Neutrosophic Beta Omega Closed Sets in Neutrosophic Topological Spaces

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129

Neutrosophic Beta Omega Closed Sets in Neutrosophic Topological Spaces

S. Pious Missier^1,b), A. Anusuya^2,a), A. Nagarajan^3,c)

1Head & Associate Professor, Department of Mathematics, Don Bosco College of Arts and Science,

(Affiliated to Manonmaniam Sundaranar University,Tirunelveli) Keela Eral, Thoothukudi, Tamil Nadu-628 908, India.

2 Research Scholar(Reg.No-19222232092024), V.O.Chidambaram College, (Affiliated to Manonmaniam Sundaranar University,Tirunelveli), Thoothukudi-628 003,

India.

3Head & Associate Professor, V. O. Chidambaram College, Thoothukudi, Tamilnadu-628 008, India. nagarajan.voc@gmail.com

b)spmissier@gmail.com

a)anuanishnakul@gmail.com

c)nagarajan.voc@gmail.com

Article Info

Page Number: 129 – 137 Publication Issue:

Vol. 71 No. 3s (2022)

Article History

Article Received: 22 April 2022 Revised: 10 May 2022

Accepted: 15 June 2022 Publication: 19 July 2022

Abstract

We introduce and examine several applications of neutrosophic beta omega closed sets namely 𝑇_{𝑁𝛽𝜔 −𝑠𝑝}, 𝑃𝑇_{𝑁𝛽𝜔 −𝑠𝑝}, 𝛽𝑇_{𝑁𝛽𝜔 −𝑠𝑝}, 𝑆𝐺𝑇_{𝑁𝛽𝜔 −𝑠𝑝}, 𝐺𝑆𝑇_{𝑁𝛽𝜔 −𝑠𝑝}in this study. We also introduce the concept of neutrosophic betaomega continuous mapping in neutrosophic topology. Furthermore, we study the relation between the neutrosophic betaomega continuous mapping with the already existing neutrosophic continuous mapping. In addition, we discuss the properties of neutrosophic betaomega continuous mapping.

Keywords:𝑇_{𝑁𝛽𝜔 −𝑠𝑝}, 𝑃𝑇_{𝑁𝛽𝜔 −𝑠𝑝}, 𝛽𝑇_{𝑁𝛽𝜔 −𝑠𝑝}, 𝑆𝐺𝑇_{𝑁𝛽𝜔 −𝑠𝑝}, 𝐺𝑆𝑇_{𝑁𝛽𝜔 −𝑠𝑝},neutrosophic beta omega continuous mapping.

AMS Mathematics Subject Classification:18B30,03E72 I. INTRODUCTION

Fuzzy set theory introduced by Zadeh[11] has laid the foundation for the new mathematical theories in the research of mathematics. Later, Attanasov[2] developed the concept of intuitionistic fuzzy sets. Smarandache[7] was the first to coin the term "neutrosophic set.".

Neutrosophic operations and neutrosophic continuous functions have been investigated by Salama[10]. Here, we shall introduce the applications of neutrosophic beta omega closed set and neutrosophic continuity of neutrosophic-beta-omega sets. Also we present the implications of neutrosophic –beta-omega-continuous mapping.

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II. PRELIMINARIES

Definition 2.1.[7] If ∆_𝑁 is a nonempty fixed set. A neutrosophic set (N-Set) G_N is a form factor G_N = {𝜁, μ_{G N}(𝜁), σ_G_N(𝜁), υ_G_N(𝜁): 𝜁∆_𝑁} where μ

G N(𝜁), σ_G_N(𝜁) and υ_G_N(𝜁)represents the membership level, non-determination level and non-membership level respectively of each element 𝜁∆_𝑁to the set 𝐺_𝑁. A Neutrosophic set 𝐺_𝑁 = {𝜁, μ

G N(𝜁), σ_G_N(𝜁), υ_G_N(𝜁): 𝜁∆_𝑁} can be recognized as a triple order μ

G N, σ_G_N, υ_G_Nin ]0,1⁺[on

∆_𝑁.

Definition 2.2. [1] For any two sets G_Nand H_N,

1. G_N ⊆H_Nμ_{G N}(𝜁) μ_{H N} (𝜁), σ_G_N(𝜁) σ_H_N(𝜁) and υ_G_N(𝜁) υ_H_N(𝜁)  𝜁∆_𝑁 2. G_N⋂ H_N = 𝜁, μ_{G N}(𝜁) μ_{H N}(𝜁), σ_G_N(𝜁) σ_H_N(𝜁), υ_G_N(𝜁) υ_H_N(𝜁) 

3. G_N⋃ H_N = 𝜁, μ_{G N}(𝜁) μ_{H N}(𝜁), σ_G_N(𝜁) σ_H_N(𝜁), υ_G_N(𝜁) υ_H_N(𝜁)  4. G_N^C = {𝜁, υ_G_N(𝜁), 1- σ_G_N(𝜁), μ

G N(𝜁): 𝜁∆_𝑁} 5. 0_N = {𝜁, 0, 0, 1: 𝜁∆_𝑁}

6. 1N = {𝜁, 1, 1, 0: 𝜁∆_𝑁}

Definition 2.3. [9] A neutrosophic topology (NT) is a family 𝜏_𝑁 of neutrosophic subsets of∆_𝑁whose member satisfy the following axioms:

1. 0_𝑁, 1_𝑁 ∈𝜏_𝑁

2. ℘_𝑁 ∩ ℛ_𝑁 ∈𝜏_𝑁for any ℘_𝑁, ℛ_𝑁 ∈ 𝜏_𝑁 3. ∪ ℛ_{𝑁 𝑖} ∈ 𝜏_𝑁∀{ ℛ_{𝑁 𝑖} : i ∈J} ⊆ 𝜏_𝑁

Any neutrosophic set in 𝜏_𝑁 is a neutrosophic open set (NO-set). If a neutrosophic set G_N is a neutrosophic open set,then its complement G_N^c is a neutrosophic closed set(NC-set) in

∆_𝑁.Also, neutrosophic topological space (NTS) is denoted by (∆_𝑁, 𝜏_𝑁).

