Decisive convergence spaces

Abstract

We examine a class of convergence spaces characterized by the property that ultrafilters which converge to a given point x are in a certain sense independent of those which do not. Such spaces are said to be decisive. This definition gives every indication of being a natural one as the following facts will testify: (a) The set of all decisive convergence structures on a set S is a sublattice of the set of all convergence structures on S; (b) Any product of decisive convergence structures is decisive; (c) The image of a decisive convergence space under a quotient map is again decisive. Decisiveness shares with the property of being "almost pretopological" the distinction of being a necessary condition for a convergence space to be pretopologically coherent, and, in what is perhaps our main result, we obtain a complete solution to Problem 1 of [1] by showing that a convergence space is pretopologically coherent if and only if it is simultaneously decisive and almost pretopological. We also give simple characterizations for decisive pretopological spaces and decisive T 1 topological spaces.