The Kuratowski Closure-complement Theorem

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. The most convenient approach to investigating the Kuratowski closure-complement theorem is via operator notation. If (X,T) is a topological space we will consider the complement operator a acting on the set of subsets ofX defined by a(A) = X\A and the closure operator b defined by b(A) = Ā. We will also use the symbol i to denote the interior of a set. We prefer the neutral symbols a and b for complement and closure since expressions built from iterated dashes and bars look awkward, while the notation C(A) could mean either complement or closure. The investigation of topology from the point of view of topological closure operators on a set was begun by Kuratowski. Given a set O := {oi : i ∈ I} of operators on a set S we may construct some possibly new operators by composing the members of O. As composition is associative, the set of distinct operators that arises forms a monoid under composition, with identity element id. Here two operators o1 and o2 are distinct if there is an s ∈ S such that o1(s) 6= o2(s). We will refer to the monoid of operators generated by closure and complement on a topological space (X,T) as the Kuratowski monoid of (X,T) and to the operators themselves as Kuratowski operators. Since we will almost always be considering operators on topological spaces it will be useful to refer to an operator that always produces a closed set (or open set) as a closed operator (or open operator respectively). There is a natural partial order on a monoid {oi : i ∈ I} of operators on the subsets of a set X defined by o1 ≤ o2 if for every A ⊆ X we have o1(A) ⊆ o2(A). For example, we have id ≤ b. In general we say that an operator o is isotonic if X ⊆ Y ⇒ o(X) ⊆ o(Y ) and is idempotent if o = o. With these ideas in hand it is now quite easy to give a proof of the Kuratowski theorem (see also [29, 21, 30, 15, 8]).