On complete partially ordered sets and compatible topologies (Q1096918)

The authors describe two complete partially ordered sets which are the intersection of complete linear orderings but which have no compatible Hausdorff topology. One is two-dimensional, while the second is countable, and leads to an example of a countable, compact, \(T_ 1\) space with a countable base which is not the continuous image of any compact Hausdorff space.