Semi-θ-Continuous and Semi-θ-Open Multifunctions in Bitopological Spaces: Characterizations, Duality, and Topological Invariance

Source abstract

In this paper, we introduce and systematically investigate upper and lower semi-θ-continuous multifunctions and semi-θ-open multifunctions in the setting of bitopological spaces. The central concept underlying our approach is that of semi-ϕθ-open sets, defined via the θ-closure and θ-interior operators associated with the two topologies. We establish pointwise and global characterizations of upper semi-θ-continuity in terms of the semi-θ-interior and semi-θ-closure operators together with the upper and lower inverse mappings. Analogous characterizations are obtained for lower semi-θ-continuity, and we demonstrate through explicit examples that upper and lower semi-θ-continuity are independent notions, neither implying the other. We then introduce the dual concept of semi-θ-open multifunctions and derive several equivalent global characterizations, including the analogue of the classical result that a function is open precisely when the image of every open set is open. The interplay between upper semi-θ-continuity and semi-θ-openness is examined: the two notions are shown to be independent for multifunctions, while for bijective single-valued functions they coincide via the inverse mapping. Finally, we extend classical compactness, connectedness, and Hausdorffness to the semi-θ setting, establish their preservation and reflection properties under the relevant classes of multifunctions, and show via counterexamples that the multifunction analogues of certain single-valued results fail in the absence of disjoint-valuedness. The results unify and extend several known characterizations in the literature.