On Neutrosophic αψ-Closed Sets

The aim of this paper is to introduce the concept of αψ-closed sets in terms of neutrosophic topological spaces. We also study some of the properties of neutrosophic αψ-closed sets. Further, we introduce continuity and contra continuity for the introduced set. The two functions and their relations are studied via a neutrosophic point set.


Introduction
Zadeh [1] introduced and studied truth (t), the degree of membership, and defined the fuzzy set theory.The falsehood (f), the degree of nonmembership, was introduced by Atanassov [2][3][4] in an intuitionistic fuzzy set.Coker [5] developed intuitionistic fuzzy topology.Neutrality (i), the degree of indeterminacy, as an independent concept, was introduced by Smarandache [6,7] in 1998.He also defined the neutrosophic set on three components (t, f , i) = (truth, f alsehood, indeterminacy).The Neutrosophic crisp set concept was converted to neutrosophic topological spaces by Salama et al. in [8].This opened up a wide range of investigation in terms of neutosophic topology and its application in decision-making algorithms.Arokiarani et al. [9] introduced and studied α-open sets in neutrosophic topoloical spaces.Devi et al. [10][11][12] introduced αψ-closed sets in general topology, fuzzy topology, and intutionistic fuzzy topology.In this article, the neutrosophic αψ-closed sets are introduced in neutrosophic topological space.Moreover, we introduce and investigate neutrosophic αψ-continuous and neutrosophic contra αψ-continuous mappings.
We provide some of the basic definitions in neutrosophic sets.These are very useful in the sequel.

Definition 1. [6]
A neutrosophic set (NS) A is an object of the following form where the mappings µ U : X → I, ν U : X → I, and ω U : X → I denote the degree of membership (namely µ U (x)), the degree of indeterminacy (namely ν U (x)), and the degree of nonmembership (namely ω U (x)) for each element x ∈ X to the set U, respectively, and 0 ≤ µ U (x) + ν U (x) + ω U (x) ≤ 3 for each a ∈ X.
Definition 2. [6] Let U and V be NSs of the form We will use the notation otherwise.
Let f be a mapping from an ordinary set X into an ordinary set Y.
for each y ∈ Y.

Definition 3. [8]
A neutrosophic topology (NT) in a nonempty set X is a family τ of NSs in X satisfying the following axioms: Definition 5. [8] Let p (r,s,t) be an NP in NTS X.An NS U in X is called a neutrosophic neighborhood (NN) of p (r,s,t) if there exists an NOS V in X such that p (r,s,t) ∈ V ⊆ U. Definition 6. [9] A subset U of a neutrosophic space (X, τ) is called The pre-closure (respectively, semi-closure and α-closure) of a subset U of a neutrosophic space (X, τ) is the intersection of all pre-closed (respectively, semi-closed, α-closed) sets that contain U and is denoted by N pcl(U) (respectively, Nscl(U) and Nαcl(U)).Proof.Let U be an Nsemi-closed set in (X, τ), then

On Neutrosophic αψ-Closed
The converses of the above theorems are not true, as can be seen by the following counter example.
Example 1.Let X = {u, v, w} and neutrosophic sets G 1 , G 2 , G 3 , G 4 be defined by Here, G 6 is an Nα open set, and Nψcl(G 5 ) ⊆ G 6 .Then G 5 is Nαψ-closed in (X, τ) but is not Nα-closed; thus, it is not N-closed and G 7 is Nαψ-closed in (X, τ), but not Nsemi-closed.Theorem 3. Let (X, τ) be an NTS and let U ∈ NS(X).I f U is an Nαψ-closed set and U ⊆ V ⊆ Nψcl(U), then V is an Nαψ-closed set.