Definition 2.4. [8] A N-Set𝐺_𝑁 of (∆_𝑁,𝜏_𝑁) is calledneutrosophic beta omegaclosed(𝑁𝛽𝜔𝐶) if 𝛽𝑐𝑙_𝑁(𝐺_𝑁) 𝑈_𝑁 whenever 𝐺_𝑁𝑈_𝑁 and 𝑈_𝑁 is N𝜔𝑂 in (∆_𝑁,𝜏_𝑁).

Definition 2.5. [8]For every set 𝐺_𝑁 ⊆ ∆_𝑁, 𝜏_𝑁, we define 1. 𝛽𝜔𝑐𝑙_𝑁(𝐺_𝑁)= ∩{𝑉_𝑁: 𝐺_𝑁⊆ 𝑉_𝑁, 𝑉_𝑁 ∈ 𝑁𝛽𝜔C∆_𝑁, 𝜏_𝑁}.

2. 𝛽𝜔𝑖𝑛𝑡_𝑁(𝐺_𝑁) = ∪{𝑈_𝑁: 𝑈_𝑁 ⊆ 𝐺_𝑁 and 𝑈_𝑁 ∈ 𝑁𝛽𝜔𝑂∆_𝑁, 𝜏_𝑁}

Definition2.6. [6]A mapping𝑓: ∆_𝑁, 𝜏_𝑁 → 𝛤_𝑁, 𝜎_𝑁 is said to be

1. Neutrosophic-continuous(𝑁_{𝑐𝑜𝑛𝑡}) if 𝑓⁻¹(𝐺_𝑁) is 𝑁𝐶set in ∆_𝑁, 𝜏_𝑁for each𝑁𝐶-set 𝐺_𝑁 in (𝛤_𝑁, 𝜎_𝑁).

2. Neutrosophic-pre-continuous(𝑁𝑃_{𝑐𝑜𝑛𝑡}) if 𝑓⁻¹(𝐺_𝑁) is 𝑁𝑃𝐶set in ∆_𝑁,𝜏_𝑁for each𝑁𝐶-set 𝐺_𝑁 in (𝛤_𝑁, 𝜎_𝑁).

3. Neutrosophic-beta-continuous(𝑁𝛽_{𝑐𝑜𝑛𝑡}) if 𝑓⁻¹(𝐺_𝑁) is 𝑁𝛽Cset in ∆_𝑁, 𝜏_𝑁for each𝑁𝐶-set 𝐺_𝑁 in (𝛤_𝑁, 𝜎_𝑁)

4. Neutrosophic-alpha-generalized-continuous(𝑁𝛼𝐺_{𝑐𝑜𝑛𝑡}) if 𝑓⁻¹(𝐺_𝑁) is 𝑁𝛼𝐺𝐶set in ∆_𝑁, 𝜏_𝑁for each𝑁𝐶-set 𝐺_𝑁 in (𝛤_𝑁, 𝜎_𝑁).

III. APPLICATIONS OF NEUTROSOPHIC BETA OMEGA CLOSED SETS Definition 3.1.A neutrosophic topological space(∆_𝑁,𝜏_𝑁) is called a

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131 2. Pre-𝑇_𝑁𝛽𝜔-space(𝑃𝑇_{𝑁𝛽𝜔−𝑠𝑝}) if all𝑁𝛽𝜔𝐶set is NPC.

3. Beta-𝑇_𝑁𝛽𝜔-space(𝛽𝑇_{𝑁𝛽𝜔−𝑠𝑝}) if all𝑁𝛽𝜔𝐶set is N𝛽𝐶.

4. Semi-generalized-𝑇_𝑁𝛽𝜔-space(𝑆𝐺𝑇_{𝑁𝛽𝜔−𝑠𝑝}) if all𝑁𝛽𝜔𝐶set is NSGC.

5. Generalized-semi-𝑇_{𝑁𝛽 𝜔}-space(𝐺𝑆𝑇_{𝑁𝛽𝜔−𝑠𝑝}) if all𝑁𝛽𝜔𝐶set is NGSC.

Example 3.1.Let ∆_𝑁 = [0, 1], 𝜏_𝑁 = {0_𝑁, 𝐺_𝑁^𝑐 λ , 𝐻_𝑁^𝑐 λ , 1_𝑁},(∆_𝑁, 𝜏_𝑁) = {0_𝑁, 𝐺_𝑁(λ), 𝐻_𝑁 λ , 1_𝑁}. where

𝐺_𝑁(λ) =

< 𝜆, 𝜆, 1 − 𝜆 > 𝑤𝑕𝑒𝑟𝑒 0 ≤ 𝜆 ≤¹

2

< 1 − 𝜆, 1 − 𝜆, 𝜆 >¹

2≤ 𝜆 ≤ 1 and 𝐻_𝑁(λ) =

< 𝜆, 𝜆, 1 − 𝜆 > 𝑤𝑕𝑒𝑟𝑒 0 ≤ 𝜆 ≤¹

2

< ¹

2, ¹

2> ¹

2≤ 𝜆 ≤ 1 . Then 𝜏_𝑁 is a NT. Here, (∆_𝑁, 𝜏_𝑁) is 𝑇_{𝑁𝛽𝜔−𝑠𝑝}.

Example 3.2. Let ∆_𝑁 = {𝜆₁,𝜆₂, 𝜆₃}, 𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1_𝑁} where 𝐺_𝑁 = < 𝜁, ^𝜆¹

0.7,^𝜆²

0.6,^𝜆³

0.5 ,

𝜆₁ 0.6,^𝜆²

0.6,^𝜆³

0.7 , ^𝜆¹

0.3,^𝜆²

0.5,^𝜆³

0.4 >.Then 𝜏_𝑁 is a NT. Then 𝑁𝛽𝜔𝐶(∆_𝑁, 𝜏_𝑁) = (∆_𝑁, 𝜏_𝑁)-{ 𝑃_𝑁 / 𝐺_𝑁⊆𝑃_𝑁 ⊂ 1_𝑁} = 𝑁𝑃𝐶(∆_𝑁, 𝜏_𝑁). Therefore,every 𝑁𝛽𝜔𝐶set is𝑁𝑃𝐶. Hence(∆_𝑁,𝜏_𝑁) is 𝑃𝑇_{𝑁𝛽𝜔−𝑠𝑝}.

Example 3.3. Let ∆_𝑁 = {𝜆₁,𝜆₂, 𝜆₃}, 𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1_𝑁} where 𝐺_𝑁 = < 𝜁, ^𝜆¹

0.6,^𝜆²

0.6,^𝜆³

0.6 ,

𝜆₁ 0.7,^𝜆²

0.7,^𝜆³

0.7 , ^𝜆¹

0.3,^𝜆²

0.3,^𝜆³

0.4 >.Then 𝜏_𝑁 is a NT. Then 𝑁𝛽𝜔𝐶(∆_𝑁, 𝜏_𝑁) = (∆_𝑁, 𝜏_𝑁)-{ 𝑃_𝑁 / 𝐺_𝑁⊆𝑃_𝑁 ⊂ 1_𝑁} = 𝑁𝛽𝐶(∆_𝑁, 𝜏_𝑁). Therefore,every 𝑁𝛽𝜔𝐶set is NβC. Hence(∆_𝑁, 𝜏_𝑁) is 𝛽𝑇_{𝑁𝛽𝜔−𝑠𝑝}.

Example 3.4. Let ∆_𝑁 = {𝜆₁, 𝜆₂, 𝜆₃}, 𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1_𝑁} where 𝐺_𝑁 = < 𝜁, ^𝜆¹

0.3,^𝜆²

0.4,^𝜆³

0.4 ,

𝜆₁ 0.2,^𝜆²

0.3,^𝜆³

0.1 , ^𝜆¹

0.8,^𝜆²

0.8,^𝜆³

0.9 >.Then 𝜏_𝑁 is a NT. Then 𝑁𝛽𝜔𝐶(∆_𝑁, 𝜏_𝑁) = (∆_𝑁, 𝜏_𝑁) = 𝑁𝐺𝑆𝐶(∆_𝑁, 𝜏_𝑁). Therefore every 𝑁𝛽𝜔𝐶set is 𝑁𝐺𝑆𝐶. Hence(∆_𝑁, 𝜏_𝑁) is 𝐺𝑆𝑇_{𝑁𝛽𝜔−𝑠𝑝}.

Example 3.5. Let ∆_𝑁 = {𝜆₁, 𝜆₂, 𝜆₃},𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1_𝑁} where 𝐺_𝑁 = < 𝜁, ^𝜆¹

0.3,^𝜆²

0.3,^𝜆³

0.3 ,

𝜆₁ 0.1,^𝜆²

0.3,^𝜆³

0.1 , ^𝜆¹

0.8,^𝜆²

0.8,^𝜆³

0.8 >.Then 𝜏_𝑁 is a NT. Then 𝑁𝛽𝜔𝐶(∆_𝑁, 𝜏_𝑁) = (∆_𝑁, 𝜏_𝑁) = 𝑁𝑆𝐺𝐶(∆_𝑁, 𝜏_𝑁). Therefore every 𝑁𝛽𝜔𝐶set is 𝑁𝑆𝐺𝐶. Hence(∆_𝑁, 𝜏_𝑁)is 𝑆𝐺𝑇_{𝑁𝛽𝜔−𝑠𝑝}.

Theorem3.1.

(a). Every 𝑃𝑇_{𝑁𝛽𝜔 −𝑠𝑝} is 𝛽𝑇_{𝑁𝛽𝜔 −𝑠𝑝}. (b). Every 𝑇_{𝑁𝛽𝜔 −𝑠𝑝} is 𝑃𝑇_{𝑁𝛽𝜔 −𝑠𝑝}. (c). Every 𝑇_{𝑁𝛽𝜔 −𝑠𝑝} is 𝛽𝑇_{𝑁𝛽𝜔 −𝑠𝑝}. (d). Every 𝑇_{𝑁𝛽𝜔 −𝑠𝑝} is 𝐺𝑆𝑇_{𝑁𝛽𝜔 −𝑠𝑝}. (e). Every 𝑇_{𝑁𝛽𝜔 −𝑠𝑝} is 𝑆𝐺𝑇_{𝑁𝛽𝜔 −𝑠𝑝}.

Proof :

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(a). Let(∆_𝑁, 𝜏_𝑁) be𝑃𝑇_{𝑁𝛽𝜔 −𝑠𝑝}. We know that, every 𝑁𝑃𝐶 set is 𝑁𝛽𝐶 set and since every 𝑁𝛽𝜔𝐶set is 𝑁𝑃𝐶, we get 𝑁𝛽𝜔𝐶set is 𝑁𝛽𝐶 set.Hence(∆_𝑁, 𝜏_𝑁) is a 𝛽𝑇_{𝑁𝛽𝜔 −𝑠𝑝}.

(b), (c), (d) follow from the facts that every 𝑁𝐶 set is 𝑁𝑃𝐶 set, 𝑁𝛽𝐶 set, 𝑁𝐺𝑆C set.

𝑁𝑆𝐺𝐶 set respectively.

Remark 3.1.The examples below show that the converse of the preceding theorems does not have to be true.

Example 3.6.Take∆_𝑁 = {𝜆₁, 𝜆₂, 𝜆₃}, 𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1_𝑁} where 𝐺_𝑁 = < 𝜁, ^𝜆¹

0.3,^𝜆²

0.3,^𝜆³

0.4 ,

𝜆₁ 0.3,^𝜆²

0.3,^𝜆³

0.3 , ^𝜆¹

0.6,^𝜆²

0.6,^𝜆³

0.6 >.Then 𝜏_𝑁 is a NT. Then 𝑁𝛽𝜔𝐶(∆_𝑁, 𝜏_𝑁) = (∆_𝑁, 𝜏_𝑁) = 𝑁𝛽𝐶(∆_𝑁, 𝜏_𝑁). But 𝑁𝑃𝐶(∆_𝑁, 𝜏_𝑁) = (∆_𝑁, 𝜏_𝑁)-{𝑃_𝑁 / 𝐺_𝑁 ⊂ 𝑃_𝑁 ⊂ 𝐺_𝑁^𝐶}-{𝑄_𝑁 / 𝐺_𝑁 ⊂ 𝑄_𝑁 ⊈ 𝐺_𝑁^𝐶} .Hence(∆_𝑁, 𝜏_𝑁)is 𝛽𝑇_{𝑁𝛽𝜔−𝑠𝑝}but not 𝑃𝑇_{𝑁𝛽𝜔−𝑠𝑝}.

Example 3.7. Let ∆_𝑁 = {𝜆₁, 𝜆₂, 𝜆₃}, 𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1_𝑁} where 𝐺_𝑁 = < 𝜁, ^𝜆¹

0.7,^𝜆²

0.6,^𝜆³

0.5 ,

𝜆₁ 0.6,^𝜆²

0.6,^𝜆³

0.7 , ^𝜆¹

0.3,^𝜆²

0.5,^𝜆³

0.4 >.Then 𝜏_𝑁 is a NT. Then 𝑁𝛽𝜔𝐶(∆_𝑁, 𝜏_𝑁) = (∆_𝑁, 𝜏_𝑁)-{𝑃_𝑁 / 𝐺_𝑁⊆ 𝑃_𝑁 ⊂ 1_𝑁} = 𝑁𝑃𝐶(∆_𝑁, 𝜏_𝑁)and 𝑁𝐶(∆_𝑁, 𝜏_𝑁)={0_𝑁, 𝐺_𝑁^𝐶, 1_𝑁}. Hence(∆_𝑁, 𝜏_𝑁)is 𝑃𝑇_{𝑁𝛽𝜔−𝑠𝑝}but not 𝑇_{𝑁𝛽𝜔 −𝑠𝑝}.

Example 3.8. Let ∆_𝑁 = {𝜆₁,𝜆₂, 𝜆₃}, 𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1_𝑁} where 𝐺_𝑁 = <ζ, ^𝜆¹

0.6,^𝜆²

0.6,^𝜆³

0.6 ,

𝜆₁ 0.7,^𝜆²

0.7,^𝜆³

0.7 , ^𝜆¹

0.3,^𝜆²

0.3,^𝜆³

0.4 >.Then 𝜏_𝑁 is a NT. Then 𝑁𝛽𝜔𝐶(∆_𝑁, 𝜏_𝑁)= (∆_𝑁, 𝜏_𝑁)-{𝑃_𝑁 / 𝐺_𝑁⊆ 𝑃_𝑁 ⊂ 1_𝑁} = 𝑁𝛽𝐶(∆_𝑁, 𝜏_𝑁) and 𝑁𝐶(∆_𝑁, 𝜏_𝑁) ={0_𝑁, 𝐺_𝑁^𝐶, 1_𝑁}. Hence(∆_𝑁, 𝜏_𝑁)is 𝛽𝑇_{𝑁𝛽𝜔−𝑠𝑝}but not 𝑇_{𝑁𝛽𝜔 −𝑠𝑝}.

Example 3.9. Let ∆_𝑁 = {𝜆₁, 𝜆₂, 𝜆₃}, 𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1_𝑁} where 𝐺_𝑁 = <ζ, ^𝜆¹

0.3,^𝜆²

0.4,^𝜆³

0.4 ,

𝜆₁ 0.2,^𝜆²

0.3,^𝜆³

0.1 , ^𝜆¹

0.8,^𝜆²

0.8,^𝜆³

0.9 >.Then 𝜏_𝑁 is a NT. Then 𝑁𝛽𝜔𝐶(∆_𝑁, 𝜏_𝑁) = (∆_𝑁, 𝜏_𝑁)= 𝑁𝐺𝑆𝐶(∆_𝑁, 𝜏_𝑁) and 𝑁𝐶(∆_𝑁, 𝜏_𝑁) ={0_𝑁, 𝐺_𝑁^𝐶, 1_𝑁}. Hence(∆_𝑁, 𝜏_𝑁)is 𝐺𝑆𝑇_{𝑁𝛽𝜔−𝑠𝑝}but not 𝑇_{𝑁𝛽𝜔 −𝑠𝑝}.

Example 3.10.Let ∆_𝑁 = {𝜆₁, 𝜆₂, 𝜆₃}, 𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1_𝑁} where 𝐺_𝑁 = < 𝜁, ^𝜆¹

0.3,^𝜆²

0.3,^𝜆³

0.3 ,

𝜆₁ 0.1,^𝜆²

0.3,^𝜆³

0.1 , ^𝜆¹

0.8,^𝜆²

0.8,^𝜆³

0.8 >.Then 𝜏_𝑁is a NT. Then 𝑁𝛽𝜔𝐶(∆_𝑁, 𝜏_𝑁) = (∆_𝑁, 𝜏_𝑁)= 𝑁𝑆𝐺𝐶(∆_𝑁, 𝜏_𝑁) and 𝑁𝐶(∆_𝑁, 𝜏_𝑁) ={0_𝑁,𝐺_𝑁^𝐶, 1_𝑁}. Therefore every 𝑁𝛽𝜔𝐶 set is 𝑁𝑆𝐺𝐶. Hence(∆_𝑁, 𝜏_𝑁) is 𝑆𝐺𝑇_{𝑁𝛽𝜔−𝑠𝑝}but not 𝑇_{𝑁𝛽𝜔 −𝑠𝑝}.

IV. NEUTROSOPHIC BETA OMEGACONTINUOUS MAPPING

Definition4.1. A mapping 𝑓: ∆_𝑁, 𝜏_𝑁 → 𝛤_𝑁, 𝜎_𝑁 is calledneutrosophic betaomegacontinuous(𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}) if 𝑓⁻¹(𝐺_𝑁) is 𝑁𝛽𝜔𝐶set in (∆_𝑁, 𝜏_𝑁)for all𝑁𝐶-set 𝐺_𝑁 in (𝛤_𝑁, 𝜎_𝑁).

Example 4.1. Let ∆_𝑁 = {𝜆₁, 𝜆₂, 𝜆₃}, Γ_N= {𝛿₁, 𝛿₂, 𝛿₃},𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1N} and 𝜎_𝑁 = {0_𝑁, 𝐻_𝑁, 𝑊_𝑁, 1N} where 𝐺_𝑁 = < 𝜁, ^𝜆¹

0.6,^𝜆²

0.7,^𝜆³

0.6 , ^𝜆¹

0.7,^𝜆²

0.8,^𝜆³

0.7 , ^𝜆¹

0.4,^𝜆²

0.3,^𝜆³

0.4 >, 𝐻_𝑁 = <

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133 𝜁, ^𝛿¹

0.7,^𝛿²

0.7,^𝛿³

0.7 , ^𝛿¹

0.8,^𝛿²

0.8,^𝛿³

0.7 , ^𝛿¹

0.3,^𝛿²

0.2,^𝛿³

0.2 >, 𝑊_𝑁 =

< 𝜁, ^𝛿¹

0.8,^𝛿²

0.7,^𝛿³

0.8 , ^𝛿¹

0.9,^𝛿²

0.8,^𝛿³

0.7 , ^𝛿¹

0.2,^𝛿²

0.2,^𝛿³

0.2 >.Then 𝜏_𝑁 and 𝜎_𝑁 are NTs. Define a mapping 𝑓: 𝛥_𝑁, 𝜏_𝑁 → 𝛤_𝑁, 𝜎_𝑁 by 𝑓(𝜆₁) = 𝛿₁, 𝑓(𝜆₂) = 𝛿₂ and 𝑓 𝜆₃ = 𝛿₃. Then 𝑓 ⁻¹(𝑉_𝑁) is NβωC in (∆_𝑁, 𝜏_𝑁) for every 𝑁𝐶set 𝑉_𝑁in (𝛤_𝑁, 𝜎_𝑁). Hence𝑓 is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}.

Theorem 4.1.

1. Every 𝑁_{𝑐𝑜𝑛𝑡} mapping is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping.

2. Every Nβ_cont mapping is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping.

3. Every 𝑁𝐺^∗_{𝑐𝑜𝑛𝑡} mappingis 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping.

4. Every 𝑁𝜓_{𝑐𝑜𝑛𝑡} mapping is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping.

5. Every 𝑁𝑊G^∗_{𝑐𝑜𝑛𝑡} mapping is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡} mapping.

6. Every 𝑁𝑃_{𝑐𝑜𝑛𝑡}mapping is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡} mapping.

7.

Proof:

1. Taking𝑓as a𝑁_{𝑐𝑜𝑛𝑡}mappingand𝑉_𝑁 be any 𝑁𝐶-setin (𝛤_𝑁, 𝜎_𝑁). Since,𝑓 is 𝑁_{𝑐𝑜𝑛𝑡} mapping, we get 𝑓⁻¹(𝑉_𝑁) is 𝑁𝐶-set in (∆_𝑁, 𝜏_𝑁) implies 𝑓⁻¹(𝑉_𝑁) is 𝑁𝛽𝜔𝐶set. Hence 𝑓 is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping in (∆_𝑁, 𝜏_𝑁).

2. Let 𝑓 be any Nβ_{𝑐𝑜𝑛𝑡}mapping and 𝑉_𝑁 be any 𝑁𝐶-set in (𝛤_𝑁, 𝜎_𝑁). Since 𝑓 is Nβ_{𝑐𝑜𝑛𝑡}mapping, we get 𝑓⁻¹(𝑉_𝑁) is 𝑁𝛽𝐶 in ∆_𝑁, 𝜏_𝑁 .Since every𝑁𝛽𝐶set is𝑁𝛽𝜔𝐶,𝑓⁻¹(𝑉_𝑁) is 𝑁𝛽𝜔𝐶. Hence 𝑓 is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping in (∆_𝑁, 𝜏_𝑁).

