Insertion of continuous real functions on spaces, bispaces, ordered spaces and pointfree spaces -- a common root
type: Item
instance of: scholarly article
title: Insertion of continuous real functions on spaces, bispaces, ordered spaces and pointfree spaces -- a common root
zbMATH Open document ID: 1242.06011
author: Maria João Ferreira
author: Javier Gutiérrez García
author: Jorge Picado
published in: Applied Categorical Structures
publication date: 2011-05-11T00:00:00Z
review text: A biframe is a triple consisting of a frame (called the total part of the biframe) and two subframes which together generate the total part. In the same way that a frame has its co-frame of sublocales which can be made into a frame by reverse inclusion, a biframe also has what could informally be called a ``biframe of sublocales''. For a given biframe, the authors define a real function (not necessarily continuous) on the biframe to be a real function on the first part of the ``biframe of sublocales'' which is a biframe homomorphism from the biframe of reals to the given biframe. This done, they then characterise normal and extremally disconnected biframes in terms of insertion of continuous real functions between given lower and upper semicontinuous real functions. The various insertion-type theorems (and characterisations of normality and extremal disconnectedness) for frames, topological spaces, ordered topological spaces and bispaces appear as special cases of similar results for biframes. This fully justifies the appendage ``a common root'' in the title of the paper.
reviewed by: Themba Dube
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54D15
Mathematics Subject Classification ID: 54E55
Mathematics Subject Classification ID: 54F05
zbMATH DE Number: 5886918
zbMATH Keywords: frame
zbMATH Keywords: biframe
zbMATH Keywords: bitopological space
zbMATH Keywords: ordered topological space
zbMATH Keywords: locale
zbMATH Keywords: localic real function
zbMATH Keywords: normal biframe
zbMATH Keywords: extremally disconnected biframe
MaRDI profile type: Publication
full work available at URL: s10485-009-9209-0
OpenAlex ID: W2082634990
cites work: Q4396195
cites work: BIFRAMES AND BISPACES
cites work: A separation theorem for semicontinuous functions on preordered topological spaces
cites work: Completely normal frames and real-valued functions
cites work: Monotone insertion and monotone extension of frame homomorphisms
cites work: Lower and upper regularizations of frame semicontinuous real functions
cites work: Pointfree forms of Dowker's and Michael's insertion theorems
cites work: Localic real functions: A general setting
cites work: On the algebraic representation of semicontinuity
cites work: Q3965241
cites work: On Real-Valued Functions in Topological Spaces
cites work: Bitopological Spaces
cites work: Q4180946
cites work: Constructive insertion theorems and extension theorems on extremally disconnected frames
cites work: Q4955925
cites work: Q3989734
cites work: Q5340517
cites work: A new look at localic interpolation theorems
cites work: Q3539850
cites work: Separation Theorems for Semi-Continuous Functions on Normally Ordered Topological Spaces
cites work: A note on extremal disconnectedness in bitopological spaces
cites work: Normality for biframes
cites work: Some characterizations of normal and perfectly normal spaces
cites work: Boundedness Properties in Function-Lattices
DOI: 10.1007/S10485-009-9209-0
description: scientific article; zbMATH DE number 5886918