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Dcpo models of T 1 spaces

A poset model of a topological space X is a poset P together with a homeomorphism φ : X−→Max(P ) (Max(P ) is the subspace of the Scott space ΣP consisting of maximal points of P ). In [11] (also in [2]), it was proved that every T1 space has a bounded complete algebraic poset model. It is, however still unclear whether each T1 space has a dcpo model. In this paper we give a positive answer to this problem. In section 1, we show that every T1 space has a dcpo model. In section 2, we prove that a T1 space is sober if and only if its dcpo model constructed in section 1 is a sober dcpo. These results provide us with a method to construct non-sober dcpos from any non-sober T1 spaces. In section 3, for some special spaces we construct a more concrete dcpo model.