Any T1 space has a continuous poset model

In this paper we prove that any T1 subspace of a continuous dcpo with the relative Scott topology can be “modeled” by a continuous poset. Using this result we are able to show that any T1 topological space (X,τ) is homeomorphic to the space of maximal elements of a continuous poset. We also find a bitopological characterization of a topological space (X,τ) that can be modeled by a continuous poset. It is proved that for any T1 topological space (X,τ) there is a T1 topology τ∗ on X such that (X,τ,τ∗) is pairwise completely regular.