Neutrosophic generalized semi pre regular and Normal Space by Shravan

Neutrosophic Generalized Semi Pre-Regular and Normal Space

Chandramathi N 1 and Rajeshwaran N

1 Department of Mathematics, Government Arts College, Udumalpet-642126, Tamil Nādu, India. Email: drmathimaths@gmail.com

2 Research Scholar, Department of Mathematics, Government Arts College, Udumalpet-642126, Tamilnadu, India.

E-mail: rajeshw851@gmail.com ***

Abstract -In this manuscript, we inaugurate Neutrosophic generalized semi-pre regular and normal space. We investigate its properties. Also, we add some improvisation of Neutrosophic generalized semi pre regular and normal space.

Keywords:Neutrosophic generalized semi pre closed sets; Neutrosophic generalized semi pre regular space, Neutrosophic generalized semi pre normal space.

1.INTRODUCTION

In 2014, the pioneering work of Salama, Smarandache, and Valeri [10] introduced the concept of Neutrosophic closed sets and Neutrosophic continuous functions. Subsequent advancements by Salama and Alblowi [10] led to the development of generalized Neutrosophic sets and generalized Neutrosophic topological spaces. Notably, the idea of NOS (Neutrosophic Open Sets) gained prominence through the contributions of Wadel and Smarandache [14]. Furthermore, Ishwarya and Bageerathi [8] offered insights into the perspective of NSO (Neutrosophic Sets) within the framework of Neutrosophic topological spaces. In their publication [25], Rajeshwaran N and Chandramathi N presented the novel idea of Neutrosophic generalized semi pre closed sets within the realm of Neutrosophic topological spaces. Similarly, in another work [26], they introduced the concept of Neutrosophic generalized semi pre Homeomorphisms in the same context. This manuscript seeks to define and investigate the concept of Neutrosophic generalized semi pre-connected space, delving into its inherent properties. The study encompasses an exploration of various related notions and introduces a collection of noteworthy theorems within this domain.

2. Preliminaries

Definition 2.1: [10] A neutrosophic topology (NT for short) a non-empty set X is a family τN of neutrosophic subsets in X adheres the following axioms (NT1)0N,1N ∈ τN (NT2)G1 ∩G2 ∈ τN (NT3)∪Gi ∈ τN ,∀{Gi:i∈J}⊆ τN Here (X,τN) is called a neutrosophic topological space (NTS for short).

Definition 2.2: [10] Let A1��A2 be two Neutrosophic Sets (NS for Short) of the form A1 ={⟨X,μA1 (X),σA1(X),γA1(X)⟩:xϵX} , A2 =

{⟨X,μA2 (X),σA2(X),γA2(X)⟩:xϵX} .

(a) A1 ⊆A2 if and only if μA1 (X) ≤ μA2 (X),σA1(X)≤ σA2(X)and γA1 (X)≥ γA2 (X)forallx∈X

(b) A1 C ={⟨X,γA1 (X),1 σA1(X),μA1(X)⟩:xϵX}

(c)A1 ∩A2 =

{⟨X,μA1 (X)⋀μA2 (X),σA1(X)⋀ σA2(X),γA1(X)⋁γA2(X)⟩:xϵX}

(d)A1 ∪A2 =

{⟨X,μA1 (X)⋁μA2 (X),σA1(X)⋁ σA2(X),γA1(X)⋀γA2(X)⟩:xϵX}

We can use the symbol A1 ={⟨X,μA(X),σA(X),γA(X)⟩:xϵX}

Definition 2.3: [25] Let (Ҳ ,τƝ) be a neutrosophic topological space. A subset Ą of (Ҳ ,τƝ) is called Neutrosophic generalized semi pre closed [ƝƓŚƤ -closed] set if spclƝ (Ą) ⊆ Մ , whenever Ą ⊆ Մ and Մ is Neutrosophic open set.

3.Generalized Semi Pre Regular and Normal Space in Neutrosophic Topological Spaces

In this paper we introduce the new concept namely Neutrosophic generalized semi pre regular space in neutrosophic topological spaces. We delve into the foundations of Neutrosophic generalized semi-pre regular space.

Definition 3.1.1: A topological space (Ҳ ,τƝ) is said to be ƝƓŚƤ regular if for each ƝƓŚƤ closed set M of (Ҳ ,τƝ) and each point x∈ X M, there exist disjoint open sets P and Q of (Ҳ ,τƝ) such that x∈PandM⊆Q. Since every ƝƓ-closed set is ƝƓŚƤ-closed set so every ƝƓŚƤ regular space is ƝƓ regular space.

