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More Functions Associated with Neutrosophic gsα*- Closed Sets in Neutrosophic Topological Spaces

By P. Anbarasi Rodrigo and S. Maheswari

Submitted: July 9th 2021Reviewed: July 15th 2021Published: September 13th 2021

DOI: 10.5772/intechopen.99464

Downloaded: 20

Abstract

The concept of neutrosophic continuous function was very first introduced by A.A. Salama et al. The main aim of this paper is to introduce a new concept of Neutrosophic continuous function namely Strongly Neutrosophic gsα* - continuous functions, Perfectly Neutrosophic gsα* - continuous functions and Totally Neutrosophic gsα* - continuous functions in Neutrosophic topological spaces. These concepts are derived from strongly generalized neutrosophic continuous function and perfectly generalized neutrosophic continuous function. Several interesting properties and characterizations are derived and compared with already existing neutrosophic functions.

Keywords

  • Neutrosophic gsα*- closed set
  • Neutrosophic gsα*- open set
  • Strongly Neutrosophic gsα*- continuous function
  • Perfectly Neutrosophic gsα*- continuous function
  • Totally Neutrosophic gsα*- continuous function

1. Introduction

The concept of Neutrosophic set theory was introduced by F. Smarandache [1] and it comes from two concept, one is intuitionistic fuzzy sets introduced by K. Atanassov’s [2] and the other is fuzzy sets introduced by L.A. Zadeh’s [3]. It includes three components, truth, indeterminancy and false membership function. R. Dhavaseelan and S. Jafari [4] has discussed about the concept of strongly generalized neutrosophic continuous function. Further he also introduced the topic of perfectly generalized neutrosophic continuous function. The real life application of neutrosophic topology is applied in Information Systems, Applied Mathematics etc.

In this paper, we introduce some new concepts related to Neutrosophic gsαcontinuous function namely Strongly Neutrosophic gsαcontinuous function, Perfectly Neutrosophic gsαcontinuous function, Totally Neutrosophic gsαcontinuous function.

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2. Preliminaries

Definition 2.1:[5] Let Pbe a non-empty fixed set. A Neutrosophic set Ҥ on the universe Pis defined as Ҥ=ptҤpiҤpfҤp:pЄPwhere tҤp,iҤp,fҤprepresent the degree of membership function tҤp, the degree of indeterminacy iҤpand the degree of non-membership function fҤprespectively for each element pЄPto the set Ҥ. Also, tҤ,iҤ,fҤ: P]0,1+[and 0 tҤp+iҤp+fҤp3+. Set of all Neutrosophic set over Pis denoted by Neu(P).

Definition 2.2:[8] Let Pbe a non-empty set. Ⱥ=ptȺpiȺpfȺp:pЄPand Ƀ=ptɃpiɃpfɃp:pЄPare neutrosophic sets, then

  1. Ⱥ ⊆ Ƀ if tȺptɃp,iȺpiɃp,fȺpfɃpfor all pЄP.

  2. Intersection of two neutrosophic set Ⱥand Ƀis defined as ȺɃ=pmintȺptɃpminiȺpiɃpmaxfȺpfɃp:pЄP.

  3. Union of two neutrosophic set Ⱥand Ƀis defined as ȺɃ=pmaxtȺptɃpmaxiȺpiɃpminfȺpfɃp:pЄP.

  4. Ⱥc=pfȺp1iȺptȺp:pЄP.

  5. 0Neu=p001:pЄPand 1Neu=p110:pЄP.

Definition 2.3:[5] A neutrosophic topology (NeuT) on a non-empty set Pis a family τNeuof neutrosophic sets in Psatisfying the following axioms,

  1. 0Neu, 1NeuЄτNeu.

  2. Ⱥ1Ⱥ2ЄτNeufor any Ⱥ1,Ⱥ2ЄτNeu.

  3. ȺiЄτNeufor every family Ⱥi/iЄτNeu.

In this case, the ordered pair PτNeuor simply Pis called a neutrosophic topological space (NeuTS). The elements of τNeuis neutrosophic open set NeuOSand τNeucis neutrosophic closed set NeuCS.

