On Todorcevic orderings
Volume 228 / 2015
Fundamenta Mathematicae 228 (2015), 173-192
MSC: Primary 03E05; Secondary 03E40.
DOI: 10.4064/fm228-2-4
Abstract
The Todorcevic ordering $\mathbb {T}(X)$ 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 $\sigma $-finite cc and even need not have the Knaster property. We are interested in properties of $\mathbb {T}(X)$ 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 $\mathbb {T}(X)$. We study the properties of $\mathbb {T}(X)$ as a forcing notion and the homogeneity of the generated complete Boolean algebra.