Definition: 3.1.2 A Neutrosophic topology (Ҳ ,τƝ) is said to be (ƝƓŚƤ, ƝƓŚ) regular if for each ƝƓŚƤ closed set of

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(Ҳ ,τƝ) and each point x∈X M, there exists disjoint ƝƓŚ open sets P and Q of (Ҳ ,τƝ) such that x∈P and M⊆Q

Theorem 3.1.3: A Neutrosophic topological space (Ҳ ,τƝ) is a ƝƓŚƤ regular if and only if for each ƝƓŚƤ closed set M of (Ҳ ,τƝ) and each point x∈X M, there exist open sets P and Q of (Ҳ ,τƝ) such that x∈P∶M⊆Q and clƝ(P)∩clƝ(Q)=ϕ

Proof: Necessity: Let M be a ƝƓŚƤ closed set of (Ҳ ,τƝ) and x∈ X M. There exist Neutrosophic open sets P0 and Q of (Ҳ ,τƝ) such that x∈P0 ,M⊆Q and P0 ∩Q=ϕ, hence P0 ∩ clƝ(Q)=ϕ. Since (Ҳ ,τƝ) is ƝƓŚƤ regular, there exist Neutrosophic open sets G and H of (Ҳ ,τƝ) such that x∈ G,clƝ(Q) ⊆H and G∩H=ϕ, hence clƝ(G)∩H=ϕ. Now put P=P0 ∩G, then P and Q are Neutrosophic open sets of (Ҳ ,τƝ) such that x∈P,M⊆Q and clƝ(P)∩clƝ(Q)=ϕ.

Sufficiency: This is obvious.

Theorem 3.1.4: Let (Ҳ ,τƝ) be a Neutrosophic topological space then the following statements are equivalent:

(i) Let (Ҳ ,τƝ) is ƝƓŚƤ regular space.

(ii) For each point x∈(Ҳ ,τƝ) and for each ƝƓŚƤ open neighbourhood W of x, there exists a Neutrosophic open set of x, such that clƝ(Q)⊆W (iii) For each point of x∈(Ҳ ,τƝ) and for each ƝƓŚƤ closed not containing x, then there exists a Neutrosophic open set Q of X, such that clƝ(Q)∩M= ϕ.

Proof: (i)⇒(ii) Let W be a ƝƓŚƤ open neighbourhood of x Then there exists a ƝƓŚƤ open set G such that x∈X⊆W

Since X G is ƝƓŚƤ closed set and x∉X G, by hypothesis there exist Neutrosophic open sets P and Q such that X G⊆ P,x∈Q and P∩Q= ϕ and so Q⊆(X P). Now clƝ(Q) ⊆ clƝ(X P)=(X P) and (X G)⊆P implies (X P)⊆ G⊆W. Therefore clƝ(Q)⊆W. (ii)⇒(i) : Let M be any ƝƓŚƤ closed set of x∉M. Then x∈ X M and X M is ƝƓŚƤ open and so X M is a ƝƓŚƤ open neighbourhood of x. By hypothesis there exists a Neutrosophic open Q of x such that x ∈ Q and clƝ(Q) ⊆ (X M) which implies M ⊆(X clƝ(Q)). Then (X clƝ(Q)) is Neutrosophic open set containing M and Q ∩(X clƝ(Q))= ϕ . Therefore (Ҳ ,τƝ) is ƝƓŚƤ regular space. (ii)⇒(iii) : Let x ∈ X and M be a ƝƓŚƤ closed set such that x ∉ M. Then (X M) is a ƝƓŚƤ open neighbourhood of x and by hypothesis there exists a Neutrosophic open set Q of x such that clƝ(Q)⊆(X M) and therefore clƝ(Q)∩ M = ϕ (iii)⇒(ii) : Let x ∈ X and W be a ƝƓŚƤ open neighbourhood of x then there exists a Neutrosophic ƓŚƤ open set G such that x ∈ G ⊆ W. Since (X G) is ƝƓŚƤ closed and x ∉ (X G) by hypothesis there exists a Neutrosophic open set Q of x such that clƝ(Q)∩(X G)= ϕ. Therefore clƝ(Q)⊆ G ⊆ W.

