Regular relations and strictly completely regular ordered spaces

The classical theorem of Zarecki˘ı about regular relations is slightly extended and an intrinsic characterization of regularity is given. Based on the extended Zarecki˘ı theorem and the intrinsic characterization of regularity, we give a characterization of the strict complete regularity of ordered spaces by means of a certain regular relation between the closed and the open upper sets. As an application, it is shown 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.By means of regular relations we present a new proof of the strict Tychonoff embedding theorem for strictly completely regular ordered spaces.