Representing topologies using partially ordered semigroups

Abstract

Two common ways of using the partially ordered semigroup structure of the reals to model topological spaces are:Defining a distance into IR and using the balls of positive radius about a point as its basic neighborhoods, andSeeing if the space is a subspace of IR with its interval topology or its upper topology.We show that many po-semigroups can be used in place of IR and in fact:Every topological space is induced by a quasimetric into and set of positives in some po-semigroup, and is also a subspace of a po-semigroup with its upper topology.Also, the following are equivalent for any topological space:It is completely regular.It is induced by a pseudometric into and set of positives in some po-semigroup.It is a subspace of some po-semigroup with set of positives in their induced interval topology.