On Soft Bitopological Ordered Spaces

Abstract

This paper introduces soft bitopological ordered spaces, combining soft topological spaces with partial order relations. The authors extensively investigate increasing, decreasing, and balancing pairwise open and closed soft sets, analyzing their properties. They prove that the collection of increasing (decreasing) open soft sets forms an increasing (decreasing) soft topology. The paper thoroughly examines increasing and decreasing pairwise soft closure and interior operators. Notably, it introduces bi−ordered soft separation axioms, denoted as PSTi(PST∙i,PST∗i,PST∗∗i)− ordered spaces, i=0,1,2, showcasing their interrelationships through examples. It explores separation axiom distinctions in bitopological ordered spaces, referencing relevant literature such as the work of El-Shafei et al. [5]. The paper investigates new types of regularity and normality in soft bitopological ordered spaces and their connections to other properties. Importantly, it establishes the equivalence of three properties for a soft bitopological ordered space satisfying the conditions of being TP∗ -soft regularly ordered: PST2−ordered, PST1-ordered, and PST0-ordered. It introduces the concept of a bi−ordered subspace and explores its hereditary property. The authors define soft bitopological ordered properties using ordered embedding soft homeomorphism maps and verify their applicability for different types of PSTi−ordered spaces, i=0,1,2. Finally, the paper identifies the properties of being a TP∗;(PP∗)− soft T3−ordered space and a TP-soft T4-ordered space as a soft bitopological ordered property.