Any space has a continuous poset model

In this paper we prove that any T 1 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 T 1 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 T 1 topological space ( X , τ ) there is a T 1 topology τ ∗ on X such that ( X , τ , τ ∗ ) is pairwise completely regular.