) where S in S is an object and f:S --> X is a map. An S-net space is defined as a pair (X, \pi) where X is a set and \pi is a map assigning a subset of X to every S-net in X . As usual, we write (S, f) \PPI x instead of x in \pi(S, f). Given a pair of S-net spaces (X, \pi) and (Y, \rho), by a continuous map of (X, \pi) into (Y, \rho) we mean a map F:X --> Y fulfilling (S, f) \PPI x implies (S, F o f) \PPI F(x). We study the category Net_S of S-net spaces with continuous maps as morphisms and show that Net_S is a strong topological preuniverse. For S-net spaces we consider a number of axioms some of which are extensions of the usual net axioms. By imposing these axioms we obtain a number of full subcategories of Net_S most of which are shown to be cartesian closed. We also investigate relations of these subcategories to some known topological categories (subcategories and supercategories of Top). Some of the subcategories obtained are trivial, some are isomorphic to certain well-known categories, but many of them are new and useful for applications to various branches of mathematics.

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaf-76.