There are Forty Nine KURATOWSKI Lattices in CANTOR Space

Abstract

Kuratowski observed that, starting from a subset M of a topological space and applying the closure operator and the interior operator arbitrarily often, one can generate at most seven different sets. We show that there are forty nine different types of sets w.r.t. the inclusion relations between the seven generated sets. All these types really occur in Cantor space, even for subsets defined by finite automata. For a given type, it is NL-complete to decide whether a set M, accepted by a given finite automaton, is of this type. In the topological space of real numbers only 39 of the 49 types really occur.