Todorcevic orderings as examples of ccc forcings without adding random reals

摘要

In [{it Two examples of Borel partially ordered sets with the countable chain condition/}, Proc. Amer. Math. Soc. {f 112} (1991), no.~4, 1125--1128], Todorcevic introduced a ccc forcing which is Borel definable in a separable metric space. In [{it On Todorcevic orderings/}, Fund. Math., to appear], Balcar, Paz'ak and Th"ummel applied it to more general topological spaces and called such forcings {it Todorcevic orderings/}. There they analyze Todorcevic orderings quite deeply. A significant remark is that Th"ummel solved the problem of Horn and Tarski by use of Todorcevic ordering [{it The problem of Horn and Tarski/}, Proc. Amer. Math. Soc. {f 142} (2014), no.~6, 1997--2000]. This paper supplements the analysis of Todorcevic orderings due to Balcar, Paz'ak and Th"ummel in [{it On Todorcevic orderings/}, Fund.~ Math., to appear]. More precisely, it is proved that Todorcevic orderings add no random reals whenever they have the countable chain condition.