Cofinal types on ω₂

Abstract

In this paper we start the analysis of the class scriptDℵ2$\mathcal {D}_{\aleph _2}$, the class of cofinal types of directed sets of cofinality at most ℵ2. We compare elements of scriptDℵ2$\mathcal {D}_{\aleph _2}$ using the notion of Tukey reducibility. We isolate some simple cofinal types in scriptDℵ2$\mathcal {D}_{\aleph _2}$, and then proceed to find some of these types which have an immediate successor in the Tukey ordering of scriptDℵ2$\mathcal {D}_{\aleph _2}$.