Irregularity

Abstract

Regular and irregular pretopologies are studied. In particular, for every ordinal there exists a topology such that the series of its partial (pretopological) regularizations has length of that ordinal. Regularity and topologicity of standard pretopologies on cascades can be characterized in terms of their states, so that their study for such spaces reduces to that of a combinatorics of states. For example, if an iterated partial regularization r^kpi is topological for k > 0 then r pi is a regular topology. Irregularity of pretopologies of countable character can be characterized in terms of sequential cascades with standard irregular pretopologies.