Geodesic
Antichains and Finite Sets that Meet all Maximal Chains
Canadian journal of mathematics, Tome 38 (1986) no. 3, pp. 619-632

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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. 
Ginsburg, J.; Rival, I.; Sands, B. Antichains and Finite Sets that Meet all Maximal Chains. Canadian journal of mathematics, Tome 38 (1986) no. 3, pp. 619-632. doi: 10.4153/CJM-1986-031-3
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