Definition 2.4:[6] A neutrosophic set Ⱥ in a NeuTS PτNeuis called a neutrosophic generalized semi alpha star closed set NeugsαCSif NeuαintNeuαclȺNeuint(G), whenever Ⱥ ⊆ Gand Gis Neuαopen set.

Definition 2.5:[7] A neutrosophic topological space PτNeuis called a NeugsαT12space if every NeugsαCSin PτNeuis a NeuCSin PτNeu.

Definition 2.6:A neutrosophic function f:PτNeuQσNeuis said to be

  1. neutrosophic continuous [8] if the inverse image of each NeuCSin QσNeuis a NeuCSin PτNeu.

  2. Neugsαcontinuous [7] if the inverse image of each neutrosophic closed set in QσNeuis a Neugsαclosed set in PτNeu.

  3. Neugsαirresolute map [7] if the inverse image of each Neugsαclosed set in QσNeuis a Neugsαclosed set in PτNeu.

  4. strongly neutrosophic continuous [4] if the inverse image of each neutrosophic set in QσNeuis both NeuOSand NeuCSin PτNeu.

  5. perfectly neutrosophic continuous [4] if the inverse image of each NeuCSin QσNeuis both NeuOSand NeuCSin PτNeu.

Definition 2.7:[9] Let τNeu=0Neu1Neuis a neutrosophic topological space over P. Then PτNeuis called neutrosophic discrete topological space.

Definition 2.8:A neutrosophic topological space PτNeuis called a neutrosophic clopen set (Neuclopen set) if it is both NeuOSand NeuCSin PτNeu.

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3. Strongly neutrosophic gsα-continuous function

Definition 3.1:A neutrosophic function f:PτNeuQσNeuis said to be strongly Neugsαcontinuous if the inverse image of every NeugsαCSin QσNeuis a NeuCSin PτNeu. (ie) f1Ⱥis a NeuCSin PτNeufor every NeugsαCSȺ in QσNeu.

Theorem 3.2:Every strongly Neugsαcontinuous is neutrosophic continuous, but not conversely.

Proof:

Let f:PτNeuQσNeube any neutrosophic function. Let Ⱥ be any NeuCSin QσNeu. Since every NeuCSis NeugsαCS, then Ⱥ is NeugsαCSin QσNeu. Since fis strongly Neugsαcontinuous, then f1Ⱥis NeuCSin PτNeu. Therefore, fis neutrosophic continuous.

Example 3.3:Let P=pand Q=q. τNeu=0Neu1NeuȺand σNeu=0Neu1NeuɃare NeuTS on PτNeuand QσNeurespectively. Also Ⱥ=p0.6,0.4,0.4and Ƀ=q0.4,0.6,0.2are NeuPand NeuQ. Define a map f:PτNeuQσNeuby fp=q+0.2. Let Ƀc=q0.2,0.4,0.4be a NeuCSin QσNeu. Then f1Ƀc=p0.40.60.6. Now, Neuclf1Ƀc=Ⱥc1Neu=Ⱥc=f1Ƀcf1Ƀcis NeuCSin PτNeu. Therefore, fis neutrosophic continuous, but fis not strongly Neugsαcontinuous. Let Ȼ=q0.10.20.8be a NeugsαCSin QσNeu. Then f1Ȼ=p0.30.41. Now Neuclf1Ȼ=Ⱥc1Neu=Ⱥcf1Ȼf1Ȼis not NeuCSin PτNeu.

Theorem 3.4:Let f:PτNeuQσNeube strongly Neugsαcontinuous iff the inverse image of every NeugsαOSin QσNeuis NeuOSin PτNeu.

Proof:

Assume that fis strongly Neugsαcontinuous function. Let Ⱥ be any NeugsαOSin QσNeu. Then Ⱥcis NeugsαCSin QσNeu. Since fis strongly Neugsαcontinuous, then f1Ⱥcis NeuCSin PτNeuf1Ⱥcis NeuCSin PτNeuf1Ⱥis NeuOSin PτNeu. Conversely, Let Ⱥ be any NeugsαCSin QσNeu. Then Ⱥcis NeugsαOSin QσNeu. By hypothesis, f1Ⱥcis NeuOSin PτNeuf1Ⱥcis NeuOSin PτNeuf1Ⱥis NeuCSin PτNeu. Therefore, fis strongly Neugsαcontinuous.

