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Neutrosophic [alpha]gs Continuity And Neutrosophic [alpha]gs Irresolute Maps.

1. Introduction

Neutrosophic set theory concepts first initiated by F.Smarandache[11] which is Based on K. Atanassov's intuitionistic[6]fuzzy sets & L.A.Zadeh's [20]fuzzy sets. Also it defined by three parameters truth(T), indeterminacy (I),and falsity(F)-membership function. Smarandache's neutrosophic concept have wide range of real time applications for the fields of [1,2,3,4&5] Information Systems, Computer Science, Artificial Intelligence, Applied Mathematics, decision making. Mechanics, Electrical & Electronic, Medicine and Management Science etc,.

A.A.Salama[16] introduced Neutrosophic topological spaces by using Smarandache's Neutrosophic sets. I.Arokiarani.[7] et.al., introduced Neutrosophic [alpha]-closed sets.P. Ishwarya, [13]et.al., introduced and studied Neutrosophic semi-open sets in Neutrosophic topological spaces. Neutrosophic continuity functions introduced by A.A.Salama[15]. Neutrosophic [alpha]gs-closed set[8] introduced by V.Banu priya&S.Chandrasekar. Aim of this present paper is, we introduce and investigate new kind of Neutrosophic continuity is called Neutrosophic [alpha]gs Continuity maps in Neutrosophic topological spaces and also we discussed about properties and characterization Neutrosophic [alpha]gs Irresolute Maps

2. PRELIMINARIES

In this section, we introduce the basic definition for Neutrosophic sets and its operations.

Definition 2.1 [11]

Let E be a non-empty fixed set. A Neutrosophic set [lambda] writing the format is

[lambda] = {<e, [[eta].sub.[lambda]](e), [[sigma].sub.[lambda]](e),[[gamma].sub.[lambda]](e) >:e[member of]E}

Where [[eta].sub.[lambda]](e), [[sigma].sub.[lambda]](e) and [[gamma].sub.[lambda]](e) which represents Neutrosophic topological spaces the degree of membership function, indeterminacy and non-membership function respectively of each element e [member of] E to the set [lambda].

Remark 2.2 [11]

A Neutrosophic set [lambda]={<e, [[eta].sub.[lambda]](e), [[sigma].sub.[lambda]](e), [[gamma].sub.[lambda]](e) >: e[member of]E} can be identified to an ordered triple <[[eta].sub.[lambda]], [[sigma].sub.[lambda]], [[gamma].sub.[lambda]]> in [].bar]-0,1+[[.bar] on E.

Remark 2.3[11]

Neutrosophic set [lambda]={<e, [[eta].sub.[lambda]](e),[[sigma].sub.[lambda]](e),[[gamma].sub.[lambda]](e) >:e[member of]E}our convenient we can write [lambda]=<e, [[eta].sub.[lambda]], [[sigma].sub.[lambda]], [[gamma].sub.[lambda]]>.

Example 2.4 [11]

we must introduce the Neutrosophic set [0.sub.N] and [1.sub.N] in E as follows:

[0.sub.N] may be defined as:

([0.sub.1]) [0.sub.N]={<e, 0, 0, 1>: e[member of]E}

([0.sub.2]) [0.sub.N]={<e, 0, 1, 1>: e[member of]E}

([0.sub.3]) [0.sub.N] ={<e, 0, 1, 0 >:e[member of]E}

([0.sub.4]) [0.sub.N]={<e, 0, 0, 0>: e[member of]E}

[1.sub.N] may be defined as:

([1.sub.1]) [1.sub.N] = {<e, 1, 0, 0>: e[member of]E}

([1.sub.2]) [1.sub.N] = {<e, 1, 0, 1 >: e[member of]E}

([1.sub.3]) [1.sub.N] ={<e, 1, 1, 0 >: e[member of]E}

([1.sub.4]) [1.sub.N] ={<e, 1, 1, 1 >: e[member of]E}

Definition 2.5 [ii]

Let [lambda]=<[[eta].sub.[lambda]], [[sigma].sub.[lambda]],[[gamma].sub.[lambda]]> be a Neutrosophic set on E, then [[lambda].sup.C] defined as [[lambda].sup.C]={<e, [[gamma].sub.[lambda]](e),1- [[sigma].sub.[lambda]](e), [[eta].sub.[lambda]](e) >: e [member of]E}

Definition 2.6 [11]

Let E be a non-empty set, and Neutrosophic sets [lambda] and [micro] in the form

[lambda] ={<e, [[eta].sub.[lambda]](e), [[sigma].sub.[lambda]](e), [[gamma].sub.[lambda]](e)>:e[member of]E} and

[micro] ={<e, [eta][micro](e), [sigma][micro](e), [gamma][micro](e)>: e[member of]E}.

Then we consider definition for subsets ([lambda][??][micro]).

[lambda][??][micro] defined as: [lambda][??][micro] [??][[eta].sub.[lambda]](e) [less than or equal to] [[eta].sub.[mu]](e), [[sigma].sub.[lambda]](e) [less than or equal to] [[sigma].sub.[mu]](e) and [[gamma].sub.[lambda]](e) [greater than or equal to] [[gamma].sub.[mu]](e) for all e[member of]E

Proposition 2.7 [11]

For any Neutrosophic set [lambda], then the following condition are holds:

(i) [0.sub.N][??][lambda], [0.sub.N][??] [0.sub.N]

(ii) [lambda][??][1.sub.N], [1.sub.N][??] [1.sub.N]

Definition 2.8 [11]

Let E be a non-empty set, and [lambda]=<e, [[eta].sub.[mu]](e),[[sigma].sub.[lambda]](e), [[gamma].sub.[lambda]](e)>, [micro] =<e, [[eta].sub.[mu]](e), [[sigma].sub.[mu]](e), [[gamma].sub.[mu]](e)> be two Neutrosophic sets. Then

