Regular relations and strictly completely regular ordered spaces.
Zareckiĭ's theorem about regular relations is slightly generalized and an intrinsic characterization of regularity of relations is obtained. With the help of these results the authors characterize strict complete regularity of partially ordered topological spaces by means of a certain regular relation between the closed and the open upper sets. As an application they conclude that a quasicontinuous domain endowed with the Lawson topology is strictly completely regular provided that the Lawson-open lower sets are contained in the lower topology. Finally they present a new proof of the strict Tychonoff embedding theorem for strictly completely regular ordered spaces which makes use of regular relations. Reviewer's remark: A binary relation \(\rho:X\rightharpoonup Y\) is called regular if \(\exists \sigma: Y\rightharpoonup X\) such that \(\rho\circ \sigma \circ \rho =\rho.\) Zareckiĭ's theorem says that a binary relation \(\rho\) on a set \(X\) is regular if and only if the complete lattice of all images of \(\rho\) under the order of set inclusion is completely distributive.
scientific article; zbMATH DE number 2037224