Theorem 3.5:Every strongly Neugsαcontinuous is Neugsαcontinuous, but not conversely.

Proof:

Let f:PτNeuQσNeube any neutrosophic function. Let Ⱥ be any NeuCSin QσNeu. Then Ⱥ is NeugsαCSin QσNeu. Since fis strongly Neugsαcontinuous, then f1Ⱥis a NeuCSin PτNeuf1Ⱥis NeugsαCSin PτNeu. Therefore, fis Neugsαcontinuous.

Example 3.6:Let P=pand Q=q. τNeu=0Neu1NeuȺand σNeu=0Neu1NeuɃare NeuTS on PτNeuand QσNeurespectively. Also Ⱥ =p0.4,0.5,0.7and Ƀ =q0.6,0.8,0.4are NeuPand NeuQ. Define a map f:PτNeuQσNeuby fp=q. Let Ƀc=q0.4,0.2,0.6be a NeuCSin QσNeu. Then f1Ƀc=p0.40.20.6. NeuαOS=NeuαOS=0Neu1NeuȺand NeuαCS=0Neu1NeuȺc. Neuαclf1Ƀc=Ⱥc1Neu=Ⱥc. Now, NeuαintNeuαclf1Ƀc=ȺNeuint1Neu=1Neu,whenever f1Ƀc1Neuf1Ƀcis NeugsαCSin PτNeu. Therefore, fis Neugsαcontinuous. But fis not strongly Neugsαcontinuous. Let Ȼ=q0.30.10.7be a NeugsαCSin QσNeu. Then f1Ȼ=p0.30.10.7. Now Neuclf1Ȼ=Ⱥc1Neu=Ⱥcf1Ȼf1Ȼis not NeuCSin PτNeu.

Theorem 3.7:Every strongly neutrosophic continuous is strongly Neugsαcontinuous, but not conversely.

Proof:

Let f:PτNeuQσNeube any neutrosophic function. Let Ⱥ be any NeugsαCSin QσNeu. Since fis strongly neutrosophic continuous, then f1Ⱥis both NeuOSand NeuCSin PτNeuf1Ⱥis NeuCSin PτNeu. Hence, fis strongly Neugsαcontinuous.

Example 3.8:Let P=pand Q=q. τNeu=0Neu1NeuȺȻand σNeu=0Neu1NeuɃare NeuTS on PτNeuand QσNeurespectively. Also Ⱥ=p0.4,0.6,0.2,Ȼ=p0.4,1[0.61]0,0.2and Ƀ=q0.4,0.6,0.2are NeuPand NeuQ. Define a map f:PτNeuQσNeuby fp=q. Let Ⱦ=q00.200.40.41be a NeugsαCSin QσNeu. Then f1Ⱦ=p00.20,0.40.4,1. Now Neuclf1Ⱦ=ȺcȻc1Neu=Ȼc=f1Ⱦ. Therefore, fis strongly Neugsαcontinuous. But fis not strongly neutrosophic continuous. Let Ɇ=q0.40.60.2be a neutrosophic set in QσNeu. Then f1Ɇ=p0.40.60.2. Now Neuintf1Ɇ=0NeuȺ=Ⱥ=f1Ɇf1Ɇis NeuOSin PτNeu. Also Neuclf1Ɇ=1Neuf1Ɇf1Ɇis not NeuCSin PτNeu. Therefore, f1Ɇis not both NeuOSand NeuCSin PτNeu.

Remark 3.9: Every strongly neutrosophic continuous is Neugsαcontinuous, but not conversely. (by Theorem 3.5 & 3.7).

Theorem 3.10:Let f:PτNeuQσNeube neutrosophic function and QσNeube NeugsαT12space. Then the following are equivalent.

  1. fis strongly Neugsαcontinuous.

  2. fis neutrosophic continuous.

