On functions connected with sequential absolute, Cantor completion and classical ring of quotients (Q1119924)

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.