Kuratowski's Theorem for Two Closure Operators

A celebrated 1922 theorem of Kuratowski states that there are at most 14 distinct sets arising from applying the operations of complementation and closure, any number of times, in any order, to a subset of a topological space. In this paper we consider the case of complementation and two abstract closure operators. In contrast to the case of a single closure operation, we show that infinitely many distinct sets can be generated, even when the closure operators commute.