On completion and connectedness properties of Csaszar frames

Abstract

A Cs´ asz´ ar frame is a point free version of syntopogenous space, itself a concept that is attributed to ´ Akos Cs´ asz´ ar [14]. In his two papers, Chung ([12] and [13]) characterised few types of Cs´ asz´ ar frames and extended Hong’s construction [21] to the Cauchy completions in Cs´ asz´ ar frames. From his results, we anchored objectives of our study on the actions of certain frame homomorphisms on proximal Cs´ asz´ ar frames, as well as co-reflective subcategories of Cauchy complete Cs´ asz´ ar frames. We conclude the dissertation by constructing the compactification of proximal Cs´ asz´ ar frames by applying the methods of Banaschewski and Mulvey [7]. We introduce a weak notion of connectedness of Cs´ asz´ ar frames and show, following the approach of Baboolal and Banaschewski [4], that most of the standard results on connectedness are do-able in the setting of Cs´asz'ar.