Generalized Quasi-Continuities of Multifunctions on Bitopological Spaces

The aim of the article is to introduce a few variants of the generalized quasi-continuity of multifunctions defined on a bitopological space and to study their mutual relationship. The results known for functions are extended to multifunctions, which provide a wider range of relationships, mainly in terms of upper and lower semi-continuities and corresponding continuities with respect to a dual bitopology. The proof procedures are based on a notion of the pseudo-refinement of two topologies and the Baire property in a bitopological space. A characterization of some continuities depending on two topologies by continuities depending only on one topology and the structure of the sets of semi-discontinuity points are given. The equivalence between the upper and lower Baire continuity and upper and lower quasi-continuity (upper and lower continuity with respect to ideal topology) of compact-valued multifunction from a Baire space into a regular one has an important position. The end of this article is dedicated to several interpretations that facilitate and clarify orientation in the achieved results.