The aim of this study was to originate six types of generalized compactness and Lindelöfness in the frame of supra soft topological spaces (or SSTSs) based on the approaches of supra soft somewhere dense sets (or SS-sd-sets) and the SS-sd-closure operator, named SS-sd-almost compact (Lindelöf) spaces, SS-sd-approximately compact (Lindelöf) spaces and SS-sd-mildly compact (Lindelöf) spaces. The essential properties of each type of the aforementioned notions were studied. Specifically, we studied the invariance of these notions under specific types of soft mappings. Moreover, the relationships among these notions and between corresponding notions were discussed. Furthermore, the equivalence among them was proved under the SS-sd-partition condition. Finally, we provided a diagram to summarize these relationships with the support of concrete counterexamples.

In the fields of mathematics and information sciences, binary relations are vital. Fuzzy sets (FSs), rough sets (RSs), and soft sets (SSs) are mathematical strategies that effectively handle ambiguous and imprecise data in practical situations. This work presents various properties of the roughness of fuzzy sets regarding foresets (F-sets) and aftersets (A-sets) using soft binary relations (SBRs). Initially, two pairs of fuzzy soft sets (FSSs) are obtained by approximating a fuzzy subset using an SBR, and their distinctive axiomatic systems are explored. Additionally, two types of fuzzy topologies that result from soft reflexive relations (SRRs) are examined. Numerous similarity relations allied with SBRs are also investigated. In addition, we present the accuracy measure and roughness measure for a fuzzy subset based on the mass assignment of the fuzzy subset through soft relations. Next, we outline a decision-making (DM) approach within the context of the proposed method. In addition, we provide two algorithms and decision phases. Ultimately, an applied example is used to evaluate the reliability of the decision processes. An extensive comparison study confirms the proposed method's feasibility and superiority over current DM methods.

This project aimed to introduce the notion of supra soft ${\delta }_i$-open sets in supra soft topological spaces. Also, we declared the differences between the new concept and other old generalizations. We presented new operators such as supra soft ${\delta }_i$-interior, supra soft ${\delta }_i$-closure, supra soft ${\delta }_i$-boundary and supra soft ${\delta }_i$-cluster. We found out many deviations to our new operators; to name a few: If $in{t}_{{\delta }_i}^s(F,E)=(F,E)$, then it doesn't imply that $(F,E)\in SO{S}_{{\delta }_i}(X)$. Furthermore, we applied this notion to define new kinds of mappings, like supra soft ${\delta }_i$-continuous mappings, supra soft ${\delta }_i$-irresolute mappings, supra soft ${\delta }_i$-open mappings and supra soft ${\delta }_i$-closed mappings. We studied their main properties in special to distinguish between our new notions and the previous generalizations. It has been pointed out in this work that many famous previous studies have been investigated here; in fact, I believe that this is an extra justification for the work included in this manuscript.

Our purpose in this work is to present a new generalized soft open set in supra soft topological spaces, named supra soft sd-sets. With deep discussion, we found out that they contain almost all kinds of weaker supra soft open sets which have been discussed in earlier studies, as shown in the following figure.

So, directly we can notice the value of the introduced results. Also, the notion of a supra soft sc-set is presented, and many of its basic properties are explored. Furthermore, we show that the new family fails to form soft topology or supra soft topology. In addition, the definitions of the supra soft sd-closure operator, supra soft sd-cluster operator, and supra soft sd- interior operator are introduced, and many of their interesting properties are explored. Finally, we prove that the property of being a supra soft sd-set is a supra soft topological property.

In this work, we introduce a very wide category of open sets in topological spaces, called $\mathrm{\aleph }$-open sets. We study the category of $\mathrm{\aleph }$-open sets that contains $\beta$-open sets in addition to ${\beta }^{\ast }$-open and $e^{\ast }$-open sets. We present the essential properties of this class and disclose its relationships with many different classes of open sets with the help of concrete counterexamples. In addition, we introduce the $\mathrm{\aleph }$-interior and $\mathrm{\aleph }$-closure operators. Moreover, we study the concept of $\mathrm{\aleph }$-continuity of functions inspired by the classes of $\mathrm{\aleph }$-open and $\mathrm{\aleph }$-closed sets. Also, we discuss some kinds of separation axioms and some theorems related to the graph of functions.