Theorem 3.1.5: A Neutrosophic topological space (Ҳ ,τƝ) is a ƝƓŚƤ regular space if and only if given any x∈ P and any Neutrosophic open set P of (Ҳ ,τƝ) there is a ƝƓŚƤ open set Q such that x∈Q⊆gspclƝ(Q) ⊆P

Proof: Let P be a Neutrosophic open set, x∈P. So X P is closed set such that x∉P. Since X is a ƝƓŚƤ regular space then there exist a Neutrosophic ƓŚƤ open sets Q1 and Q2 such that Q1 ∩Q2 =ϕ, X P⊆Q2, x∈Q1. Since Q1 ∩Q2 =ϕ,

we have gspclƝ(Q1)⊆gspclƝ(X Q2)=X Q2. Since X P⊆Q2, we have X Q2 ⊆P. Hence we have x∈Q1 ⊆ gspclƝ(Q1)⊆X Q2 ⊆P

Conversely, let M be a Neutrosophic closed set in X and x∈ X M. So X M is a Neutrosophic open set such that x∈X M. Hence there exists a ƝƓŚƤ open set P such that x∈P⊆ gspclƝ(P) ⊆(X M). Let Q=X gspclƝ(Q). So Q is a ƝƓŚƤ open set which contains M and P∩Q=ϕ. Hence X is a ƝƓŚƤ regular space.

Theorem 3.1.6: Let (Ҳ ,τƝ) and (Ƴ,σƝ) be a Neutrosophic topological space and (Ƴ,σƝ) is a regular. If φƝ ∶(Ҳ ,τƝ) → (Ƴ,σƝ) is Neutrosophic closed ƓŚƤ irresolute and one to one then X is a ƝƓŚƤ regular space.

Proof: Let M be a closed set in X, x∉M. Since φƝ is closed mapping, then φƝ(M) is closed set in (Ƴ,σƝ), φƝ(x)=y∉

φƝ(M). But (Ƴ,σƝ) is ƝƓŚƤ regular space then there are two Neutrosophic open sets P and Q in (Ƴ,σƝ) such that φƝ(M)⊆ Q, y∈P,P∩Q=ϕ. Since φƝ is ƝƓŚƤ-irresolute mapping and one to one so φƝ 1(P),φƝ 1(Q) are two Neutrosophic open sets in X and x∈φƝ 1(P), M∈φƝ 1(Q), φƝ 1(P) ∈ φƝ 1(Q) =ϕ. Hence X is ƝƓŚƤ regular space.

Theorem 3.1.7: A Neutrosophic topological space (Ҳ ,τƝ) is a (ƝƓŚƤ, ƝƓŚ) regular space if and only if given ƝƓŚƤ open set P with x∈P, there exists ƝƓŚ open sets Q such that x∈ Q⊆sclƝ(Q)⊆P

Proof: Let Ƥ be a ƝƓŚƤ open set, x∈Ƥ. So Ҳ Ƥ is a ƝƓŚƤ closed set such x∉ Ҳ Ƥ. Since (Ҳ ,τƝ) is (ƝƓŚƤ, ƝƓŚ) regular space then there exist ƝƓŚ open sets Q1 and Q2 such that Q1 ∩Q2 =ϕ, Ҳ Ƥ⊆Q2, x∈Q1. Since Q1 ∩Q2 =ϕ, we have sclƝ(Q1)⊆sclƝ(Ҳ Q2)= Ҳ Q2. Since Ҳ Ƥ⊆ Q2 we have Ҳ Q2 ⊆Ƥ. Hence we have x∈Q1 ⊆ sclƝ(Q1)⊆ Ҳ Q2 ⊆Ƥ. Conversely, let M be a ƝƓŚƤ closed set in (Ҳ ,τƝ) and x∈ Ҳ M. So Ҳ M is a ƝƓŚ open set such that x∈ Ҳ M. Hence there exists a ƝƓŚ open set Ƥ such that x∈Ƥ⊆sclƝ(Q) ⊆ Ҳ M. Let Q= Ҳ gspclƝ(Q). So Q is a ƝƓŚ open set which contains M and Ƥ∩ Q=ϕ. Hence Ҳ is a (ƝƓŚƤ, ƝƓŚ) regular space.

3.2 ƝŚƤ, ƝƓŚƤ normal spaces

In this section, we delve into the fundamental concepts and properties of generalized semi pre normal space in Neutrosophic topology, exploring their significance in Neutrosophic topology.

Definition 3.2.1: A Neutrosophic topological space (Ҳ ,τƝ) is said to be ƝƓŚƤ normal if for any pair of disjoint ƝƓŚƤ closed sets Ą and Ƀ, there exist disjoint Neutrosophic open sets Մ and Ṿ such that Ą ⊂Մ,Ƀ⊂Ṿ Since every ƝƓ-closed set is ƝƓŚƤ-closed set so every ƝƓŚƤ normal space is ƝƓ normal space.

