Poset models of topological spaces

We consider poset models of topological spaces and show that every T1-space has an bounded complete algebraic poset model, thus give a positive answer to a question asked in a recent paper by Waszkiewicz. It is also proved that every T1-space is homeomorphic to the maximal point space of a d-space. In the classic general topology, people are mainly interested in the spaces which satisfy at least T1 separation axiom. One possible reason for this phenomena is that in the early time, people considered only those spaces which are subspaces of Euclidean n-spaces. Another reason is that there is no meaningful natural examples of non T1 spaces, most of the existing non T1 spaces are artificially constructed. It is, probably, only until the appear of domain theory which has a deep root in computer science, people begin to be interested in T0 spaces. The most important examples of T0 spaces are the Scott spaces, defined originally for complete lattices by Dana Scott. For every complete lattice L, Scott introduced a topology, σ(L) on L, which is always T0 but not T1. Later this topology was defined for directed complete posets (dcpos), and more recently, for arbitrary posets. The Scott topology on a poset is not T1 unless the poset has the discrete order. On the first look, it seems that Scott spaces are too weak in separation to be interested by classical topologists. However, to certain extend, two results on Scott spaces proved in the past decades have change one’s views on the theoretical importance of such spaces. The first one was proved by Dana Scott in [10], which characterizes the injective T0 spaces–these as exactly the continuous lattices with their Scott topology. The second significant result was proved in [1]: every complete metric space is homeomorphic to 2000 Mathematics Subject Classification. 54E30, 06A06.