Proof. Let G be an
Conversely, suppose that the condition is satisfied.Then Nψint(U) ⊆ V whenever V is an Nα-open set and U ⊆ V.This implies that Nψcl(U) ⊆ V = G, where G is Nα-open and U ⊆ G. Therefore, U is Nαψ-closed and hence U is Nαψ-open.Theorem 6.Let U be an Nαψ-closed subset of (X, τ).Then Nψcl(U) − U does not contain any non-empty Nαψ-closed set.
Proof.Assume that U is an Nαψ-closed set.Let F be a non-empty Nαψ-closed set, such that i.e., F ⊆ φ.Therefore, F is empty.
Proof.The proof follows from the Theorem 3.9.Proof.Since U is both an Nψ-open and Nαψ-closed set in X, then Nψcl(U) ⊆ U. We also have U ⊆ Nψcl(U).Thus, Nψcl(U) = U.Therefore, U is an Nψ-closed set in X. (i) g(NαψNcl(U)) ⊆ Ncl(g(U)), for all neutrosophic sets U in X;
Theorem 10.Let g be a function from an NTS (X, τ) to an NTS (Y, σ).Then the following statements are equivalent.
(i) g is a neutrosophic αψ-continuous function; (ii) For every NP p (r,s,t) ∈ X and each NN U of g(p (r,s,t) ), there exists an Nαψ-open set V such that p (r,s,t) ∈ V ⊆ g −1 (U).
(iii) For every NP p (r,s,t) ∈ X and each NN U of g(p (r,s,t) ), there exists an Nαψ-open set V such that p (r,s,t) ∈ V and g(V) ⊆ U.
Proof.(i) ⇒ (ii).If p (r,s,t) is an NP in X and if U is an NN of g(p (r,s,t) ), then there exists an NOS W in Y such that g(p (r,s,t) ) ∈ W ⊂ U. Thus, g is neutrosophic αψ-continuous, V = g −1 (W) is an NαψOset, and Thus, (ii) is a valid statement.
(ii) ⇒ (iii).Let p (r,s,t) be an NP in X and let U be an NN of g(p (r,s,t) ).Then there exists an NαψOset U such that p (r,s,t) ∈ V ⊆ g −1 (U) by (ii).Thus, we have p (r,s,t) ∈ V and g(V) ⊆ g(g −1 (U)) ⊆ U. Hence, (iii) is valid.
Therefore, we know that g −1 (V) is an NαψOset in X.Thus, g is neutrosophic αψ-continuous.Definition 11.A function is said to be a neutrosophic contra αψ-continuous function if the inverse image of each NOS V in Y is an NαψC set in X.
Theorem 11.Let g : (X, τ) → (Y, σ) be a function.Then the following assertions are equivalent: (i) g is a neutrosophic contra αψ-continuous function; (ii) g −1 (V) is an Nαψ C set in X, for each NOS V in Y.
Proof.(i) ⇒ (ii) Let g be any neutrosophic contra αψ-continuous function and let V be any NOS in Y. Then V is an NCS in Y. Based on these assumptions, g −1 (V) is an NαψOset in X.Hence, g −1 (V) is an NαψCset in X.
The converse of the theorem can be proved in the same way.

Sets Definition 8 .Definition 9 .Theorem 1 .Theorem 2 .
A neutrosophic αψ-closed (Nαψ-closed) set is defined as if Nψcl(U) ⊆ G whenever U ⊆ G and G is an Nα-open set in (X, τ).Its complement is called a neutrosophic αψ-open (Nαψ-open) set.Let U be an NS in NTS X.Then Nαψint(U) = ∪{O : O is an NαψOS in X and O ⊆ U} is said to be a neutrosophic αψ-interior of U; Nαψcl(U) = ∩{O : O is an NαψCS in X and O ⊇ U} is said to be a neutrosophic αψ-closure of U.All Nα-closed sets and N-closed sets are Nαψ-closed sets.Proof.Let U be an Nα-closed set, then U = Nαcl(U).Let U ⊆ G, where G is Nα-open.Since U is Nα-closed, Nψcl(U) ⊆ Nαcl(U) ⊆ G. Thus, U is Nαψ-closed.Every Nsemi-closed set in a neutrosophic set is an Nαψ-closed set.

Theorem 4 .Theorem 5 .
Let U be an Nαψ-open set in X and Nψint(U)⊆ V ⊆ U, then V is Nαψ-open.Proof.Suppose U is Nαψ-open in X and Nψint(U) ⊆ V ⊆ U. Then U is Nαψ-closed and U ⊆ V ⊆ Nψcl(U).Then U is an Nαψ-closed set by Theorem 3.5.Hence, V is an Nαψ-open set in X.An NS U in an NTS (X, τ) is an Nαψ-open set i f and only i f V ⊆ Nψint(U) whenever V is an Nα-closed set and V ⊆ U.Proof.Let U be an Nαψ-open set and let V be an Nα-closed set such that

Theorem 7 .
If U is both Nψ-open and Nαψ-closed, then U is Nψ-closed.

Theorem 8 .Theorem 9 .
Contra αψ-Continuity Definition 10.A function f : X → Y is said to be a neutrosophic αψ-continuous (briefly, Nαψ-continuous) function if the inverse image of every open set in Y is an Nαψ-open set in X.Let g : (X, τ) → (Y, σ) be a function.Then the following conditions are equivalent.(i)g is Nαψ-continuous; (ii) The inverse f −1 (U) of each N-open set U in Y is Nαψ-open set in X.Proof.The proof is obvious, since g −1 (U) = g −1 (U) for each N-open set U of Y.If g : (X, τ) → (Y, σ) is an Nαψ-continuous mapping, then the following statements hold: