On functions connected with sequential absolute, Cantor completion and classical ring of quotients
type: Item
instance of: scholarly article
title: On functions connected with sequential absolute, Cantor completion and classical ring of quotients
zbMATH Open document ID: 0671.54011
DOI: 10.1007/BF01848057
published in: Periodica Mathematica Hungarica
publication date: 1988-01-01T00:00:00Z
review text: The author gives a complete set of proofs of the previously announced solutions of the three functional characterization problems asking for functional descriptions of: the Cantor completion of the vector lattice C of all bounded continuous real-valued functions on a completely regular space X; the uniform completion of the bounded part of the classical ring of quotients of the ring C (called the Fine-Gillman-Lambek extension of C); and the sequential absolute of X. The three solutions involve a common set of functions, K, on X called the classically quasi-continuous functions (actually K is the set of equivalence classes of such functions under a certain equivalence relation). In particular C can be canonically embedded into the vector-lattice K, and a proof is given that this embedding is the Cantor completion of C. This embedding is also proved to be isomorphic to the Fine-Gillman-Lambek extension of C. In addition, the sequential absolute of X is shown to be that perfect (cozero-preserving) preimage S of X which is uniquely determined by the existence of a certain bijection between K and a space of bounded real-valued functions on S.
Mathematics Subject Classification ID: 54C40
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54E99
Mathematics Subject Classification ID: 46E99
Mathematics Subject Classification ID: 46A40
zbMATH DE Number: 4098269
zbMATH Keywords: Cantor completion
zbMATH Keywords: vector lattice C of all bounded continuous real-valued functions
zbMATH Keywords: uniform completion
zbMATH Keywords: Fine-Gillman-Lambek extension
zbMATH Keywords: sequential absolute
zbMATH Keywords: classically quasi-continuous functions
author: V. K. Zakharov
reviewed by: R. A. McCoy
MaRDI profile type: Publication
cites work: Q5835185
cites work: Order-Cauchy Completions of Rings and Vector Lattices of Continuous Functions
cites work: Sequence completion of lattice moduls
cites work: Q5519434
cites work: Q5509706
cites work: Q5519166
cites work: Order convergence and topological completion of commutative lattice-groups
description: scientific article; zbMATH DE number 4098269