Theorem 3.2.2: A Neutrosophic topological space (Ҳ ,τƝ) is a ƝƓŚƤ normal space if and only if any disjoint ƝƓŚƤ closed sets P and Q of (Ҳ ,τƝ), there exist Neutrosophic open sets M and N of (Ҳ ,τƝ) such that P⊂M,Q⊂N and clƝ(M)∩ clƝ(N) =φ

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Proof: Necessity: Let P and Q be any disjoint ƝƓŚƤ-closed sets of (Ҳ ,τƝ). There exist Neutrosophic open sets M0 and N of (Ҳ ,τƝ) such that P⊂M0, Q⊂N and M0 ∩N=φ hence M0 ∩clƝ(N)=φ. Since (Ҳ ,τƝ) is ƝƓŚƤ normal there exist Neutrosophic open sets G and H of (Ҳ ,τƝ) such that P⊂G, clƝ(N)⊂H and G∩H=φ, hence clƝ(G)∩H=φ. Now put M=M0 ∩G, then M and N are Neutrosophic open sets of (Ҳ ,τƝ) such that P⊂M,Q⊂N and clƝ(M)∩clƝ(N) =φ. Sufficiency: Obvious.

Theorem 3.2.3: A Neutrosophic topological space (Ҳ ,τƝ) is said to be a ƝƓŚƤ normal space if and only for every Neutrosophi closed set F and for every Neutrosophic open set G contain F there exist ƝƓŚƤ open set M such that F⊂M⊂ gspclƝ(M)⊂G

Proof: Let F be a Neutrosophic closed set in (Ҳ ,τƝ) and G be a Neutrosophic open set in (Ҳ ,τƝ) such that F⊂M, X G is a Neutrosophic closed set and (X G)∩F=φ. Since (Ҳ ,τƝ) is ƝƓŚƤ normal space then there exist open sets M and N of (Ҳ ,τƝ) such that M∩N=φ, X G⊂N and F⊂M, M⊂ (X N). Since every Neutrosophic open set is ƝƓŚƤ open set and hence M and N are ƝƓŚƤ open sets of (Ҳ ,τƝ) such that gspclƝ(M)⊂ gspclƝ(X N)=X N. Hence F⊂M⊂ gspclƝ(N) ⊂(X N)⊂G.

Theorem 3.2.4: If φƝ ∶(Ҳ ,τƝ) →(Ƴ,σƝ) is a open ƝƓŚƤirresolute bijection and (Ҳ ,τƝ) is ƝƓŚƤ normal, then (Ƴ,σƝ) is ƝƓŚƤ normal.

Proof: Let P and Q be any disjoint ƝƓŚƤ closed sets of (Ƴ,σƝ). Since φƝ is ƝƓŚƤ-irresolute, φƝ 1(P) and φƝ 1(Q) are disjoint ƝƓŚƤ closed sets (Ҳ ,τƝ). Since (Ҳ ,τƝ) is ƝƓŚƤ normal then there exists disjoint Neutrosophic open sets M and N such that φƝ 1(P)⊂ M and φƝ 1(Q)⊂ N. Since φƝ is Neutrosophic open and bijectivity, we obtain P ⊂ φƝ(M), Q ⊂ φƝ(N), φƝ(M)∩ φƝ(N)= φ and also φƝ(M) and φƝ(N) are Neutrosophic open sets of (Ƴ,σƝ). This shows that (Ƴ,σƝ) is ƝƓŚƤ normal.

Theorem 3.2.5: The following properties are equivalent for a space (Ҳ ,τƝ) (i) (i). (Ҳ ,τƝ) is (ƝŚƤ ,ƝƓŚƤ)- normal (ii) (ii). For any pair of disjoint Neutrosophic semi pre closed sets P and Q of (Ҳ ,τƝ), there exist disjoint ƝƓŚƤ open sets M and N such that P ⊂ M and Q ⊂ N (iii) (iii). For any Neutrosophic semi pre closed set P and Neutrosophic semi pre open set N containing P, there exists ƝƓŚƤ open set M such that P ⊂ M ⊂ spclƝ(M)⊂ N

Proof: (i)⇒(ii) This proof is obvious since every ƝŚƤ open set is ƝƓŚƤ open set.

