Bulletin of the Belgian Mathematical Society - Simon Stevin

Insertion and extension results for pointfree complete regularity

Javier Gutiérrez García and Jorge Picado

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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.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 4 (2013), 675-687.

First available in Project Euclid: 22 October 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 06D22: Frames, locales {For topological questions see 54-XX}
Secondary: 54C30: Real-valued functions [See also 26-XX] 54D15: Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)

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


Gutiérrez García, Javier; Picado, Jorge. Insertion and extension results for pointfree complete regularity. Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 4, 675--687. doi:10.36045/bbms/1382448188. https://projecteuclid.org/euclid.bbms/1382448188

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