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 σ-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.
In 1948 A. Horn and A. Tarski asked whether the notions of a σ-finite cc and a σ-bounded cc ordering are equivalent. We give a negative answer to this question.When analyzing Boolean algebras carrying a measure, Horn and Tarski [HT48] defined the following two notions:Definition 1. An ordering P is called (i) σ-bounded cc if P = n∈ω P n , where each P n has the n + 2-cc.(ii) σ-finite cc if P = n∈ω P n , where each P n has the ω-cc.Here an ordering or its subset has the κ-cc (κ-chain condition) for a cardinal κ if it contains no antichain (set of pairwise orthogonal elements) of size κ.
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