Neutrosophic Bipolar Vague Resolvable and Neutrosophic Bipolar Vague Irresolvable Spaces

In this paper the concepts of neutrosophic bipolar vague resolvable, neutrosophic bipolar vague irresolvable, neutrosophic bipolar vague open hereditarily irresolvable spaces, maximally neutrosophic bipolar vague irresolvable spaces and neutrosophic bipolar vague submaximal spaces are introduced. Also we study several interesting properties of the neutrosophic bipolar vague open hereditarily irresolvable spaces besides giving characterization of these spaces by means of somewhat neutrosophic bipolar vague continuous and somewhat neutrosophic bipolar vague open functions.


Introduction
The fuzzy concept had invaded almost all branches of mathematics ever since the introduction of fuzzy sets by L. A. Zadeh [17]. The theory of fuzzy topological space was introduced and developed by C. L. Chang [3] and since then various notions in classical topology have been extended to fuzzy topological space. In 1995, the denition of Smarandaches neutrosophic set, neutrosophic sets and neutrosophic logic have been useful in many real applications to handle improbability. Neutrosophy is a branch of philosophy which studies the source, nature and scope of neutralities, as well as their interactions with different ideational scales [14]. The neutrosophic set uses one single value to indicate the truth-membership grade, ISSN: 2456-8686, 5(1), 2021:076-087 https://doi.org/10.26524/cm94 indeterminacy-membership degree and falsity membership grade of an element in the universe X. The conception of Neutrosophic Topological space was introduced by A.A.Salama and S.A.Alblowi [11]. Bipolar-valued fuzzy sets, which was introduced by Lee [8,9] is an extension of fuzzy sets whose membership degree range is extended from the interval [0, 1] to [-1, 1]. The membership degrees of the Bipolar valued fuzzy sets signify the degree of satisfaction to the property analogous to a fuzzy set and its counter-property in a bipolar valued fuzzy set, if the membership degree is 0 it means that the elements are unrelated to the corresponding property. Furthermore if the membership degree is on (0, 1] it indicates that the elements somewhat fulfil the property, and if the membership degree is on [-1,0) it indicates that elements somewhat satisfy the entire counter property. After that, Deli et al. [4] announced the concept of bipolar neutrosophic sets, as an extension lead of neutrosophic sets. Neutrosophic vague set is a combination of neutrosophic set and vague set which was well-defined by Shawkat Alkhazaleh [13]. Neutrosophic vague theory is a useful tool to practise incomplete, indeterminate and inconsistent information. Satham Hussain [12] introduced Neutrosophic bipolar vague sets. Mohana K and Princy R [10] have introduced Neutrosophic Bipolar Vague sets in topological spaces. The concepts of resolvability and irresolvability in topological spaces was introduced by E. Hewit [7]. The concept of open hereditarily irresolvable spaces in the classical topology was introduced by A. Gelikin [6]. The concept of fuzzy resolvable and irresolvable spaces was introduced by G. Thangaraj and G. Balasubramanian [16]. In this paper the concept of neutrosophic bipolar vague resolvable, neutrosophic bipolar vague irresolvable, neutrosophic bipolar vague open hereditarily irresolvable spaces and maximally neutrosophic bipolar vague irresolvable spaces are introduced. 1,0] are the two neutrosophic bipolar vague sets then their union, intersection and complement are well-defined as follows:

Preliminaries
Definition 2.2 [10] Suppose A and B be two neutrosophic bipolar vague sets defined over a universe of disclosure X. We say that A ⊆ B if and only if

Definition 2.3 [10]
A neutrosophic bipolar vague topology (in short NBVT)on a non-empty set X is a family τ of Neutrosophic bipolar vague set in X sustaining the following axioms: 1. 0, 1 ∈ τ . 2. G 1 ∩ G 2 ∈ τ for any G 1 , G 2 ∈ τ . 3. ∪G i ∈ τ for any arbitrary family {G i : G i ∈ τ, i ∈ I}. Under such case the pair (X,τ ) is known as the neutrosophic bipolar vague topological space (In short NBVTS) and any NBVS in (X,τ ) is known as bipolar vague open set (In short NBVOS) in X . The complement of a neutrosophic bipolar vague open set A c in a neutrosophic bipolar vague topological space (X, τ ) is referred as a neutrosophic bipolar vague closed set(In short NBVCS) in (X, τ ).

Neutrosophic Bipolar Vague Resolvable and Irresolvable
Hence there exists a neutrosophic bipolar vague dense set B in (X,T), such that N BV int(B c ) = 0 ∼ . Hence the neutrosophic bipolar vague topological space (X,T) is called a neutrosophic bipolar vague resolvable.
Which is a contradiction. Hence (X,T) has a pair of neutrosophic bipolar vague dense set A 1 and A 2 such that A 1 ⊆ A c 2 . Converse, suppose that the neutrosophic bipolar vague topological space (X,T) has a pair of neutrosophic bipolar vague dense sets A 1 and A 2 , such that A 1 ⊆ A c 2 . Suppose that (X,T) is a neutrosophic bipolar vague irresolvable space. Then for all neutrosophic bipolar vague dense set A 1 and A 2 in (X,T), we have N BV cl(A c 1 ) = 1 ∼ . Then N BV cl(A c 2 ) = 1 ∼ implies that there exists a neutrosophic bipolar vague closed set B in (X,T) such that A c Which is a contradiction. Hence (X,T) is a neutrosophic bipolar vague resolvable space. Conversely, N BV int(A) = 0 ∼ , for all neutrosophic bipolar vague dense set A in (X,T). Suppose that (X,T) is a neutrosophic bipolar vague resolvable. Then there exists a neutrosophic bipolar vague dense set A in (X,T), such that N BV cl(A c ) = 1 ∼ implies that (N BV int(A)) c = 1 ∼ , implies N BV int(A) = 0 ∼ . Which is a contradiction. Hence (X,T) is a neutrosophic bipolar vague irresolvable space.       N BV cl(A) = 1 ∼ , which implies that N BV int(N BV cl(A)) = 1 ∼ = 0 ∼ . Since (X,T) is neutrosophic bipolar vague open hereditarily irresolvable, we have N BV int(A) = 0 ∼ . Therefore by Proposition 3.6, N BV int(A) = 0 ∼ for all neutrosophic bipolar vague dense set in (X,T), implies that (X,T) is neutrosophic bipolar vague irresolvable. The converse of the above proposition need not be true by the following example.    f (A)). Since f is somewhat neutrosophic bipolar vague continuous, there exists a C ∈ T , such that C = 0 ∼ and C ⊆ f −1 (B). Hence C ⊆ f −1 (B) ⊆ A, which implies that N BV int(A) = 0 ∼ . Which is a contradiction. Hence N BV int(f (A)) = 0 ∼ in (Y,S).