Neutrosophic αψ -Homeomorphism in Neutrosophic Topological Spaces

: In this article, the concept of neutrosophic homeomorphism and neutrosophic αψ homeomorphism in neutrosophic topological spaces is introduced. Further, the work is extended as neutrosophic αψ ∗ homeomorphism, neutrosophic αψ open and closed mapping and neutrosophic T αψ space in neutrosophic topological spaces and establishes some of their related attributes.


Introduction
Zadeh [1] introduced the fuzzy set in 1965, the elements used of which in the fuzzy set had only the degree of membership. Later, Atanassav [2] introduced the intuitionistic fuzzy set in 1983. Both the fuzzy set and intuitionistic fuzzy set used all the elements that had the degree of membership and degree of non-membership. Salama and Alblowi [3] introduced the new concept of neutrosophic topological space (NTS) in 2012, which had been investigated recently. In the neutrosophic set, all the elements have the degree of membership, indeterminacy and degree of non-membership. The neutrosophic closed sets and neutrosophic continuous functions were introduced by Salama, Smarandache and Valeri [4] in 2014. Arokiarani et al. [5] introduced the neutrosophic α closed set in NTS. The neutrosophic ω closed sets in NTS were introduced by Santhi et al. [6] in 2016. The intuitionistic homeomorphism was introduced by Lee and Lee [7]. Jeevitha and Parimala [8] studied the concept of minimal αψ closed sets in minimal structure spaces also Parimala et al. [9] studied the concept of neutrosophic αψ closed sets.
The purpose of this article is to introduce the idea of neutrosophic homeomorphism and neutrosophic αψ homeomorphism in neutrosophic topological spaces and establish some of their attributes. It also establishes the notion of neutrosophic αψ * homeomorphism, neutrosophic αψ open and closed mapping and neutrosophic T αψ space. The present study demonstrates some of the related theorems, results and properties.

Preliminaries
The neutrosophic set in (X, τ N ) has the form S = {< x, M S (x), I S (x), N S (x) >: x ∈ (X, τ N )}, where the functions M S : S → [0, 1], I S : S → [0, 1],N S : S → [0, 1] denote the degree of membership, indeterminacy and degree of non-membership. The neutrosophic set S = {< x, M S (x), I S (x), N S (x) >: x ∈ (X, τ N )} is called a subset of T = {< x, M T (x), I T (x), N T (x) >: x ∈ (X, τ N )} (in short S ⊂ T) if the degree of membership and indeterminacy is minimum in S and the degree of non-membership is maximum in S or the degree of membership is minimum and the degree of non-membership and indeterminacy is maximum in S. The complement to NTS S = {< x, M S (x), I S (x), N S (x) >: x ∈ (X, τ N )} is S c = {< x, N S (x), I S (x), M S (x) >: x ∈ (X, τ N )}. Definition 1 ([10]). Let (X, τ N ) be a non-empty fixed set. A neutrosophic set S is an object having the form S = {< x, M S (x), I S (x), N S (x) >: x ∈ (X, τ N )}; where M S (x), I S (x), N S (x), which represent the degree of membership, the degree of indeterminacy and the degree of non-membership of each element x ∈ (X, τ N ) to the set S.

Definition 2 ([3]). Let S and T be NS of the form S
A neutrosophic topology on a non-empty set X is a family τ of neutrosophic subsets in X satisfying the following axioms: ). Let (X, τ N ) be NTS and S = {< x, M S (x), I S (x), N S (x) >: x ∈ (X, τ N )} be an NS in X. Then, the neutrosophic closer and neutrosophic interior of S are defined by: (1) Ncl(S) = ∩{K : K is an neutrosophic closed set in X and S ⊆ K}.

Lemma 1 ([3]
). Let (X, τ N ) be an NTS and A, B be two neutrosophic sets in X. Then, the following properties hold: Definition 5 ([7]). A bijection f : (X, τ 1 ) → (Y, τ 1 ) is said to be an intuitionistic homeomorphism if f is both an intuitionistic continuous and intuitionistic open function in X.
is a function and X and Y a two neutrosophic topological space, then g is said to be neutrosophic continuous if the preimage of each neutrosophic closed set in (Y, τ N2 ) is a neutrosophic set in (X, τ N1 ).
is an Nαψ closed set in (X, τ N1 ) for every neutrosophic closed set U of (Y, τ N2 ).
is a function and X and Y a two neutrosophic topological space, then g is said to be neutrosophic open if the image of each neutrosophic set in (X, τ N1 ) is a neutrosophic set in (Y, τ N2 ).

