Abstract In this paper we start the analysis of the class D ℵ 2 $\mathcal {D}_{\aleph _2}$ , the class of cofinal types of directed sets of cofinality at most ℵ2. We compare elements of D ℵ 2 $\mathcal {D}_{\aleph _2}$ using the notion of Tukey reducibility. We isolate some simple cofinal types in D ℵ 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 ℵ 2 $\mathcal {D}_{\aleph _2}$ .
