On functions connected with absolute Dedekind completion and divisible envelope (Q1103224)

The author presents direct proofs of his earlier results published in Usp. Mat. Nauk 35, No.3, 153-157 (1980; Zbl 0439.54016) and Tr. Mosk. Mat. O.-va. 45, 68-104 (1982; Zbl 0547.46011)]. A real-valued function f on a space X is called quasicontinuous if for every natural n there exists a dense open set U and an open covering \(\{G_ i\}\) of U such that the oscillation of f on any set \(G_ i\) is less than 1/n. The family of all the bounded quasi-continuous functions is denoted by Q *(X). Functions f,g\(\in Q\) *(X) are called equivalent (f\(\sim g)\) if for every n there exists a nowhere dense set R such that \(| f(x)- g(x)| <1/n\) for all \(x\not\in R\). \(Q\) \(*_ r(X)\) is a set of equivalence classes. The main theorem: Let \(\alpha\) T be the absolute of T and \(\tau\) : \(\alpha\) \(T\to T\) be the canonical mapping. Then (a) there exists a bijection \(\phi\) : Q\({}\) \(*_ r(T)\to C\) *(\(\alpha\) T) such that for each \(\bar x\in Q\) \(*_ r(T)\phi\) (x)\(\sim \bar x\circ \tau\); (b) \(\alpha\) T as a perfect preimage of T is completely determined by the property (a).