Neutrosophic αψ Homeomorphism
We introduce the following definition.
is called a neutrosophic homeomorphism if g and g −1 are neutrosophic continuous. Here, g(p) = p, g(q) = q, g(r) = r, and assume Hence, g is neutrosophic continuous, and (g −1 ) −1 : X → Y is said to be neutrosophic continuous. If A c is a neutrosophic closed set in X, then the image g(A c ) = F c is neutrosophic closed in Y. Hence, g and g −1 are neutrosophic continuous; therefore, it is a neutrosophic homeomorphism.
Here, g(p) = p, g(q) = q, g(r) = r, and assume S = y, is a neutrosophic closed set in Y, then g −1 (S) is neutrosophic αψ closed in X. Hence, g is neutrosophic αψ continuous and g −1 is neutrosophic αψ continuous if A c is a neutrosophic αψ closed set in X, then the image g(A c ) = F c is neutrosophic closed in Y. Hence, g and g −1 are neutrosophic αψ continuous, which is therefore a neutrosophic αψ homeomorphism.
is called a neutrosophic ψ homeomorphism if g and g −1 are neutrosophic ψ continuous.
Hence, g is neutrosophic ψ continuous and g −1 is neutrosophic ψ continuous if A c is a neutrosophic ψ closed set in X, then the image g(A c ) = F c is neutrosophic closed in Y. Hence, g and g −1 are neutrosophic ψ continuous; therefore, it is a neutrosophic ψ homeomorphism.
Proof. Let a bijection mapping g : (X, τ N1 ) → (Y, τ N2 ) be neutrosophic homeomorphism, in which g and g −1 are neutrosophic continuous. We know that every neutrosophic continuous function is N αψ continuous; hence, g and g −1 N αψ continuous. Therefore, g is an N αψ homeomorphism.
Let a neutrosophic αψ homeomorphism be not a neutrosophic homeomorphism by the following example.
Here, g(p) = p, g(q) = q, g(r) = r, and assume S = y, is a neutrosophic closed set in Y, then g −1 (S) is neutrosophic αψ closed in X. Hence, g is neutrosophic αψ continuous and g −1 is neutrosophic αψ continuous if A c is a neutrosophic αψ closed set in X, then the image g(A c ) = F c is neutrosophic closed in Y. Hence, g and g −1 are neutrosophic αψ continuous; therefore, it is a neutrosophic αψ homeomorphism. However, here, S is a neutrosophic closed set in Y, but it is not a neutrosophic closed set in X. Therefore, it is not neutrosophic continuous. Therefore, it is not a neutrosophic homeomorphism.
Let a neutrosophic αψ homeomorphism be not a neutrosophic ψ homeomorphism by the following example.
Here, g(p) = p, g(q) = q, g(r) = r, and assume S = y, ( p 0.6 , Here, g and g −1 are neutrosophic αψ continuous; therefore, it is a neutrosophic αψ homeomorphism, but it is not a neutrosophic ψ homeomorphism because S is neutrosophic ψ continuous.  2 ) is a neutrosophic closed set in X, then g(S) is neutrosophic αψ closed in Y. Therefore, g is a neutrosophic αψ closed mapping. Theorem 3. Each neutrosophic closed mapping is a neutrosophic αψ closed mapping.
Proof. Let us assume that g : (X, τ N1 ) → (Y, τ N2 ) is a neutrosophic closed mapping, such that H is a neutrosophic closed set in X. Since g is a neutrosophic closed mapping, g(H) is neutrosophic closed in Y. We know that every neutrosophic closed set is a neutrosophic αψ closed set. Therefore, g(H) is an N αψ closed set in Y. Hence, g is an N αψ closed mapping.
Let a neutrosophic αψ closed mapping be not a neutrosophic closed mapping by the following example.
Here, g(p) = p, g(q) = q, g(r) = r, and assume is a neutrosophic closed set in X, then g(S) is a neutrosophic αψ closed in Y. Therefore, g is a neutrosophic αψ closed mapping. However, it is not a neutrosophic closed mapping because g(S) is not neutrosophic closed in Y. Proof. (a) ⇒ (b) Let us assume that g is a bijective mapping and a neutrosophic αψ closed mapping. Hence, g −1 is a neutrosophic αψ continuous mapping. We know that each neutrosophic open set in X is a neutrosophic αψ open set in Y. Hence, g is a neutrosophic αψ open mapping.
(b) ⇒ (c) Let g be a bijective and neutrosophic open mapping. Furthermore, g −1 is a neutrosophic αψ continuous mapping. Hence, g and g −1 are neutrosophic αψ continuous. Therefore, g is a neutrosophic αψ homeomorphism.
(c) ⇒ (a) Let g be a neutrosophic αψ homeomorphism, then g and g −1 are neutrosophic αψ continuous. Since each neutrosophic closed set in X is a neutrosophic αψ closed set in Y, hence g is a neutrosophic αψ closed mapping.
Definition 14. Let (X, τ N1 ) be a neutrosophic topological spaces said to be a neutrosophic T αψ space if every neutrosophic αψ closed set is neutrosophic closed in X.
Proof. Through the assumption that H is a neutrosophic closed set in Y, then g −1 (H) is a neutrosophic αψ closed set in X. Since X is an NT αψ space, g −1 (H) is a neutrosophic closed set in X. Therefore, g is neutrosophic continuous. By hypothesis, g −1 is neutrosophic αψ continuous. Let G be a neutrosophic closed set in X. Then, (g −1 ) −1 (G) = g(G) is a neutrosophic closed set in Y, by presumption. Since Y is an NT αψ space, g(G) is a neutrosophic closed set in Y. Hence, g −1 is neutrosophic continuous. Hence, g is a neutrosophic homeomorphism. Since Y is an NT αψ space, so g(int(H)) is a neutrosophic open set in Y. Therefore, g(int(H)) = int(g(int(H))) ⊆ cl(int(g(H))).
(c) ⇒ (a) Let H be a neutrosophic closed set in X. Then, H c is a neutrosophic open set in X. In the supposed way, g(int(H c )) ⊆ cl(int(g(H c ))). Hence, g(H c ) ⊆ cl(int( f (H c ))). Therefore, g(H c ) is neutrosophic αψ open set in Y. Therefore, g(H) is a neutrosophic αψ closed set in X. Hence, g is a neutrosophic closed mapping.
Proof. Let H be a neutrosophic closed set in X. Since f is neutrosophic αψ closed and f (H) is a neutrosophic αψ closed set in Y, by assumption, f (H) is a neutrosophic closed set in Y. Since g is neutrosophic αψ closed, then g( f (H)) is neutrosophic αψ closed in (τ N1 , τ N3 ) and g( f (H)) = g • f (H). Therefore, g • f is neutrosophic αψ closed.

