Insertion and extension results for pointfree complete regularity

• Main Author: Gutiérrez García, Javier
• Publication Date: 2013
• Other Authors: Picado, Jorge
• Format: Article
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: https://hdl.handle.net/10316/89466; https://doi.org/10.36045/bbms/1382448188
• topic: Frame, locale, sublocale, completely separated sublocales, compact sublocale, compact-like real function, complete regular frame, upper semicontinuous, lower semicontinuous, insertion, insertion theorem, C-embedding, C∗-embedding

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 weakerclass of frames introduced in this paper (called 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.