Antichains and Finite Sets that Meet all Maximal Chains

Abstract

This paper is inspired by two apparently different ideas. Let P be an ordered set and let M(P) stand for the set of all of its maximal chains. The collection of all sets of the formandwhere x ∊ P, is a subbase for the open sets of a topology on M(P). (Actually, it is easy to check that the B(x) sets themselves form a subbase.) In other words, as M(P) is a subset of the power set 2|p| of P, we can regard M(P) as a subspace of 2|p| with the usual product topology. M. Bell and J. Ginsburg [1] have shown that the topological space M(P) is compact if and only if, for each x ∊ P, there is a finite subset C(x) of P all of whose elements are noncomparable to x and such that {x} ∪ C(x) meets each maximal chain.