Partial Order Structures and Topological Spaces

Abstract

In the work ahead we will be interested in Boolean algebras that are associated with certain partial order structures (Definition 5.4) and Boolean algebras of regular open sets of certain topological spaces. Quite often we find that the Boolean algebra associated with a particular partial order structure is the same algebra as that of the regular open sets of a certain topological space even though there appears to be no connection between the partial order structure and the topological space. In this section we will establish such a connection. For a given partial order structure we will define a topological space of ultrafilters for the partial order structure (Definitions 5.2, 5.3, and 5.6). We will show that in general this topological space is a T₁-space (Theorem 5.7). If, however, the partial order structure is one associated with a Boolean algebra, then the topological space is in fact Hausdorff (Theorem 5.8).