Let P be an infinite partially
ordered set with 0 and 1. A subset B of P is called a π-base 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 π-base 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 well-ordered or dually well-ordered subset of cardinality κ}.
We establish the following result: Theorem. |P|≤ hπ(P)d(P). Various corollaries areobtained which imply and extend several known results on the cardinality of partiallyordered sets, for example: Corollary. (a) |P|≤ 2hπ(P). (b) |P|≤ w(P)d(P). (c) If Bis a Boolean algebra then |B|≤ 2w(B).
