Proximity Biframes and Nachbin Spaces

Abstract

In our previous paper, in order to develop the pointfree theory of compactifications of ordered spaces, we introduced the concept of a proximity on a biframe as a generalization of the concept of a strong inclusion on a biframe. As a natural next step, we introduce the concept of a proximity morphism between proximity biframes. Like in the case of de Vries algebras and proximity frames, we show that the proximity biframes and proximity morphisms between them form a category PrBFrm in which composition is not function composition. We prove that the category KRBFrm of compact regular biframes and biframe homomorphisms is a proper full subcategory of PrBFrm that is equivalent to PrBFrm. We also show that PrBFrm is equivalent to the category PrFrm of proximity frames, and give a simple description of the concept of regularization using the language of proximity biframes. Finally, we describe the dual equivalence of PrBFrm and the category Nach of Nachbin spaces, which provides a direct way to construct compactifications of ordered spaces.