In [Two examples of Borel partially ordered sets with the countable chain condition, Proc. Amer. Math. Soc. 112 (1991), no. 4, 1125–1128], Todorcevic introduced a ccc forcing which is Borel definable in a separable metric space. In [On Todorcevic orderings, Fund. Math., to appear], Balcar, Pazák and Thümmel applied it to more general topological spaces and called such forcings Todorcevic orderings. There they analyze Todorcevic orderings quite deeply. A significant remark is that Thümmel solved...
We introduce a generalization of a Dowker space constructed from a Suslin tree by Mary Ellen Rudin, and the rectangle refining property for forcing notions, which modifies the one for partitions due to Paul B. Larson and Stevo Todorčević and is stronger than the countable chain condition. It is proved that Martin's Axiom for forcing notions with the rectangle refining property implies that every generalized Rudin space constructed from Aronszajn trees is non-Dowker, and that the same can be forced...
It is proved that ideal-based forcings with the side condition method of Todorcevic (1984) add no random reals. By applying Judah-Repický's preservation theorem, it is consistent with the covering number of the null ideal being ℵ₁ that there are no S-spaces, every poset of uniform density ℵ₁ adds ℵ₁ Cohen reals, there are only five cofinal types of directed posets of size ℵ₁, and so on. This extends the previous work of Zapletal (2004).
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