There is a lot of ambiguous information in the real world that crisp values can't manage. The fuzzy set theory was proposed by Zadeh. It is an age-old and excellent tool for dealing with uncertain information. As a result, intuitionistic fuzzy set theory was suggested. However, these theories are incapable of dealing with all forms of uncertainty, such as indeterminate and inconsistent data in various decision-making situations. To address this shortfall, Smarandache proposed the neutrosophic set theory by introducing a degree of indeterminacy as an independent component. In this current decade, neutrosophic environments are mainly interested by different fields of researchers. Recently, Smarandache introduced theNeutroAlgebra and AntiAlgebras. NeutroAlgebras and AntiAlgebras represent a new research subject that is based on real-world examples. In this chapter, some properties of NeutroTopological space are introduced and studied with examples. Several definitions of NeutroInterior, NeutroClosure, and NeutroBoundary are defined, and the authors also studied its properties with examples.
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The concept of a fuzzy set was first developed by Zadeh (1965), where the concept of the membership function is defined and discussed the concept of uncertainty using a fuzzy set. Atanassov (1986) proposed the intuitionistic fuzzy set by generalizing the concept of fuzzy sets and introducing the degree of non-membership as a component. Chang (1968) was the first to introduce fuzzy topology, while Coker (1997) defined intuitionistic fuzzy topological space. Salama et al. (2014) investigated topology using neutrosophic sets. Kelly (1963) introduced the concept of bitopological space. The concept of fuzzy bitopological space was investigated by Kandil et al. (1995). Florentin Smarandache (1998) defined the idea of neutrosophic logic and the concept of neutrosophic set. After that, the concepts of the neutrosophic set have been applied in many branches of sciences and technology. The concept of neutrosophic topological space was introduced by Salama and Alblowi (2012). Devi et al. (2017) discussed separation axioms in ordered neutrosophic bitopological space. Mwchahary et al. (2020) did their work in neutrosophic bitopological space. After defining the neutrosophic group, Sumathi et al. (2016) defined the concept of the topological group structure of the neutrosophic set and also, Sumathi and Arockiarani (2015) studied the fuzzy neutrosophic group.
In recent years, there has been a surge in academic interest in neutrosophic set theory. The concept of neutro-structures and anti-structures was first defined by Florentin Smarandache (2019, 2020). Şahin et al. (2021) discussed the idea of neutro-topological space and anti-topological space. Smarandache (2020) studied NeutroAlgebra as a generalization of partial algebra. Agboola (2020) investigated the idea of NeutroRings, NeutroGroups, and finite NeutroGroups of type-NG. Smarandache (2020) proposed the generalizations and alternatives of Classical Algebraic Structures to NeutroAlgebraic Structures and AntiAlgebraic Structures. Al-Tahan et al. (2021) studied the NeutroOrderedAlgebra, NeutroHyper structures, and their properties.
Blizard (1989) investigated the concept of multiset theory and neutrosophic multi groups and applications were studied by Bakbak (2019). Basumatary et al. (2020) studied on interval-valued triangular neutrosophic linear programming problem also Basumatary et al. (2021) investigated some properties of the neutrosophic multi topological group. Yager (1986) introduced the concept of fuzzy multiset (fuzzy bag) to generalize the fuzzy set, and Miyamoto (2001) studied Fuzzy Multisets and Their Generalizations. A fuzzy multiset element can appear more than once, with the same or different membership values. Onasanya et al. (2018) investigated the Algebraic properties of alpha-level subsets topology of a fuzzy subset, and the authors Onasanya et al. (2019) and Al Tahan et al. (2020) studied fuzzy multi-polygroups. Al Tahan et al. (2019) studied the fuzzy multi-Hv-ideals of Hv-rings fuzzy multi-Hv-ideals and their properties. Sebastian et al. (2011) investigated a novel type of fuzzy set (fuzzy multiset). The concept of multi-fuzzy complex numbers and multi-fuzzy complex sets was initially introduced by Dey et al. (2015). Yong et al. (2013) recently suggested the notion of the multi-fuzzy soft set for its application to decision making, which is a more broad fuzzy soft set. Basumatary (2016) investigated the concept of fuzzy closure and the fuzzy boundary based on reference function.
Smarandache (2019, 2020) founded and studied the concept of neutro-structures and anti-structures. From the concepts of NeutroAlgebra, he showed that if a statement (theorem, lemma, consequence, property, etc.) is totally true in a classical Algebra, it does not mean that it is also totally true in a NeutroAlgebra or an AntiAlgebra. It depends on the operations and axioms (if they are totally true, partially true, totally false, or partially or totally indeterminate) it is based upon.