3. Let 𝑓 be𝑁𝐺^∗_{𝑐𝑜𝑛𝑡} mapping and 𝑉_𝑁 be any 𝑁𝐶-set in (𝛤_𝑁, 𝜎_𝑁). Since 𝑓 is 𝑁𝐺^∗_{𝑐𝑜𝑛𝑡}mapping, we get 𝑓⁻¹(𝑉_𝑁) is 𝑁𝐺*C in (∆_𝑁, 𝜏_𝑁). Therefore, 𝑓⁻¹(𝑉_𝑁) is 𝑁𝛽𝜔𝐶, because every 𝑁𝐺^∗𝐶 set is 𝑁𝛽𝜔𝐶 set. Hence 𝑓 is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping in (∆_𝑁, 𝜏_𝑁) 4. Let 𝑓 be 𝑁𝜓_{𝑐𝑜𝑛𝑡}mapping and 𝑉_𝑁 be any 𝑁𝐶-set in (𝛤_𝑁, 𝜎_𝑁). Since 𝑓 is 𝑁𝜓_{𝑐𝑜𝑛𝑡}mapping, we

get 𝑓⁻¹(𝑉_𝑁) is 𝑁𝜓𝐶in (∆_𝑁, 𝜏_𝑁). Since all𝑁𝜓𝐶 set is 𝑁𝛽𝜔𝐶,𝑓⁻¹(𝑉_𝑁) is NβωC. Hence 𝑓 is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping in (∆_𝑁, 𝜏_𝑁).

5. Let 𝑓 be 𝑁𝑊G^∗_{𝑐𝑜𝑛𝑡}mapping and 𝑉_𝑁 be any 𝑁𝐶-set in (𝛤_𝑁, 𝜎_𝑁). Since 𝑓 is 𝑁𝑊G^∗_{𝑐𝑜𝑛𝑡}mapping, we get 𝑓⁻¹(𝑉_𝑁) is 𝑁𝑊𝐺^∗𝐶 in (∆_𝑁, 𝜏_𝑁).We know that every 𝑁𝑊𝐺^∗𝐶 set is 𝑁𝛽𝜔 set,𝑓⁻¹(𝑉_𝑁) is 𝑁𝛽𝜔𝐶. Hence 𝑓 is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping in (∆_𝑁, 𝜏_𝑁).

6. Let 𝑓 be 𝑁𝑃_{𝑐𝑜𝑛𝑡}mapping in (∆_𝑁, 𝜏_𝑁). Let 𝑉_𝑁 be any 𝑁𝐶-set in (𝛤_𝑁, 𝜎_𝑁)). Since 𝑓 is 𝑁_{𝑐𝑜𝑛𝑡}mapping, we get 𝑓⁻¹(𝑉_𝑁) is 𝑁𝑃𝐶in (∆_𝑁, 𝜏_𝑁). Every 𝑁𝑃𝐶 is 𝑁𝛽𝜔𝐶 that implies𝑓⁻¹(𝑉_𝑁) is𝑁𝛽𝜔𝐶. Hence 𝑓 is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping in (∆_𝑁, 𝜏_𝑁).

Example 4.2. Let ∆_𝑁 = {𝜆₁, 𝜆₂, 𝜆₃}, 𝛤_𝑁 = {𝛿₁, 𝛿₂, 𝛿₃},𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1N} and 𝜎_𝑁 = {0_𝑁, 𝐻_𝑁, 1N} where 𝐺_𝑁 = < 𝜁, ^𝜆¹

0.2,^𝜆²

0.2,^𝜆³

0.1 , ^𝜆¹

0.7,^𝜆²

0.7,^𝜆³

0.7 , ^𝜆¹

0.7,^𝜆²

0.7,^𝜆³

0.7 >, 𝐻_𝑁 =

< 𝜁, ^𝛿¹

0.2,^𝛿²

0.2,^𝛿³

0.1 , ^𝛿¹

0.2,^𝛿²

0.1,^𝛿³

0.2 , ^𝛿¹

0.7,^𝛿²

0.8,^𝛿³

0.8 >.Then 𝜏_𝑁 and 𝜎_𝑁 are NTs. Define a mapping 𝑓: 𝛥_𝑁, 𝜏_𝑁 → 𝛤_𝑁, 𝜎_𝑁 by 𝑓(𝜆₁) = 𝛿₁, 𝑓(𝜆₂) = 𝛿₂ and 𝑓 𝜆₃ = 𝛿₃. Then 𝑓 is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping but not 𝑁𝛽_{𝑐𝑜𝑛𝑡} mapping.

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Example 4.3. Let ∆_𝑁 = {𝜆₁, 𝜆₂, 𝜆₃}, 𝛤_𝑁 = {𝛿₁, 𝛿₂, 𝛿₃},𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1N} and 𝜎_𝑁 = {0_𝑁, 𝐻_𝑁, 𝑊_𝑁 1_N} where 𝐺_𝑁 = < 𝜁, _0.5^𝜆¹,_0.6^𝜆²,_0.5^𝜆³ , _0.4^𝜆¹,_0.3^𝜆²,_0.4^𝜆³ , _0.6^𝜆¹,_0.8^𝜆²,_0.7^𝜆³ >, 𝐻_𝑁 = < 𝜁,

𝛿₁ 0.6,^𝛿²

0.9,^𝛿³

0.7 , ^𝛿¹

0.8,^𝛿²

0.8,^𝛿³

0.7 , ^𝛿¹

0.3,^𝛿²

0.4,^𝛿³

0.3 >, 𝑊_𝑁 =

< 𝜁, ^𝛿¹

0.7,^𝛿²

0.9,^𝛿³

0.8 , ^𝛿¹

0.8,^𝛿²

0.9,^𝛿³

0.7 , ^𝛿¹

0.2,^𝛿²

0.4,^𝛿³

0.2 >.Then 𝜏_𝑁 and 𝜎_𝑁 are NTs. Define a mapping 𝑓: 𝛥_𝑁, 𝜏_𝑁 → 𝛤_𝑁, 𝜎_𝑁 by 𝑓(𝜆₁) = 𝛿₁, 𝑓(𝜆₂) = 𝛿₂ and 𝑓 𝜆₃ = 𝛿₃. Then 𝑓 is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping but not 𝑁𝐺^∗_{𝑐𝑜𝑛𝑡}mapping.