(ii)⇒(iii) Let P be any ƝŚƤ closed set and N be a ƝŚƤ open set containing P. Since P and X-N are disjoint ƝŚƤ closed set of (Ҳ ,τƝ), since P and X-N are disjoint ƝŚƤ closed sets of (Ҳ ,τƝ), then there exist ƝƓŚƤ open sets M, W of (Ҳ ,τƝ) such that P ⊂ M, X N ⊂ W and M ∩ N = φ,since M ∩ spintƝ(W)= φ. We have spclƝ(M)∩ spintƝ(W)= φ and hence spclƝ(M)⊂ X spintƝ(W)⊂ N. Therefore, we obtain P ⊂ M ⊂ spcl(M)⊂ N

(iii)⇒(i) Let P and Q be any disjoint ƝŚƤ closed sets of (Ҳ ,τƝ). Since X-Q is a ƝŚƤ open set containing P, there exist a ƝƓŚƤ open set G, such that P ⊂ G ⊂ spclƝ(G) ⊂ X Q, we have P ⊂ spclƝ(G). Put M = spintƝ(G) and N = X spclƝ(G). Then M and N are disjoint ƝŚƤ open sets and hence are disjoint ƝƓŚƤ open sets such that P ⊂ M and Q ⊂ N. Therefore (Ҳ ,τƝ) is (ƝŚƤ,ƝƓŚƤ)- normal.

Definition: 3.2.6 A function φƝ ∶(Ҳ ,τƝ)→(Ƴ,σƝ) is called Neutrosophic pre generalized semi pre closed (brifly, ƝƤ ƓŚƤ closed) if for each Neutrosophic semi pre closed set Ɗ of (Ҳ ,τƝ), φƝ(Ɗ) is ƝƓŚƤ closed set in (Ƴ,σƝ)

Theorem 3.2.7: A surjective function φƝ ∶(Ҳ ,τƝ) → (Ƴ,σƝ) is a ƝƤ ƓŚƤ-closed if and only if for each subset Ɗ of (Ƴ,σƝ), and ƝŚƤ open set M of (Ҳ ,τƝ) containing φƝ 1(Ɗ), there exists a ƝƓŚƤ open set N of (Ƴ,σƝ) such that Ɗ ⊂ N and φƝ 1(N)⊂ M

Proof: Necessity: Suppose that φƝ is ƝƤ ƓŚƤ closed. Let Ɗ be any subset of (Ƴ,σƝ) and M be ƝŚƤ open set of (Ҳ ,τƝ) containing φƝ 1(Ɗ). Put N= Y φƝ (X M). Then N is ƝƓŚƤ open in (Ƴ,σƝ), Ɗ⊂N and φƝ 1(N)⊂M

Sufficiency: Let W be any ƝŚƤ closed set of (Ҳ ,τƝ). Put Ɗ=Y φƝ(W), then we have φƝ 1(Ɗ) ⊂X W and X-W is ƝŚƤ open in (Ҳ ,τƝ). There exists a ƝƓŚƤ open set N of (Ƴ,σƝ) such that Ɗ=Y φƝ(W))⊂N and φƝ 1(N) ⊂X W. Therefore, we obtain φƝ(W)=Y N and hence φƝ(W) is ƝƤ ƓŚƤ closed in (Ƴ,σƝ). This show that φƝ is ƝƤ ƓŚƤ closed.

Theorem 3.2.8: If φƝ ∶(Ҳ ,τƝ) →(Ƴ,σƝ) is a Neutrosophic semi pre-irresolute pre ƓŚƤ-closed surjection and (Ҳ ,τƝ) is semi pre normal. Then (Ƴ,σƝ) is (ƝŚƤ ,ƝƓŚƤ)- normal Proof: Let P and Q be any disjoint ƝƓŚƤ closed sets of (Ƴ,σƝ). Then φƝ 1(P) and φƝ 1(Q) are disjoint Neutrosophic semi pre closed sets of (Ҳ ,τƝ), as φƝ is Neutrosophic semi pre-irresolute. Since (Ҳ ,τƝ) is Neutrosophic semi pre normal exist disjoint Neutrosophic semi pre open sets M and N of (Ҳ ,τƝ) such that φƝ 1(P)⊂M and φƝ 1(Q)⊂N. Since φƝ is Neutrosophic pre ƓŚƤ-closed. By theorem 3.2.7 there exists ƝƓŚƤ open sets G and H such that P⊂G, Q⊂��, φƝ 1(��)⊂M and φƝ 1(H)⊂N. Since M and N are disjoint, we have ��∩�� =φ This show that (Ƴ,σƝ) is (ƝŚƤ ,ƝƓŚƤ)- normal

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