Non-Hausdorff convergence spaces

Abstract

In this paper we will establish a method of removing the Hausdorff assumption from certain convergence space theorems. As specific applications the precise form of the closure of a compact set in a regular non-Hausdorff space is given and the exact relationship between cl and cl 2 in a non-Hausdorff compact regular space is obtained. Necessary and sufficient conditions that the transition space for this procedure be topological or pretopological are found and a few embedding theorems are obtained.