Example 4.4. Let ∆_𝑁 = {𝜆₁, 𝜆₂, 𝜆₃}, 𝛤_𝑁 = {𝛿₁, 𝛿₂, 𝛿₃},𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1N} and 𝜎_𝑁 = {0_𝑁, H_N, 1N} where 𝐺_𝑁 = < 𝜁, ^𝜆¹

0.3,^𝜆²

0.3,^𝜆³

0.1 , ^𝜆¹

0.2,^𝜆²

0.2,^𝜆³

0.2 , ^𝜆¹

0.8,^𝜆²

0.8,^𝜆³

0.7 >, 𝐻_𝑁 = < 𝜁,

𝛿₁ 0.6,^𝛿²

0.6,^𝛿³

0.6 , ^𝛿¹

0.5,^𝛿²

0.6,^𝛿³

0.6 , ^𝛿¹

0.4,^𝛿²

0.4,^𝛿³

0.4 >.Then 𝜏_𝑁and 𝜎_𝑁 are NT. Define a mapping 𝑓: 𝛥_𝑁, 𝜏_𝑁 → 𝛤_𝑁, 𝜎_𝑁 by 𝑓(𝜆₁) = 𝛿₁, 𝑓(𝜆₂) = 𝛿₂ and 𝑓 𝜆₃ = 𝛿₃. Then 𝑓 is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping but not 𝑁𝑊G^∗_{𝑐𝑜𝑛𝑡}mapping.

Remark 4.1.The converse of the preceding theorem does not have to be true.

1. The example 4.1. shows that the converse of theorem 4.1(1) is not true.

2. Theexample 4.3. shows that the converse of the 4.1(4) does not exist.

3. The example 4.2 shows that 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}need not be𝑁𝑃_{𝑐𝑜𝑛𝑡}.

Remark 4.2. The following examples show that 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping set and 𝑁𝐺𝑆_{𝑐𝑜𝑛𝑡} mapping are independent in (∆_𝑁, 𝜏_𝑁).

Example 4.5. Let ∆_𝑁 = {𝜆₁, 𝜆₂, 𝜆₃}, 𝛤_𝑁 = {𝛿₁, 𝛿₂, 𝛿₃}, 𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1N} and 𝜎_𝑁 = {0_𝑁, 𝐻_𝑁, 𝑊_𝑁 1N}, where 𝐺_𝑁 = < 𝜁, ^𝜆¹

0.9,^𝜆²

0.8,^𝜆³

0.9 , ^𝜆¹

0.8,^𝜆²

0.8,^𝜆³

0.7 , ^𝜆¹

0.3,^𝜆²

0.2,^𝜆³

0.2 >, 𝐻_𝑁 = < 𝜁,

𝛿₁ 0.7,^𝛿²

0.7,^𝛿³

0.8 , ^𝛿¹

0.7,^𝛿²

0.7,^𝛿³

0.6 , ^𝛿¹

0.4,^𝛿²

0.3,^𝛿³

0.7 >, 𝑊_𝑁 =

< 𝜁, ^𝛿¹

0.6,^𝛿²

0.6,^𝛿³

0.8 , ^𝛿¹

0.6,^𝛿²

0.6,^𝛿³

0.6 , ^𝛿¹

0.5,^𝛿²

0.4,^𝛿³

0.7 >.Then 𝜏_𝑁 and 𝜎_𝑁 are NT. Define a mapping 𝑓: 𝛥_𝑁, 𝜏_𝑁 → 𝛤_𝑁, 𝜎_𝑁 by 𝑓(𝜆₁) = 𝛿₁, 𝑓(𝜆₂) = 𝛿₂ and 𝑓 𝜆₃ = 𝛿₃. Then 𝑓 is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping not𝑁𝐺𝑆_{𝑐𝑜𝑛𝑡} mapping.

Example 4.6. Let ∆_𝑁 = {𝜆₁, 𝜆₂, 𝜆₃}, 𝛤_𝑁 = {𝛿₁, 𝛿₂, 𝛿₃},𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1_N} and 𝜎_𝑁 = {0_𝑁, 𝐻_𝑁, 1_N}, where 𝐺_𝑁 = < 𝜁, ^𝜆¹

0.9,^𝜆²

0.8,^𝜆³

0.9 , ^𝜆¹

0.8,^𝜆²

0.7,^𝜆³

0.7 , ^𝜆¹

0.2,^𝜆²

0.2,^𝜆³

0.2 >, 𝐻_𝑁 = < 𝜁,

𝛿₁ 0.1,^𝛿²

0.1,^𝛿³

0.1 , ^𝛿¹

0.1,^𝛿²

0.1,^𝛿³

0.1 , ^𝛿¹

0.9,^𝛿²

0.9,^𝛿³

0.9 >.Then 𝜏_𝑁 and 𝜎_𝑁 are NT. Define a mapping 𝑓: 𝛥_𝑁, 𝜏_𝑁 → 𝛤_𝑁, 𝜎_𝑁 by 𝑓(𝜆₁) = 𝛿₁, 𝑓(𝜆₂) = 𝛿₂ and 𝑓 𝜆₃ = 𝛿₃.Then 𝑓 is𝑁𝐺𝑆_{𝑐𝑜𝑛𝑡}mapping but not 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}mapping.

Remark 4.3. Consider the examples4.5. and 4.6., these examples supports the fact that𝑁𝐺_{𝑐𝑜𝑛𝑡} and 𝑁𝛼𝐺_{𝑐𝑜𝑛𝑡} mappings are independent with 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}in (∆_𝑁, 𝜏_𝑁).

