Vol.14 / Issue 80 / Oct / 2023 International Bimonthly (Print) – Open Access ISSN: 0976 – 0997
RESEARCH ARTICLE
Generalized Semi Pre Connected Space in Neutrosophic Topological Spaces
Chandramathi N1 and Rajeshwaran N2 *
1Department of Mathematics, Government Arts College, Udumalpet-642126, Tiruppur, Tamil Nādu, India.
2Research Scholar, Department of Mathematics, Government Arts College,Udumalpet-642126, Tiruppur Tamil Nadu, India.
Received: 16 Aug 2023
Revised: 30 Aug 2023
*Address for Correspondence Rajeshwaran N Research Scholar, Department of Mathematics, Government Arts College,Udumalpet-642126, Tiruppur Tamil Nadu, India. E. Mail: rajeshw851@gmail.com
Accepted: 04 Sep 2023
This is an Open Access Journal / article distributed under the terms of the Creative Commons Attribution License (CC BY-NC-ND 3.0) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. All rights reserved.
Yashoda
ABSTRACT
In this manuscript, we inaugurate Neutrosophic generalized semi-pre connected space. We investigate its properties. Also, we add some improvisation of neutrosophic generalized semi pre connected space.
Keywords: Neutrosophic Topology, Neutrosophic generalized semi pre closed sets; Neutrosophic generalized semi pre continuous; Neutrosophic generalized semi pre connected space.
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 [11] led to the development of generalized Neutrosophic sets and generalized 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.
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
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Vol.14 / Issue 80 / Oct / 2023 International Bimonthly (Print) – Open Access ISSN: 0976 – 0997
Chandramathi
and Rajeshwaran
(NT1)0 ,1 ∈ τ
(NT2)G ∩G ∈ τ (NT3)∪G ∈ τ ,∀{G:i∈J}⊆ τ Here (X , τ ) is called a neutrosophic topological space (NTS for short).
Definition 2.2: [10] Let A A be two Neutrosophic Sets (NS for Short) of the form A ={⟨X, µ (X), σ (X), γ (X)⟩:xϵX} , A ={⟨X,µ (X), σ (X), γ (X)⟩:xϵX}
A ⊆A if and only if µ (X)≤ µ (X), σ (X)≤ σ (X)and γ (X)≥ γ (X)forallx∈X
(b) A ={⟨X, γ (X),1 σ (X), µ (X)⟩:xϵX}
(c)A ∩A ={⟨X,µ (X)⋀µ (X), σ (X)⋀ σ (X), γ (X)⋁γ (X)⟩:xϵX}
(d)A ∪A ={⟨X,µ (X)⋁µ (X), σ (X)⋁ σ (X), γ (X)⋀γ (X)⟩:xϵX}
Definition 2.3: [25] Let (X , τ ) be a neutrosophic topological space. A subset A of (X , τ ) is called Neutrosophic generalized semi pre closed set [Neutrosophic gsp-closed] if Nspcl (A ) ⊆ W, whenever A ⊆ W and W is NOS.
Definition 2.4: [25] A NT (X , τ ) is said to be an Neutrosophic C5-connected (NC5-connected for short) space if the only NS which are both a NOS and a NCS are 0 and1
Definition 2.5: [26] A bijection g:(X , τ ) →(X , σ ) is denoted NGSP homeomorphism if g is both NGSP continuous and NGSP open map.
Neutrosophic Generalized Semi Pre Connected Space
Definition.3.1: A Neutrosophic topology Ҳ , τƝ issaidtobe Neutrosophic generalized semi pre connected space (ƝGSƤ connected space for short) if the only NSs which are both a NGSPOS and a NGSPCS are 0Ɲ and1Ɲ
Example3.2: Let Ҳ ={a,b} and τƝ ={0Ɲ,1Ɲ,Ⱪ} be a NT on Ҳ ,τƝ where Ⱪ ={x,〈05,06,05〉,〈05,04,05〉} .Then Ҳ ,τƝ is a NGSP Connected space.
Theorem: 3.3: Every NGSP connected space is NC5- connected space but not conversely. Proof: Let Ҳ ,τƝ be NGSP-connected space. Suppose Ҳ , τƝ is not an NC5- connected space, then there exists a proper NS Ɗ which is both NOS and a NCS in Ҳ ,τƝ . That is, Ɗ is both NGSPOS and a NGSPCS in Ҳ ,τƝ . So, Ҳ ,τƝ is not a NGSP connected space. This is a contradiction. Therefore Ҳ ,τƝ be a NC5- connected space. Example3.4: Let Ҳ ={a,b} and Ⱪ ={〈05,06,05〉,〈05,04,05〉} .Then τƝ ={0Ɲ,1Ɲ,Ⱪ } is a NT on Ҳ ,τƝ . Then Ҳ , τƝ is a NC - connected space but not NGSP connected space, since NS Ⱪ in Ҳ , τƝ is both a NGSPCS and NGSPOS in Ҳ ,τƝ
Theorem 3.5: Every NGSP connected space is NGO-connected space but not conversely.