Proof:

  1. (2), Proof follows from theorem 3.2.

  2. (1), Let Ⱥ be any NeugsαCSin QσNeu. Since QσNeuis NeugsαT12space, then Ⱥ is NeuCSin QσNeu. Since fis neutrosophic continuous, then f1Ⱥis NeuCSin PτNeu. Therefore, fis strongly Neugsαcontinuous.

Theorem 3.11:Let f:PτNeuQσNeube Neugsαcontinuous. Both PτNeuand QσNeuare NeugsαT12space, then fis strongly Neugsαcontinuous.

Proof:

Let Ⱥ be any NeugsαCSin QσNeu. Since QσNeuis NeugsαT12space, then Ⱥ is NeuCSin QσNeu. Since fis Neugsαcontinuous, then f1Ⱥis NeugsαCSin PτNeu. Since PτNeuis NeugsαT12space, then f1Ⱥis NeuCSin PτNeu. Therefore, fis strongly Neugsαcontinuous.

Theorem 3.12:Let f:PτNeuQσNeube strongly Neugsαcontinuous, then fis Neugsαirresolute.

Proof:

Let Ⱥ be any NeugsαCSin QσNeu. Since fis strongly Neugsαcontinuous, then f1Ⱥis NeuCSin PτNeuf1Ⱥis NeugsαCSin PτNeu. Hence, fis Neugsαirresolute.

Theorem 3.13:Let f:PτNeuQσNeube Neugsαirresolute and PτNeube NeugsαT12space, then fis strongly Neugsαcontinuous.

Proof:

Let Ⱥ be any NeugsαCSin QσNeu. Since fis Neugsαirresolute, then f1Ⱥis NeugsαCSin PτNeu. Since PτNeuis NeugsαT12space, then f1Ⱥis NeuCSin PτNeu. Therefore, fis strongly Neugsαcontinuous.

Theorem 3.14:Let f:PτNeuQσNeuand g:QσNeuRγNeube strongly Neugsαcontinuous, then gof:PτNeuRγNeuis strongly Neugsαcontinuous.

Proof:

Let Ⱥ be any NeugsαCSin RγNeu. Since gis strongly Neugsαcontinuous, then g1Ⱥis NeuCSin QσNeug1Ⱥis NeugsαCSin QσNeu. Since fis strongly Neugsαcontinuous, then f1g1Ⱥ=gof1Ⱥis NeuCSin PτNeu. Therefore, gofis strongly Neugsαcontinuous.

Theorem 3.15:Let f:PτNeuQσNeube strongly Neugsαcontinuous and g:QσNeuRγNeube Neugsαcontinuous, then gof:PτNeuRγNeuis neutrosophic continuous.

Proof:

Let Ⱥ be any NeuCSin RγNeu. Since gis Neugsαcontinuous, then g1Ⱥis NeugsαCSin QσNeu. Since fis strongly Neugsαcontinuous, then f1g1Ⱥ=gof1Ⱥis NeuCSin PτNeu. Therefore, gofis neutrosophic continuous.

Theorem 3.16:Let f:PτNeuQσNeube strongly Neugsαcontinuous and g:QσNeuRγNeube Neugsαirresolute, then gof:PτNeuRγNeuis strongly Neugsαcontinuous.

Proof:

Let Ⱥ be any NeugsαCSin RγNeu. Since gis Neugsαirresolute, then g1Ⱥis NeugsαCSin QσNeu. Since fis strongly Neugsαcontinuous, then f1g1Ⱥ=gof1Ⱥis NeuCSin PτNeu. Therefore, gofis strongly Neugsαcontinuous.

Theorem 3.17:Let f:PτNeuQσNeube Neugsαcontinuous and g:QσNeuRγNeube strongly Neugsαcontinuous, then gof:PτNeuRγNeuis Neugsαirresolute.

Proof:

Let Ⱥ be any NeugsαCSin RγNeu. Since gis strongly Neugsαcontinuous, then g1Ⱥis NeuCSin QσNeu. Since fis Neugsαcontinuous, then f1g1Ⱥ=gof1Ⱥis NeugsαCSin PτNeu. Hence, gofis Neugsαirresolute.

