The reals as full and balanced biframe

This paper is the first part of a two-part investigation. It introduces full and balanced biframes which capture useful properties of the reals viewed as a biframe (or bitopological space). The subsequent paper will apply these concepts to the study of completions of quasi-nearness biframes. rt with the smallest dense quotient for biframes. Next we discuss the reals as a biframe and introduce the key ideas of balanced, full and stable biframes. The crucial tool here is the frame pseudocomplement. We include a discussion of the relations between the newly introduced ideas and regularity. Order topology biframes are all regular, normal and balanced but not necessarily full. We consider the plane and various examples related to zero-dimensionality. We provide methods of transferring fullness and balancedness from domain to codomain and conversely under various kinds of maps. ticular importance to our later study of completions is the idea of a biframe map whose right adjoint preserves the first and second parts of the biframe. We give a result providing sufficient conditions for a map to have a part-preserving right adjoint. We present an example of a dense onto map (which is in fact a compactification) between normal, regular biframes whose right adjoint is not part-preserving. The paper concludes with internal properties of full and balanced biframes showing the particularly close connection between the first and second parts and ends with a final visit to the biframe of reals.