Proof: Let Ҳ ,τƝ is NGSP-connected space. Suppose Ҳ , τƝ is not a NGO- connected space, then there exists a proper NS Ɗ which is both a NGOS and NGCS in Ҳ ,τƝ Ɗ is both NGSPOS and NGSPCS in Ҳ , τƝ . This implies that Ҳ , τƝ is not a NGSP connected space. This is a contradiction. Therefore Ҳ , τƝ be a NGOconnected space.
Example 3.6: In Example 3.4, Ҳ , τƝ is a NGO-connected space but not a NGSP connectedspace.
Theorem 3.7: The NT Ҳ ,τƝ is a NGSP-connected space if and only if there exists no non-zero NGSPOS A and Ɗ in Ҳ , τƝ such that A = ƊC
Proof: Necessity: Assume that Ҳ , τƝ is an NGSP-connected space. Let A and Ɗ be two NGSPOS in (X1 , τN) such that A ≠ 0Ɲ ≠Ɗ and A =ƊC. Therefore ƊC is a NGSPCS. Since A ≠ 0Ɲ, Ɗ≠ 1Ɲ This implies Ɗ is a proper NS which is both a NGSPOS and a NGSPCS in Ҳ , τƝ Hence Ҳ , τƝ is not a NGSP connected space. But this is a contradiction to our hypothesis. Thus there exists no non-zero NGSPOS A and Ɗ in Ҳ , τƝ such that A =ƊC
Sufficiency: Let A both a NGSPOS and NGSPCS in Ҳ , τƝ such that 1Ɲ ≠ A ≠ 0Ɲ Now let Ɗ = AC. Then Ɗ is a NGSPOS and Ɗ ≠ 1Ɲ. This implies Ɗ = AC ≠ 0Ɲ which is a contradiction to our hypothesis. Therefore Ҳ , τƝ a NGSP connected space.
Vol.14 / Issue 80 / Oct / 2023 International Bimonthly (Print) – Open Access ISSN: 0976 – 0997
Chandramathi and Rajeshwaran
Theorem 3.8: A NT Ҳ,τƝ is an NGSP connected space if and only if there exists no non-zero NGSPOSs A and Ɗ in Ҳ,τƝ such that A =Ɗ , Ɗ =(Nspcl(A)) ,A=(Nspcl(Ɗ)) .
Proof: Necessity: Assume that there exist NSs A and Ɗ such that A≠0Ɲ ≠Ɗ , Ɗ =A ,Ɗ =(Nspcl(A)) , A= (Nspcl(Ɗ)) . Since (Nspcl(A)) and (Nspcl(Ɗ)) are NGSPOSs in Ҳ,τƝ A and Ɗ are NGSPOSs in Ҳ,τƝ . This implies Ҳ,τƝ is not a NGSP connected space, which is a contradiction. Therefore there exists no non-zero NGSPOSs A and Ɗ in Ҳ,τƝ such that A =Ɗ , Ɗ =(Nspcl(A)) ,A=(Nspcl(Ɗ)) .
Sufficiency: Let A be both a NGSPOS and a NGSPCS in Ҳ,τƝ such that 1Ɲ ≠A≠0Ɲ Now by taking Ɗ =A , we obtain a contradiction to our hypothesis. Hence Ҳ,τƝ is a NGSP connected space.
Remark 3.9: Every Neutrosophic semi-pre T / space is an Neutrosophic semi-pre T / space but not conversely. Proof: Let Ҳ,τƝ be a Neutrosophic semi-pre T / space. Let Ɗ be an in- Neutrosophic generalized semi-pre closed set in Ҳ,τƝ By hypothesis Ɗ is a Neutrosophic closed set. Since every Neutrosophic closed set is a Neutrosophic semi-preclosedset, Ɗ isaNeutrosophic semi-preclosedset in Ҳ,τƝ Hence Ҳ,τƝ is a Neutrosophic semi-pre T / space.
Example 3.10: In Example 3.4 the NTS Ҳ,τƝ is a NT / space but not a NSPT / space, since the NS Ɗ is a NGSP closed set in Ҳ,τƝ but not a NC in Ҳ,τƝ , since Ncl(Ɗ)=Ɗ ≠Ɗ
Theorem 3.11: If φƝ ∶ Ҳ,τƝ → Ƴ,σƝ is a NGSP continuous surjection and Ҳ,τƝ is a NGSP connected space, then Ƴ,σƝ is a NC5-connected space.
