Topological Characterizations of Posets

We find out suitable conditions on a To-principal topology, under which the associated partial order is a partial order with nontransitive incomparability, that is an interval order, a partial semiorder or a semiorder. In order to perform these characterizations, only a T 1 separation axiom is needed. The settheoretical approach allows us to give a simple proof of a fundamental theorem due to Fish burn, concerning the numerical representation of interval orders. We also introduce a class of planar interval orders, called strong interval orders. Although planar posets, as well as interval orders, have arbitrary finite dimension, we prove that a strong interval order is the intersection of at most two linear orders.