Insertion and extension results for pointfree complete regularity

Abstract

There are insertion-type characterizations in pointfree topology that extend well known insertion theorems in point-set topology for all relevant higher separation axioms with one notable exception: complete regularity. In this paper we fill this gap. The situation reveals to be an interesting and peculiar one: contrarily to what happens with all the other higher separation axioms, the extension to the pointfree setting of the classical insertion result for completely regular spaces characterizes a formally weaker class of frames introduced in this paper (called \emph{completely c-regular frames}). The fact that any compact sublocale (quotient) of a completely regular frame is a $C$-sublocale ($C$-quotient) is obtained as a corollary.