Proof: Let Ҳ,τƝ be a NGSP connected space. Suppose Ƴ,σƝ is not an NC5-connected space, then there exists a proper NS Ɗ which is both a NOS and a NCS in Ƴ,σƝ . Since φƝ is a NGSP continuous mapping, φƝ (Ɗ) is both a NGSPOS and NGSPCS in Ҳ,τƝ This is a contradiction to our hypothesis. Hence Ƴ,σƝ be a NC5-connected space.
Theorem 3.12: If φƝ ∶ Ҳ,τƝ → Ƴ,σƝ is a NGSP irresolute surjection and Ҳ,τƝ is NGSP connected space, then Ƴ,σƝ is also NGSP connected space.
Proof: Suppose Ƴ,σƝ is not a NGSP connected space, then there exists a proper NS Ɗ such that Ɗ is both NGSPOS and NGSPCS in Ƴ,σƝ Since φƝ is a NGSP irresolute mapping, φƝ (Ɗ) is both NGSPOS and NGSPCS in Ҳ,τƝ This is a contradiction to our hypothesis. Hence Ƴ,σƝ is NGSP connected space.
Theorem 3.13: If a NT Ҳ,τƝ is NGSP connected between two NS A and B, then it is NC5-connected between two NSs A and B but the converse may not be true in general.
Proof: Suppose Ҳ,τƝ is not NC5-connected between A and B, then there exists a NOS Ɗ in Ҳ,τƝ such that A⊆Ɗ and Ɗq . Since every NOS is a NGSPOS,there exists a NGSPOS Ɗ in Ҳ,τƝ such that A⊆Ɗ and Ɗq
This implies Ҳ,τƝ is not NGSP connected between A and B, a contradiction to our hypothesis. So, Ҳ,τƝ is NC5-connected between A and B
Example3.14: Let Ҳ={a,b} and Ⱪ ={x,〈05,04,05〉,〈05,06,05〉} .Then τƝ ={0Ɲ,1Ɲ,Ⱪ } is a NT on Ҳ,τƝ . Let A={x,〈04,04,04〉,〈06,06,06〉}and B={x,〈03,03,03〉,〈04,04,04〉} be two NS in Ҳ,τƝ Then Ҳ,τƝ is a NC5connected between A and B, since there exists no NOS Ɗ in Ҳ,τƝ But it is not NGSP connected between the two NS A and B, since there exists an NGSPOS Ɗ={〈04,04,04〉,〈05,05,05〉} in Ҳ,τƝ such that A ⊆Ɗ and Ɗq
Theorem 3.15: A NT Ҳ,τƝ is NGSP connected between two NS A and B if and only if there is no NGSPOS and NGSPCS Ɗ in Ҳ,τƝ such thatA⊆Ɗ⊆B
Proof: Necessity: Let Ҳ,τƝ be NGSP connected between A and B. Suppose that there exists a NGSPOS and NGSPCS Ɗ in Ҳ,τƝ such that A⊆Ɗ⊆B , then Ɗ q B. andA⊆Ɗ This implies Ҳ,τƝ is not NGSP connected between A and B, by Definition 3.2 A contradiction to our hypothesis. Therefore there exists no NGSPOS and a NGSPCS Ɗ in Ҳ,τƝ such that A⊆Ɗ⊆B .
Sufficiency: Suppose that Ҳ,τƝ is not NGSP connected between A and B. Then there exists a NGSPOS Ɗ in Ҳ,τƝ such that A⊆Ɗ and Ɗ q B. This implies that there exists a NGSPOS Ɗ in Ҳ,τƝ such that A⊆Ɗ⊆B This is a contradiction to our hypothesis. Hence Ҳ,τƝ is NGSP connected.
Vol.14 / Issue 80 / Oct / 2023 International Bimonthly (Print) – Open Access ISSN: 0976 – 0997
Theorem 3.16. Let Ҳ,τƝ be an NTS and A and B be NSs in Ҳ,τƝ If AqB, then Ҳ,τƝ is Neutrosophic generalizedsemi-preconnected between A andB.
Proof: Suppose Ҳ,τƝ is not Neutrosophic generalized semi-pre connected between A and B. Then there exists a Neutrosophic generalized semi-pre open set Ɗ in Ҳ,τƝ such that A⊆Ɗ and Ɗ⊆B This implies that A⊆B That is Aq B. This is a contradiction to our hypothesis. Therefore Ҳ,τƝ is Neutrosophic generalized semi-pre connected between A and B.
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