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 form and where 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.