Compactness and subsets of ordered sets that meet all maximal chains (Q760448)

The space \({\mathcal M}(P)\) of maximal chains of a chain complete ordered set P is studied. By \(P_ 1\) and \(P_ 2\) are denoted two particular ordered sets. A characterization of compactness of \({\mathcal M}(P)\) is as follows: \({\mathcal M}(P)\) is compact iff every nonempty open interval of P has a finite cutset and P contains no subset isomorphic to either of the ordered sets \(P_ 1\) or \(P_ 2\). It is proved that the space \({\mathcal M}(P)\) is compact if P satisfies any of the following conditions: (1) P contains no copy of \(P_ 1\) or \(P_ 2\) and for every \(x\in P\) there is a finite subset F of P which is coinitial above x. (2) P contains no proper N and for every \(x\in P\) there is a finite maximal antichain which contains x. (3) P contains no copy of \(P_ 1\) or its dual and all antichains in P are finite.