Denumerable Compact Spaces and Cantor Derivative

A continuous open finite-to-one surjection f preserves Cantor rank. If the domain of f is a Hausdorff compact space, Cantor degree variations are bounded by the maximal size of the finite fibres. If the fibres are infinite, we show inequalities involving the maximal and minimal rank of the fibres. A closed preorder on a denumerable Hausdorff compact space is the intersection of clopen preoders. We compute the Cantor rank of a cartesian product and build a semiring where the Cantor derivative is a derivation. The Cantor derivative was introduced by Georg Cantor in 1872 to derivate sets of convergence of trigonometric series [1]. In Model Theory, one century later, Cantor-Bendixson rank gave birth to Morley rank in ω-stable theories [6] and to Cantor rank in small ones. We shall specify some properties of this rank, well known by logicians when they refer to Morley rank. We first notice that the Cantor derivative of a topological sum and cartesian product is well-behaved. A continuous open finite-to-one surjection f preserves Cantor rank. Moreover, if the domain of f is a Hausdorff compact space, Cantor degree variations can be bounded by the maximal size of the finite fibres. If the fibres are infinite, there are still inequalities involving the maximal and minimal rank of the fibres. Thanks to Cantor rank, a closed preorder on a countable Hausdorff compact space can be shown to be the intersection of clopen preoders. This was almost noticed in [7, 4]. We build an ordered division semiring in which the Cantor derivative is a derivation. We consider when this semiring can be given a lattice structure. We finish by applying the results to first order theories having countably many types. This gives a new proof of a Theorem in [5] computing Cantor rank and degree over an algebraic tuple. Most of section 1 and 2 must be well known. However, I could not find any reference except [3]. Proposition 5, Theorem 14 and Proposition 16 along with section 3, 4 and most of 5 seem to be new. 1991 Mathematics Subject Classification. 03C5O, 03C07, 54A05, 54B99, 54D30, 54F65.