Relative compactness, cotopology and some other notions from the bitopological point of view

The paper consists of two sections. Section 1 is the introduction which, in addition to the auxiliary information, contains some interesting results on Baire-like properties. Section 2 deals with the bitopological essence of the notions of relative compactness and cotopology in general topology, C-relation, subordination of topologies and closed neighborhoods condition in analysis. A generalization of Choquetʹs theorem on Baire spaces is given and the sufficient conditions for families of (i,j)-nowhere dense sets to coincide with families of (i,j)-first category sets are established using a finite measure. A bitopological solution of one of Ulamʹs problems is obtained. The corresponding relations are almost always studied using essentially the bitopological modifications of regularity, which, as seen in various problems of general topology, analysis and potential theory, are the most natural forms of relations of two topologies defined on the same set.