Theorem 3.18:Let f:PτNeuQσNeube neutrosophic continuous and g:QσNeuRγNeube strongly Neugsαcontinuous, then gof:PτNeuRγNeuis strongly Neugsαcontinuous.

Proof:

Let Ⱥ be any NeugsαCSin RγNeu. Since gis strongly Neugsαcontinuous, then g1Ⱥis NeuCSin QσNeu. Since fis neutrosophic continuous, then f1g1Ⱥ=gof1Ⱥis NeuCSin PτNeu. Hence, gofis strongly Neugsαcontinuous.

Inter-relationship 3.19:

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4. Perfectly neutrosophic gsα-continuous function

Definition 4.1:A neutrosophic function f:PτNeuQσNeuis said to be perfectly Neugsαcontinuous if the inverse image of every NeugsαCSin QσNeuis both NeuOSand NeuCS(ie, Neuclopen set) in PτNeu.

Theorem 4.2:Every perfectly Neugsαcontinuous is strongly Neugsαcontinuous, but not conversely.

Proof:

Let f:PτNeuQσNeube any neutrosophic function. Let Ⱥ be any NeugsαCSin QσNeu. Since fis perfectly Neugsαcontinuous, then f1Ⱥis both NeuOSand NeuCSin PτNeuf1Ⱥis NeuCSin PτNeu. Therefore, fis strongly Neugsαcontinuous.

Example 4.3:Let P=pand Q=q. τNeu=0Neu1NeuȺȻand σNeu=0Neu1NeuɃare NeuTS on PτNeuand QσNeurespectively. Also Ⱥ=p0.7,0.8,0.3,Ȼ=p0.7,10.8,10,0.3and Ƀ=q0.70.80.3are NeuPand NeuQ. Define a map f:PτNeuQσNeuby fp=q. Let Ⱦ=q00.300.20.71be a NeugsαCSin QσNeu. Then f1Ⱦ=p00.30,0.20.7,1. Now Neuclf1Ⱦ=ȺcȻc1Neu=Ȼc=f1Ⱦ. Therefore, fis strongly Neugsαcontinuous. But fis not perfectly Neugsαcontinuous, because f1Ⱦis not both NeuOSand NeuCSin PτNeu. Since, Neuintf1Ⱦ=0Neuf1Ⱦf1Ⱦis not NeuCSin PτNeu. Therefore, f1Ⱦis not both NeuOSand NeuCSin PτNeu.

Theorem 4.4:Every perfectly Neugsαcontinuous is perfectly neutrosophic continuous, but not conversely.

Proof:

Let f:PτNeuQσNeube any neutrosophic function. Let Ⱥ be any NeuCSin QσNeu. Then Ⱥ is NeugsαCSin QσNeu. Since fis perfectly Neugsαcontinuous, then f1Ⱥis both NeuOSand NeuCSin PτNeu. Therefore, fis perfectly neutrosophic continuous.

Example 4.5:Let P=pand Q=q. τNeu=0Neu1NeuȺȻɆand σNeu=0Neu1NeuɃare NeuTS on PτNeuand QσNeurespectively. Also Ⱥ=p0.4,0.2,0.6, Ȼ=p0.6,0.8,0.4,Ɇ=p0,0.40,0.20.6,1and Ƀ=q0.60.80.4are NeuPand NeuQ. Define a map f:PτNeuQσNeuby fp=q. Let Ƀc=q0.40.20.6be a NeuCSin QσNeu. Then f1Ƀc=p0.4,0.2,0.6. Now Neuclf1Ƀc=ȺcȻcɆc1Neu=Ȼc=f1Ƀc. Also, Neuintf1Ƀc=ȺɆ0Neu=Ⱥ=f1Ƀcf1Ƀcis both NeuOSand NeuCSin PτNeu. Therefore, fis perfectly neutrosophic continuous. But fis not perfectly Neugsαcontinuous. Let Ⱦ =q00.400.2[0.61]be NeugsαCSin QσNeu. Then f1Ⱦ=p00.400.2[0.61]. Since, Neuintf1Ⱦ=Ɇ0Neu=Ɇ=f1Ⱦf1Ⱦis NeuOSin PτNeu. Also, Neuclf1Ⱦ=ȺcȻcɆc1Neu=Ȼcf1Ⱦf1Ⱦis not NeuCSin PτNeu. Therefore, f1Ⱦis not both NeuOSand NeuCSin PτNeu.

