Cofinal types on ω2

In this paper we start the analysis of the class D aleph 2$\mathcal {D}_{\aleph _2}$, the class of cofinal types of directed sets of cofinality at most aleph(2). We compare elements of D aleph 2$\mathcal {D}_{\aleph _2}$ using the notion of Tukey reducibility. We isolate some simple cofinal types in D aleph 2$\mathcal {D}_{\aleph _2}$, and then proceed to find some of these types which have an immediate successor in the Tukey ordering of D aleph 2$\mathcal {D}_{\aleph _2}$.