The Kuratowski Closure-Complement Theorem

The Kuratowski Closure-Complement Theorem 1.1. [29] If (X; T) is a topo- logical space and A _ X then at most 14 sets can be obtained from A by taking closures and complements. Furthermore there is a space in which this bound is attained. This remarkable little theorem and related phenomena have been the concern of many authors. Apart from the mysterious appearance of the number 14, the attraction of this theorem is that it is simple to state and can be examined and proved using concepts available after any first encounter with topology. The goal of this article is both to provide an original investigation into variations of the theorem and its relation to properties of spaces and to survey the existing literature in this direction.