Theorem 4.6:Let f:PτNeuQσNeube perfectly Neugsαcontinuous iff the inverse image of every NeugsαOSin QσNeuis both NeuOSand NeuCSin PτNeu.

Proof:

Assume that fis perfectly Neugsαcontinuous function. Let Ⱥ be any NeugsαOSin QσNeu. Then Ⱥcis NeugsαCSin QσNeu. Since fis perfectly Neugsαcontinuous, then f1Ⱥcis both NeuOSand NeuCSin PτNeuf1Ⱥcis both NeuOSand NeuCSin PτNeuf1Ⱥis both NeuOSand NeuCSin PτNeu. Conversely, Let Ⱥ be any NeugsαCSin QσNeu. Then Ⱥcis NeugsαOSin QσNeu. By hypothesis, f1Ⱥcis both NeuOSand NeuCSin PτNeuf1Ⱥcis both NeuOSand NeuCSin PτNeuf1Ⱥis both NeuOSand NeuCSin PτNeu. Therefore, fis perfectly Neugsαcontinuous.

Theorem 4.7:Let PτNeube a neutrosophic discrete topological space and QσNeube any neutrosophic topological space. Let f:PτNeuQσNeube a neutrosophic function, then the following statements are true.

  1. fis strongly Neugsαcontinuous.

  2. fis perfectly Neugsαcontinuous.

Proof:

  1. (2), Let Ⱥ be any NeugsαCSin QσNeu. Since fis strongly Neugsαcontinuous, then f1Ⱥis NeuCSin PτNeu. Since PτNeuis neutrosophic discrete topological space, then f1Ⱥis NeuOSin PτNeuf1Ⱥis both NeuOSand NeuCSin PτNeu. Therefore, fis perfectly Neugsαcontinuous.

  2. (1), Let Ⱥ be any NeugsαCSin QσNeu. Since fis perfectly Neugsαcontinuous, then f1Ⱥis both NeuOSand NeuCSin PτNeuf1Ⱥis NeuCSin PτNeu. Therefore, fis strongly Neugsαcontinuous.

Theorem 4.8:Let f:PτNeuQσNeube perfectly neutrosophic continuous and QσNeube NeugsαT12space, then fis perfectly Neugsαcontinuous.

Proof:

Let Ⱥ be any NeugsαCSin QσNeu. Since QσNeuis NeugsαT12space, then Ⱥ is NeuCSin QσNeu.Since fis perfectly neutrosophic continuous, then f1Ⱥis both NeuOSand NeuCSin PτNeu. Therefore, fis perfectly Neugsαcontinuous.

Theorem 4.9:Let f:PτNeuQσNeuand g:QσNeuRγNeube perfectly Neugsαcontinuous, then gof:PτNeuRγNeuis perfectly Neugsαcontinuous.

Proof:

Let Ⱥ be any NeugsαCSin RγNeu. Since gis perfectly Neugsαcontinuous, then g1Ⱥis both NeuOSand NeuCSin QσNeug1Ⱥis NeugsαCSin QσNeu. Since fis perfectly Neugsαcontinuous, then f1g1Ⱥ=gof1Ⱥis both NeuOSand NeuCSin PτNeu. Therefore, gofis perfectly Neugsαcontinuous.

Theorem 4.10:Let f:PτNeuQσNeube neutrosophic continuous and g:QσNeuRγNeube perfectly Neugsαcontinuous, then gof:PτNeuRγNeuis strongly Neugsαcontinuous.

Proof:

Let Ⱥ be any NeugsαCSin RγNeu. Since gis perfectly Neugsαcontinuous, then g1Ⱥis both NeuOSand NeuCSin QσNeu. Since fis neutrosophic continuous, then f1g1Ⱥ=gof1Ⱥis NeuCSin PτNeu. Therefore, gofis strongly Neugsαcontinuous.

