This paper defines the so-called pairwise r-compactness in topological and bitopological spaces. In particular, several inferred properties of the r-compact spaces and their connections with other topological and bitopological spaces are studied theoretically. As a result, several novel theorems of the r-compact space are generalized on the pairwise r-compact space. The results established in this research paper are new in the field of topology.

Fuzzy set theory is a mathematical method for dealing with uncertainty and imprecision in decision-making. Some of the challenges and complexities involved in medical diagnosis can be addressed with the help of fuzzy set theory. Ovarian cancer is a disease that affects the female reproductive system's ovaries, which also make the hormones progesterone and estrogen. The ovarian cancer stages demonstrate how far the disease has spread from the ovaries to other organs. The TOPSIS technique (Technique for Order Preference by Similarity to Ideal Solution) aids in selecting the best option from a selection of choices by taking into account a number of variables. It provides a ranking or preference order after weighing the benefits and drawbacks of each solution. Intuitionistic fuzzy soft set (IFSS) is the framework to deal with the uncertain information with the help of the parameters. The goal of this article is to develop some basic aggregation operators (AOs) based on the IFSS and then use them to diagnose the stages of the ovarian cancer using the TOPSIS technique. Furthermore, the variation of the parameters used in the developed model AOs is also observed and graphically represented.

The primary goal of this research is to initiate the pairwise c-compact concept in topological and bitopological spaces. This would make us to define the concept of c-compact space with some of its generalization, and present some necessary notions such as the H-closed, the quasi compact and extremely disconnected compact spaces in topological and bitopological spaces. As a consequence, we derive numerous theoretical results that demonstrate the relations between c-separation axioms and the c-compact spaces.

The concepts of approximately Nth-Lindelöf spaces, introduced in [15] and [16], represent weaker forms of single Lindelöf spaces and have since attracted significant attention. In this paper, we revisit these spaces through the lens of a particular type of cover, known as a normal cover. The primary objective is to explore the properties of a new generalization, termed nearly nth-Lindelöf spaces.

The main aim of this paper is to combine the connections between graph theory and rough set theory. We created graph ideal ASs by proposing the interior and closure operators, utilizing the concept of graph ideals. With the help of various graphical examples, we applied some topological concepts to the induced graph ideal ASs, including the subspace, continuous functions, lower separation axioms, and connectedness. However, simple directed graphs with or without loops are the ones that are discussed throughout the study. The obtained results are valid for any type of graph: multi-graphs or simple graphs, connected or disconnected graphs, with loops or without loops, and undirected or directed graphs.