Bounded subsets and Grothendieck's theorem for bispaces (Q2705849)

The authors study several kinds of bounded subsets in a bitopological space. In particulur, both the classical Hewitt characterizations of pseudocompactness and other well-known characterizations of these spaces due to Glicksberg and Colmez are generalized and extended. For example, the notions of 2-boundedness, doubly \(C\)-compactness and 2-\(C\)-compactness are introduced and their relations with \(C\)-compactness are established. As a nice application of their methods and results the authors obtain a characterization of the \(T_0\) topological spaces for which every lower semicontinuous function is bounded. Moreover, they characterize several kinds of bounded subsets in quasi-pseudometrizable bispaces and study thoroughly boundedness properties of some interesting quasi pseudometric spaces which appear in the field of theoretical computer science. Finally, the authors devote the last section to obtain a version of a generalization of the classical Grothendieck Theorem due to Asanov and Velichko in the setting of bispaces.