Theorem 4.11:Let f:PτNeuQσNeube perfectly Neugsαcontinuous and g:QσNeuRγNeube strongly Neugsαcontinuous, then gof:PτNeuRγNeuis perfectly Neugsαcontinuous.

Proof:

Let Ⱥ be any NeugsαCSin RγNeu. Since gis strongly Neugsαcontinuous, then g1Ⱥis NeuCSin QσNeug1Ⱥis NeugsαCSin QσNeu. Since fis perfectly Neugsαcontinuous, then f1g1Ⱥ=gof1Ⱥis both NeuOSand NeuCSin PτNeu. Therefore, gofis perfectly Neugsαcontinuous.

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5. Totally neutrosophic gsαcontinuous function

Definition 5.1:A neutrosophic function f:PτNeuQσNeuis said to be totally Neugsαcontinuous if the inverse image of every NeuCSin QσNeuis both NeugsαOSand NeugsαCS(ie, Neugsαclopen set) in PτNeu.

Definition 5.2:A neutrosophic topological space PτNeuis called a Neugsαclopen set (Neugsαclopen set) if it is both NeugsαOSand NeugsαCSin PτNeu.

Example 5.3:Let P=pand Q=q. τNeu=0Neu1NeuȺand σNeu=0Neu1NeuɃare NeuTS on PτNeuand QσNeurespectively. Also Ⱥ=p0.4,0.5,0.7and Ƀ=q0.2,0.7,0.8are NeuPand NeuQ. Define a map f:PτNeuQσNeuby fp=q. Let Ƀc=q0.8,0.3,0.2be a NeuCSin QσNeu. Then f1Ƀc=p0.80.30.2. NeuαOS=NeuαOS=0Neu1NeuȺand NeuαCS=0Neu1NeuȺc. Neuαclf1Ƀc=1Neu.Now, NeuαintNeuαclf1Ƀc=1NeuNeuint1Neu=1Neu,whenever f1Ƀc1Neuf1Ƀcis NeugsαCSin PτNeu. Also, Neuαintf1Ƀc=0Neu.Now, NeuαclNeuαintf1Ƀc=0NeuNeucl0Neu=0Neu,whenever f1Ƀc0Neuf1Ƀcis NeugsαOSin PτNeu. Therefore, fis totally Neugsαcontinuous.

Theorem 5.4:Every perfectly Neugsαcontinuous is totally Neugsαcontinuous, but not conversely.

Proof:

Let f:PτNeuQσNeube any neutrosophic function. Let Ⱥ be any NeuCSin QσNeu. Then Ⱥ is NeugsαCSin QσNeu. Since fis perfectly Neugsαcontinuous, then f1Ⱥis both NeuOSand NeuCSin PτNeuf1Ⱥis both NeugsαOSand NeugsαCSin PτNeu. Therefore, fis totally Neugsαcontinuous.

Example 5.5:Let P=pand Q=q. τNeu=0Neu1NeuȺand σNeu=0Neu1NeuɃare NeuTS on PτNeuand QσNeurespectively. Also Ⱥ =p0.2,0.4,0.6and Ƀ =q0.6,0.8,0.4are NeuPand NeuQ. Define a map f:PτNeuQσNeuby fp=q. Let Ƀc=q0.4,0.2,0.6be a NeuCSin QσNeu. Then f1Ƀc=p0.40.20.6. NeuαOS=NeuαOS=0Neu1NeuȺand NeuαCS=0Neu1NeuȺc. Neuαclf1Ƀc=Ⱥc1Neu=Ⱥc.Now, NeuαintNeuαclf1Ƀc=Ⱥ0Neu=ȺNeuint1Neu=1Neu,whenever f1Ƀc1Neuf1Ƀcis NeugsαCSin PτNeu. Also, Neuαintf1Ƀc=0Neu.Now, NeuαclNeuαintf1Ƀc=0NeuNeucl0Neu=0Neu,whenever f1Ƀc0Neuf1Ƀcis NeugsαOSin PτNeu. Therefore, fis totally Neugsαcontinuous. But fis not perfectly Neugsαcontinuous. Let Ⱦ=q0.3,0.1,0.8be NeugsαCSin QσNeu. Then f1Ⱦ=p0.3,0.1,0.8. Now, Neuintf1Ⱦ=0Neuf1Ⱦf1Ⱦis not NeuOSin PτNeu. Also, Neuclf1Ⱦ=Ⱥcf1Ⱦf1Ⱦis not NeuCSin PτNeu. Therefore, f1Ⱦis not both NeuOSand NeuCSin PτNeu.