Proof.
A map g is a neutrosophic αψ * homeomorphism. Let us assume that H is a neutrosophic closed set in Y. This shows that H is a neutrosophic αψ closed set in Y. By assumption, g −1 (H) is a neutrosophic αψ closed set in X. Hence, g is a neutrosophic αψ continuous mapping. Hence, g and g −1 are neutrosophic αψ continuous mappings. Hence g is a neutrosophic αψ homeomorphism.
Let a neutrosophic αψ homeomorphism be not a neutrosophic αψ * homeomorphism by the following example. 2 ) is neutrosophic αψ continuous, then it is neutrosophic αψ homeomorphism. However, it is not a neutrosophic αψ * homeomorphism, because it is not neutrosophic αψ irresolute.
Proof. Let us take f and g to be two neutrosophic αψ * homeomorphisms. Assume H is a neutrosophic αψ closed set in Z. Then, by the supposed way, g −1 (H) is a neutrosophic αψ closed set in Y. Then, by hypothesis, f −1 (g −1 (H)) is a neutrosophic αψ closed set in X. Hence, g • f is a neutrosophic αψ irresolute mapping. Now, let G be a neutrosophic αψ closed set in X. Then, by presumption, f (G) is a neutrosophic αψ closed set in Y. Then, by hypothesis, g( f (G)) is a neutrosophic αψ closed set in Z. This implies that g • f is a neutrosophic αψ irresolute mapping. Hence, g • f is a neutrosophic αψ * homeomorphism.

Conclusions
In this paper, the new concept of a neutrosophic homeomorphism and a neutrosophic αψ homeomorphism in neutrosophic topological spaces was discussed. Furthermore, the work was extended as the neutrosophic αψ * homeomorphism, neutrosophic αψ open and closed mapping and neutrosophic T αψ space. Further, the study demonstrated neutrosophic αψ * homeomorphisms and also derived some of their related attributes.
Author Contributions: All the authors have contributed equally to this paper. The individual responsibilities and contribution of all authors can be described as follows: the idea of this paper was put forward by M.P.; R.J., R.U. and S.J. completed the preparatory work of the paper; F.S. and S.J. analyzed the existing work; the revision and submission of this paper were completed by M.P., R.J. and R.U.