Topological Spaces for Cpos

Informally we show why a domain with a reasonable collection of properties is a topological space. We then address the question: if one wishes to do programming language semantics in a category of topological spaces instead of the category of complete partial orders (cpos) which category should be used? This question is first considered with respect to objects, i.e., replacing a cpo by a topological space(s), and then with respect to objects and morphisms, i.e., replacing cpos and partial order continuous functions. With respect to objects it is shown that for each cpo there is a complete lattice of topologies which may be used. However, with respect to objects and morphisms it is shown that in general only the Scott topologies with their associated topologically continuous functions may be used. This latter answer is confirmed categorically by the existence of an adjunction and a Galois connection of the third kind between the category of cpos and the category of order consistent topologies where the image of the category of cpos is the subcategory of Scott topologies.