Semi-θ-Continuous and Semi-θ-Open Multifunctions in Bitopological Spaces: Characterizations, Duality, and Topological Invariance
- 1Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham 44150, Thailand
- 2Department of Mathematics and Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus, Pattani 94000, Thailand
- 3Centre of Excellence in Mathematics, MHESI, Bangkok 10400, Thailand
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.
Related
- Characterizations and boundary regularity of continuous preservers of Carleson interpolating sequences
- Continuous Distributional Characterizations via a Single Characterizing Function and Lambert W Conditional Expectation Identities
- First-order approximations of multifunctions defined implicitly by "split" feasibility problems with applications to optimization
- Universal approximation of continuous maps between topological vector spaces
- Topological Characteristics for the Analysis of Vector Fields. New stable characteristics for discrete and continuous dynamical systems
BibTeX
@article{boonpok2026semiθcontinuous,
title = {{Semi-θ-Continuous and Semi-θ-Open Multifunctions in Bitopological Spaces: Characterizations, Duality, and Topological Invariance}},
author = {Chawalit Boonpok and Areeyuth Sama-Ae},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14193634},
url = {https://www.mdpi.com/2227-7390/14/19/3634},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Semi-θ-Continuous and Semi-θ-Open Multifunctions in Bitopological Spaces: Characterizations, Duality, and Topological Invariance
AU - Chawalit Boonpok
AU - Areeyuth Sama-Ae
PY - 2026
DA - 2026/10/08
JO - Mathematics
DO - 10.3390/math14193634
UR - https://www.mdpi.com/2227-7390/14/19/3634
AB - 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.
SN - 2227-7390
ER -