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2326-9865

135 Figure.1(The implications of 𝑵𝜷𝝎_{𝒄𝒐𝒏𝒕} mapping)

V. SOME THEOREMS AND PROPERTIES OF Nβω-CONTINUOUS MAPPING Theorem5.1.If 𝑓: 𝛥_𝑁, 𝜏_𝑁 → (𝛤_𝑁, 𝜎_𝑁)is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}and𝑔: (𝛤_𝑁, 𝜎_𝑁) → (Ω_𝑁, 𝜑_𝑁) 𝑖𝑠 𝑁_{𝑐𝑜𝑛𝑡} then their composition 𝑔𝑜𝑓: (∆_𝑁,𝜏_𝑁)→ (𝛺_𝑁, 𝜑_𝑁) is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}.

Proof : Let 𝐺_𝑁 be any N-set of (𝛺_𝑁, 𝜑_𝑁). Since 𝑔 is 𝑁_{𝑐𝑜𝑛𝑡}, 𝑔⁻¹(𝐺_𝑁)is NC-set in (𝛤_𝑁, 𝜎_𝑁).

Since 𝑓 is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}, 𝑓⁻¹(𝑔⁻¹ 𝐺_𝑁 ) = (𝑔𝑜𝑓)⁻¹ 𝐺_𝑁 is𝑁𝛽𝜔𝐶in (∆_𝑁,𝜏_𝑁). Therefore 𝑔𝑜𝑓 is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}.

Remark 5.1. If two mappings are𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}, Then theircompositionneed not be 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}. Example 5.1.Let ∆_𝑁 = {𝜆₁, 𝜆₂, 𝜆₃}, 𝛤_𝑁 = {𝛿₁, 𝛿₂, 𝛿₃}, Ω_𝑁 = {𝛾₁, 𝛾₂, 𝛾₃}, 𝜏_𝑁 = {0_𝑁, 𝐺_𝑁, 1N} and 𝜎_𝑁 = {0_𝑁, 𝐻_𝑁, 1N}, 𝜎_𝑁 = {0_𝑁, 𝐼_𝑁, 1N}, where 𝐺_𝑁 =

< 𝜁, ^𝜆¹

0.7,^𝜆²

0.8,^𝜆³

0.7 , ^𝜆¹

0.8,^𝜆²

0.7,^𝜆³

0.8 , ^𝜆¹

0.3,^𝜆²

0.2,^𝜆³

0.3 >, 𝐻_𝑁 =

< 𝜁, ^𝛿¹

0.9,^𝛿²

0.9,^𝛿³

0.9 , ^𝛿¹

0.9,^𝛿²

0.9,^𝛿³

0.9 , ^𝛿¹

0.1,^𝛿²

0.1,^𝛿³

0.1 >, 𝐼_𝑁 =

< 𝜁, ^𝛿¹

0.2,^𝛿²

0.1,^𝛿³

0.2 , ^𝛿¹

0.1,^𝛿²

0.1,^𝛿³

0.1 , ^𝛿¹

0.8,^𝛿²

0.8,^𝛿³

0.8 >.Then 𝜏_𝑁 and 𝜎_𝑁 are NTs. Define the mappings𝑓: 𝛥_𝑁, 𝜏_𝑁 → 𝛤_𝑁, 𝜎_𝑁 and𝑔: (𝛤_𝑁, 𝜎_𝑁) → (Ω_𝑁, 𝜑_𝑁)by 𝑓(𝜆₁) = 𝛿₁, 𝑓(𝜆₂) = 𝛿₂,𝑓 𝜆₃ = 𝛿₃, 𝑔(𝛿₁) = 𝛾₁, 𝑔(𝛿₂) = 𝛾₂ and 𝑔 𝛿₃ = 𝛾₃. Then 𝑓and 𝑔 are𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡} mapping but 𝑓 ∘ 𝑔 is not𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡} mapping.

Theorem 5.2. A map 𝑓: 𝛥_𝑁, 𝜏_𝑁 → (𝛤_𝑁, 𝜎_𝑁)is 𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡}if and only if 𝑓⁻¹(𝐺_𝑁) is 𝑁𝛽𝜔𝑂 in 𝛥_𝑁, 𝜏_𝑁 , for every 𝑁𝑂-set 𝐺_𝑁 in (𝛤_𝑁, 𝜎_𝑁).

Proof : Let 𝑓: 𝛥_𝑁, 𝜏_𝑁 → (𝛤_𝑁, 𝜎_𝑁)be𝑁𝛽𝜔_{𝑐𝑜𝑛𝑡} and 𝐺_𝑁 be a𝑁𝑂-set in (𝛤_𝑁, 𝜎_𝑁). Then 𝑓⁻¹(𝐺_𝑁^𝐶) is 𝑁𝛽𝜔𝐶in (∆_𝑁,𝜏_𝑁). But 𝑓⁻¹ 𝐺_𝑁^𝐶 =(𝑓⁻¹(𝐺_𝑁)^𝐶 and 𝑓⁻¹(𝐺_𝑁)is NβωO in (∆_𝑁,𝜏_𝑁).

Converse is similar.

Theorem 3.3.3. Let 𝑓: 𝛥_𝑁, 𝜏_𝑁 → (𝛤_𝑁, 𝜎_𝑁)be a map. Assume that NβωO 𝛥_𝑁, 𝜏_𝑁 is 𝑁𝐶-set under any union. Then the following are equivalent :

𝑁𝛽𝜔

_{𝑐𝑜𝑛𝑡}

𝑁𝛽

_{𝑐𝑜𝑛𝑡}

𝑁

_{𝑐𝑜𝑛𝑡}

𝐺𝑆

_{𝑐𝑜𝑛𝑡}

𝑁𝑃

_{𝑐𝑜𝑛𝑡}

𝑁𝑊𝐺

^∗_{𝑐𝑜𝑛𝑡}

𝑁𝜓

_{𝑐𝑜𝑛𝑡}

𝑁𝐺

^∗_{𝑐𝑜𝑛𝑡}

𝑁𝐺

_{𝑐𝑜𝑛𝑡}

𝑁𝛼𝐺

_{𝑐𝑜𝑛𝑡}