On Todorcevic orderings

Abstract

The Todorcevic ordering T(X) consists of all finite families of convergent sequences in a given topological space X. Such an ordering was defined for the special case of the real line by S. Todorcevic (1991) as an example of a Borel ordering satisfying ccc that is not -finite cc and even need not have the Knaster property. We are interested in properties of T(X) where the space X is taken as a parameter. Conditions on X are given which ensure the countable chain condition and its stronger versions for T(X). We study the properties of T(X) as a forcing notion and the homogeneity of the generated complete Boolean algebra. 1. Introduction. In this paper, we will describe a general method of constructing an ordering from a topological space. When we look at orderings from the point of view of the forcing method, the orderings satisfying the countable chain condition (ccc) are of special interest. Our method will yield mostly such orderings. The ordering is obtained in the following way. For any topological space we can consider the set of finite unions of converging sequences such that limit points are outside of this union. This set is ordered by reverse inclusion. We obtain an ordering which was considered first by Todorcevic (Tod91) for the special case of the real numbers. Some ideas of the construction can already be found in the Galvin-Hajnal example (see (CN82)). For technical reasons, we slightly modify Todorcevic's original definition as follows.