Compact compatible topologies for posets and graphs (Q1272943)

The connections between the theory of posets and aspects of general topology are both ancient as far as the subject is concerned and fundamental since topologies like diagrams can be used to characterize posets and their properties quite completely. This paper deals with topologies on the vertex set of the comparability graph of a poset and in particular with those (compatible) topologies such that connectedness in the topological sense correlates with connectedness in the graph theoretic sense for certain types of subsets noting that in the case of weak compatibility the structure of the comparability graph is completely determined. From the technical material special results for classes of posets including trees are derived which permit decision concerning the question of whether suitably compatible topologies exist. This useful paper follows a collection of articles of the same nature and in a sense provides a culmination of the approach outlined while still leading to research questions of interest for future work in this particular area.