Directed Sets and Cofinal Types

We show that 1, w, ax, u x ux and ["iF" are the only cofinal types of directed sets of size S,, but that there exist many cofinal types of directed sets of size continuum. A partially ordered set D is directed if every two elements of D have an upper bound in D. In this note we consider some basic problems concerning directed sets which have their origin in the theory of Moore-Smith convergence in topology [12, 3, 19, 9]. One such problem is to determine "all essential kind of directed sets" needed for defining the closure operator in a given class of spaces [3, p. 47]. Concerning this problem, the following important notion was introduced by J. Tukey [19]. Two directed sets D and E are cofinally similar if there is a partially ordered set C in which both can be embedded as cofinal subsets. He showed that this is an equivalence relation and that D and E are cofinally similar iff there is a convergent map from D into E and also a convergent map from E into D. The equivalence classes of this relation are called cofinal types. This concept has been extensively studied since then by various authors [4, 13, 7, 8]. Already, from the first introduction of this concept, it has been known that 1, w, ccx, w X cox and [w1]<" represent different cofinal types of directed sets of size < Kls but no more than five such types were known. The main result of this paper shows that 1, co, iox, uXw, and [u^1" are the only cofinal types of convergence in spaces of character < N, which can be constructed without additional set-theoretic assumptions. On the other hand, we shall construct many different cofinal types of directed sets of size continuum. This gives a solution to Problem 1 of J. Isbell [7]. The paper also contains several results about the structure of the class of all cofinal types, as well as a result about decomposing arbitrary partially ordered sets into directed sets. The results of this note were proved in February-March 1982 and presented to the ASL in January 1983. 1. A decomposition theorem. In this section we show that an arbitrary partially ordered set can be decomposed into a number of its directed subsets depending on the sizes of its antichains. This result is connected with an unpublished problem of F. Galvin concerning the Dilworth decomposition theorem [6] and it generalizes a similar result of E. Milner and K. Prikry [11]. The transitivity condition of a partial ordering is not used in our proof, so we state our result so as to apply to an arbitrary Received by the editors December 3, 1984. 1980 Mathematics Subject Classification. Primary 03E05, 03E35; Secondary 06A10, 18B35, 54A15. ©1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page