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Due to the widespread use of digital images of real-world objects as mathematical models, this research examines the freezing sets invariant-base properties of digital images. In contrast to earlier studies that only covered a discrete or limited collection of points, fixed points of digitally continuous functions are approved to deal with a variety of characteristics of digital images.

Let R(+)N be the idealization of the ring R by the R-module N. In this paper, we investigate when Γ(R(+)N) is a Planar graph where R is an integral domain and we investigate when Γ(Zn(+)Zm) is a Planar graph.

In this paper, we have constructed a sequence of soft points in one soft set with respect to a fixed soft point of another soft set. The convergence and boundedness of these sequences in soft ∆-metric spaces are defined and their properties are established. Further, the complete soft ∆-metric spaces are introduced by defining soft ∆-Cauchy sequences.

The main purposes of this article is to introduce new generalizations of the notion of pairwise Lindelöf spaces in bitopological spaces where new notions: pairwise strongly Lindelöf, pairwise nearly, pairwise almost and pairwise weakly strongly Lindelöf bitopological spaces depend on the new notion pairwise preopen countable covers. These covers where we focused on their importance in topology consist of countable subfamilies whose closures cover the bitopological spaces and we clarified how pairwise preopen countable covers effect on pairwise strongly Lindelöf spaces. The new concepts of pairwise strongly Lindelöf, pairwise nearly, pairwise almost and pairwise weakly strongly Lindelöf bitopological spaces are introduced and many definitions, propositions, characterizations and remarks concerning those notions are initiated, discussed and explored. Furthermore, the relationships between those bitopological spaces are examined and investigated. We illustrated the implications hold by these new bitopological spaces. We put some queries and claims, then we struggle to provide their proofs.