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This article explores the interconnections among the single-valued neutrosophic grill, single-valued neutrosophic primal and their stratification, uncovering their fundamental characteristics and correlated findings. By introducing the notion of a single-valued neutrosophic primal, a broader framework including the fuzzy primal and intuitionistic fuzzy primal is established. Additionally, the concept of a single-valued neutrosophic open local function for a single-valued neutrosophic topological space is presented. We introduce an operator based on a single-valued neutrosophic primal, illustrating that the single-valued neutrosophic primal topology is finer than the single-valued neutrosophic topology. Lastly, the concept of single-valued neutrosophic open compatibility between the single-valued neutrosophic primal and single-valued neutrosophic topologies is introduced, along with the establishment of several equivalent conditions related to this notion. [ABSTRACT FROM AUTHOR]

This article aims to present new terms of single-valued neutrosophic notions in the Sostak sense, known as single-valued neutrosophic regularity spaces. Concepts such asr-single-valued neutrosophic semi £-open,r-single-valued neutrosophic pre-£-open, r-single valued neutrosophic regular-£-open and r-single valued neutrosophic α£-open are defined and their properties are studied as well as the relationship between them. Moreover, we introduce the concept of r-single valued neutrosophic θ£-cluster point and r-single-valued neutrosophic γ £-cluster point, r-θ£- closed, and θ£-closure operators and study some of their properties. Also, we present and investigate the notions of r-single-valued neutrosophic θ£-connectedness and r-single valued neutrosophic δ£-connectedness and investigate relationship with single-valued neutrosophic almost £-regular. We compare all these forms of connectedness and investigate their properties in single-valued neutrosophic semiregular and single-valued neutrosophic almost regular in neutrosophic ideal topological spaces in Sostak sense. The usefulness of these concepts are incorporated to multiple attribute groups of comparison within the connectedness and separateness of θ£ and δ£. [ABSTRACT FROM AUTHOR]

In the current paper, particular acheivments of single-valued neutrosophic continuity on a singlevalued neutrosophic topological space (𝔗 ̃, 𝜏̃𝛾̃, 𝜏̃𝜂̃, 𝜏̃𝜇̃) are introduced. Some necessary implications between them are illustrated. The theories of r-single-valued neutrosophic compact, r-single-valued neutrosophic ideal compact, r-single-valued neutrosophic quasi H-closed and r-single-valued neutrosophic compact modulo an single-valued neutrosophic ideal I~ are presented and investigated. [ABSTRACT FROM AUTHOR]

In this paper, a new class of fuzzy ideal sets, namely the r - (τ i , τ j) - θ -generalized fuzzy ideal closed sets, is introduced for fuzzy bitopological spaces in Šostak sense. This class falls strictly in between the class of r - (τ i , τ j) - θ -fuzzy ideal closed sets and the class of r - (τ i , τ j) -generalized fuzzy ideal closed sets. Furthermore, by using the class of r - (τ i , τ j) - θ -generalized fuzzy ideal closed sets we establish a new fuzzy closure operator which generates fuzzy bitopological spaces in Šostak sense. Finally, the (i , j) strongly- θ -fuzzy ideal continuous, (i , j) - θ -generalized fuzzy ideal continuous and (i , j) - θ -generalized fuzzy ideal irresolute mappings are introduced, and we show the (i , j) - θ -generalized fuzzy ideal continuous properly fuzzy ideal bitopological spaces in Šostak sense (for short, fibtss) in between (j , i) strongly- θ -fuzzy ideal continuous and (i , j) -generalized fuzzy continuous mappings. [ABSTRACT FROM AUTHOR]

This paper aims to introduce the notion of r-single-valued neutrosophic connected sets in single-valued neutrosophic topological spaces, which is considered as a generalization of r-connected sets in Šostak's sense and r-connected sets in intuitionistic fuzzy topological spaces. In addition, it introduces the concept of r-single-valued neutrosophic separated and obtains some of its basic properties. It also tries to show that every r-single-valued neutrosophic component in single-valued neutrosophic topological spaces is an r-single-valued neutrosophic component in the stratification of it. Finally, for the purpose of symmetry, it defines the so-called single-valued neutrosophic relations. [ABSTRACT FROM AUTHOR]

Neutrosophy is a recent section of philosophy. It was initiated in 1980 by Smarandache. It was presented as the study of origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. In this paper, we introduce the notion of single-valued neutrosophic ideals sets in Šostak's sense, which is considered as a generalization of fuzzy ideals in Šostak's sense and intuitionistic fuzzy ideals. The concept of single-valued neutrosophic ideal open local function is also introduced for a single-valued neutrosophic topological space. The basic structure, especially a basis for such generated single-valued neutrosophic topologies and several relations between different single-valued neutrosophic ideals and single-valued neutrosophic topologies, are also studied here. Finally, for the purpose of symmetry, we also define the so-called single-valued neutrosophic relations. [ABSTRACT FROM AUTHOR]