Theorem 5.6:Every totally Neugsαcontinuous is Neugsαcontinuous.

Proof:

Let f:PτNeuQσNeube any neutrosophic function. Let Ⱥ be any NeuCSin QσNeu. Since fis totally Neugsαcontinuous, then f1Ⱥis both NeugsαOSand NeugsαCSin PτNeuf1Ⱥis NeugsαCSin PτNeu. Therefore, fis Neugsαcontinuous.

Example 5.7:Let P=pand Q=q. τNeu=0Neu1NeuȺand σNeu=0Neu1NeuɃare NeuTS on PτNeuand QσNeurespectively. Also Ⱥ=p0.7,0.6,0.5and Ƀ=q0.7,0.8,0.3are NeuPand NeuQ. Define a map f:PτNeuQσNeuby fp=q. Let Ƀc=q0.3,0.2,0.7be a NeuCSin QσNeu. Then f1Ƀc=p0.3,0.2,0.7. NeuαOS=NeuαOS=0Neu1NeuȺDand NeuαCS=0Neu1NeuȺcE,where D=p0.710.6100.5,E=p00.50,0.40.7,1. Neuαclf1Ƀc=ȺcF1Neu=F,where =p0.3,0.50.2,0.40.7. Now, NeuαintNeuαclf1Ƀc=0NeuNeuintȺ,NeuintD,Neuint1Neu=Ⱥ,1Neu,whenever f1ɃcȺ,1Neuf1Ƀcis NeugsαCSin PτNeu. Therefore, fis Neugsαcontinuous. But fis not totally Neugsαcontinuous, because f1Ƀcis not NeugsαOSin PτNeu. Since NeuαclNeuαintf1Ƀc=0NeuNeuclJ=Ⱥc,whenever f1ɃcJ,where J=p0,0.30,0.20.7,1f1Ƀcis not NeugsαOSin PτNeu.

Inter-relationship 5.8:

Theorem 5.9:Let f:PτNeuQσNeube totally Neugsαcontinuous and QσNeube NeugsαT12space, then fis Neugsαirresolute.

Proof:

Let Ⱥ be any NeugsαCSin QσNeu. Since QσNeuis NeugsαT12space, then Ⱥ is NeuCSin QσNeu. Since fis totally Neugsαcontinuous, then f1Ⱥis both NeugsαOSand NeugsαCSin PτNeuf1Ⱥis NeugsαCSin PτNeu. Therefore, fis Neugsαirresolute.

Theorem 5.10:Let f:PτNeuQσNeuand g:QσNeuRγNeube totally Neugsαcontinuous and QσNeube NeugsαT12space, then gof:PτNeuRγNeuis totally Neugsαcontinuous.

Proof:

Let Ⱥ be any NeuCSin RγNeu. Since gis totally Neugsαcontinuous, then g1Ⱥis both NeugsαOSand NeugsαCSin QσNeu. Since QσNeuis NeugsαT12space, then g1Ⱥis NeuCSin QσNeu. Since fis totally Neugsαcontinuous, then f1g1Ⱥ=gof1Ⱥis both NeugsαOSand NeugsαCSin PτNeu. Therefore, gofis totally Neugsαcontinuous.

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P. Anbarasi Rodrigo and S. Maheswari (September 13th 2021). More Functions Associated with Neutrosophic gsα*- Closed Sets in Neutrosophic Topological Spaces [Online First], IntechOpen, DOI: 10.5772/intechopen.99464. Available from:

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