The principal aims of this paper include the introduction of the concept of $\sigma$-compact Spaces in $n^{th}$-topological spaces, the definition of compactness, discussion of various generalizations encompassing compactness in $n^{th}$-topological space, the properties of $\sigma$-compact Spaces. Furthermore, our study extends to the analysis of the notion of countable compactness in $n^{th}$-topological spaces. Our investigation extends to various generalizations of these spaces.

In this study, the locally compactness spaces property will be investigated for three topologies, known as tri-locally compact spaces. Additionally, the features of these spaces will be investigated in terms of locally compact spaces. Many theoretical results have been presented and proven, and several well-known theorems concerning locally compact spaces have been extended to apply to three topologies. An illustrative example is provided to support the findings.

In this research, we introduce and analyze the concepts of \(G\)-compactness, \(G\)-Lindel$\ddot{o}$fness, and \(G\)-countably compactness within the framework of topological spaces, which are characterized by more rigorous conditions compared to those governing compact and Lindel$\ddot{o}$f spaces. We formulate a series of theorems and present a variety of examples to clarify the relationships among \(G\)-compactness, \(G\)-Lindel$\ddot{o}$fness, compactness, and Lindel$\ddot{o}$fness. Additionally, we define the \(G\)-separation axioms utilizing $G_\delta$ sets and explore the interrelations among these concepts.

This paper introduces and investigates D-soft compact spaces, a novel generalization of compactness in soft topological spaces using D-soft covers. We establish fundamental properties, characterizations, and relationships between D-soft compactness and other forms of soft compactness. Key results include the hereditary nature of D-soft compactness under specific conditions, the equivalence between D-soft and soft compactness in soft locally indiscrete spaces, and preservation under continuous mappings. The study provides new perspectives on compactness conditions in soft topology with potential applications in decision-making and uncertainty modeling. Several examples and counterexamples illustrate the theoretical developments, demonstrating that D-soft compactness is generally stronger than soft compactness.

In this paper, we introduce and investigate the concept of nearly α-compact topological spaces as a natural generalization of α-compact and countably α-compact spaces. We establish fundamental properties and characterizations of nearly α-compact spaces, demonstrating their relationship with various topological properties including α-continuity, separation axioms, and compactness-like properties. Several equivalent conditions for nearly α-compactness are provided, and we prove that the property is preserved under certain types of mappings. The behavior of nearly α-compact spaces under topological operations such as subspaces, products, and sums is thoroughly examined. We also introduce the notion of α-nearness and investigate its connection with nearly α-compact spaces. Additionally, we provide comprehensive examples and establish new theorems that demonstrate the richness and applicability of this concept.

Several recent papers gave numerical radius inequalities for sums and products of operators which are defined on a complex separable Hilbert space H. In this paper, we prove a numerical radius inequality which generalizes and refines a recent inequality proved by Kittaneh.