Domain Theory as a Tool for Topology - a Case Study

In this paper we adopt a certain view on continuous posets and see them as models of their spaces of maximal elements, which are most often topologies rich in structure. Adopting this perspective seems to be fruitful: we are often able to match structural properties of the modelling poset to properties of the modelled space. It was discovered by Mike Reed and Keye Martin two years ago that existence of a measurement on the model corresponds to existence of a development for the modelled topological space. We present an elementary proof of this fact and show how one can use this result to give a new proof to one of the first metrization theorems in Topology.