Let P be an infinite partially
ordered set with 0 and 1. A subset B of P is called a πbase for P if for
every element x of P with 0 < x < 1 there exist elements b, c in B such that
0 < b ≤ x ≤ c < 1. We let π(P) denote the smallest cardinality of a πbase for P.
We also let hπ(P) = sup{π(S) : S ⊆ P}. The width and depth of P are
defined as usual: w(P) = sup{κ : P contains an antichain of cardinality κ};
d(P) = sup{κ : P contains a wellordered or dually wellordered subset of cardinality κ}.
We establish the following result: Theorem. P≤ hπ(P)^{d(P)}. Various corollaries are
obtained which imply and extend several known results on the cardinality of partially
ordered sets, for example: Corollary. (a) P≤ 2^{hπ(P)}. (b) P≤ w(P)^{d(P)}. (c) If B
is a Boolean algebra then B≤ 2^{w(B)}.
