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 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.

Todorcevic orderings as examples of ccc forcings without adding random reals

“…In this paper, we will show that Todorcevic orderings add no random reals whenever they have the countable chain condition (ccc). Todorcevic [13] and Balcar, Pazák and Thümmel [2] provided two sufficient conditions for topological spaces with the cccness of Todorcevic orderings. These conditions cover a wide class of topological spaces.…”

“…2.1 Todorcevic orderings. As said in [2], when Todorcevic ordering is applied to a topological space, it is natural to require it to be sequential and have the unique limit property 1 . A topological space X is called sequential if for any Z ⊆ X, Z is closed in X iff for any A ⊆ Z and x ∈ X to which A converges, x belongs to Z.…”

“…For a subset F of a topological space, let F d denote the first Cantor-Bendixson derivative of F , that is, the set of all accumulation points of F . Definition 2.1 (Todorcevic [13], see also [2], [11]). For a topological space X, T(X) is the set of all subsets of X which are unions of finitely many converging sequences including their limit points, and for each p and q in T(X), q ≤ T(X) p iff q ⊇ p and q d ∩ p = p d .…”

“…In [2], Balcar, Pazák and Thümmel applied Todorcevic's Borel definable ccc forcing in [13] to more general topological spaces. They called such forcings Todorcevic orderings.…”

“…They called such forcings Todorcevic orderings. In [2], they analyzed Todorcevic orderings from several view points. One of them is about the countable chain condition.…”