(i) [lambda][intersection][micro] defined as :[lambda][intersection][micro] =<e, [[eta].sub.[lambda]](e)[and][[eta].sub.[mu]](e), [[sigma].sub.[lambda]](e)[and][[sigma].sub.[mu]](e),[[gamma].sub.[lambda]](e)V[[gamma].sub.[mu]](e)>

(ii) [lambda][union][micro] defined as :[lambda][union][micro] =<e, [[eta].sub.[lambda]](e)V[[eta].sub.[mu]](e), [[sigma].sub.[lambda]](e)V[[sigma].sub.[mu]](e), [[gamma].sub.[lambda]](e)[and][[gamma].sub.[mu]](e)>

Proposition 2.9 [11]

For all [lambda] and [micro] are two Neutrosophic sets then the following condition are true:

(i) [([lambda][intersection][micro]).sup.C]=[[lambda].sup.C][union][[mu].sup.C]

(ii) [([lambda][union][micro]).sup.C]=[[lambda].sup.C][intersection][[mu].sup.C].

Definition 2.10 [16]

A Neutrosophic topology is a non-empty set E is a family [[tau].sub.N] of Neutrosophic subsets in E satisfying the following axioms:

(i) [0.sub.N], [1.sub.N] [member of][[tau].sub.N],

(ii) [G.sub.1][intersection][G.sub.1][member of][[tau].sub.N] for any [G.sub.1], [G.sub.2][member of][[tau].sub.N],

(iii) [union][G.sub.1][member of][[tau].sub.N] for any family {[G.sub.i] | i[member of]J}[??][[tau].sub.N].

the pair (E, [[tau].sub.N]) is called a Neutrosophic topological space.

The element Neutrosophic topological spaces of [[tau].sub.N] are called Neutrosophic open sets.

A Neutrosophic set [lambda] is closed if and only if [[lambda].sup.C] is Neutrosophic open.

Example 2.11[l6]

Let E={e} and

[A.sub.1]= {<e, .6, .6, .5>:e[member of]E}

[A.sub.2]= {<e, .5, .7, .9>:e[member of]E}

[A.sub.3]= {<e, .6, .7, .5>:e[member of]E}

[A.sub.4]= {<e, .5, .6, .9>:e[member of]E}

Then the family IN={[0.sub.N], [1.sub.N],[A.sub.1], [A.sub.2], [A.sub.3], [A.sub.4]}is called a Neutrosophic topological space on E.

Definition 2.12[l6]

Let (E, [[tau].sub.N]) be Neutrosophic topological spaces and [lambda]={<e, [[eta].sub.[lambda]](e), [[sigma].sub.[lambda]](e), [[gamma].sub.[lambda]](e)>:e[member of]E} be a Neutrosophic set in E.

Then the Neutrosophic closure and Neutrosophic interior of [lambda] are defined by

Neu-cl([lambda])=[intersection]{D:D is a Neutrosophic closed set in E and [lambda][??]D}

Neu-int([lambda])=[union]{C:C is a Neutrosophic open set in E and C[??][lambda]}.

Definition 2.13

Let (E, [[tau].sub.N]) be a Neutrosophic topological space. Then [lambda] is called

(i) Neutrosophic regular Closed set [7] (Neu-RCS in short) if [lambda]=Neu-Cl(Neu-Int([lambda])),

(ii) Neutrosophic [alpha]-Closed set[7] (Neu-[alpha]CS in short) if Neu-Cl(Neu-Int(Neu-Cl([lambda])))[??][lambda],

(iii) Neutrosophic semi Closed set [13] (Neu-SCS in short) if Neu-Int(Neu-Cl([lambda]))[??][lambda],

(iv) Neutrosophic pre Closed set [18] (Neu-PCS in short) if Neu-Cl(Neu-Int([lambda]))[??][lambda],

Definition 2.14

Let (E, [[tau].sub.N]) be a Neutrosophic topological space. Then [lambda] is called

(i). Neutrosophic regular open set [7](Neu-ROS in short) if [lambda]=Neu-Int(Neu-Cl([lambda])),

(ii). Neutrosophic [alpha]-open set [7](Neu-[alpha]OS in short) if [lambda][??]Neu-Int(Neu-Cl(Neu-Int([lambda]))),

(iii). Neutrosophic semi open set [l3](Neu-SOS in short) if [lambda][??]Neu-Cl(Neu-Int([lambda])),

(iv).Neutrosophic pre open set [18] (Neu-POS in short) if [lambda][??]Neu-Int(Neu-Cl([lambda])),

Definition 2.15

Let (E,)[[tau].sub.N] be a Neutrosophic topological space. Then [lambda] is called

(i).Neutrosophic generalized closed set[9](Neu-GCS in short) if Neu-cl([lambda])[??]U whenever [lambda][??]U and U is a Neu-OS in E,

(ii).Neutrosophic generalized semi closed set[17] (Neu-GSCS in short) if Neu-scl([lambda])[??]U Whenever [lambda][??]U and U is a Neu-OS in E,

(iii).Neutrosophic [alpha] generalized closed set [14](Neu-[alpha]GCS in short) if Neu-[alpha]cl([lambda])[??]U whenever [lambda][??]U and U is a Neu-OS in E,

(iv).Neutrosophic generalized alpha closed set [10] (Neu-G[alpha]CS in short) if Neu-[alpha]cl([lambda])[??]U whenever [lambda][??]U and U is a Neu-[alpha]OS in E.

The complements of the above mentioned Neutrosophic closed sets are called their respective Neutrosophic open sets.

Definition 2.16 [8]

Let (E, [[tau].sub.N]) be a Neutrosophic topological space. Then [lambda] is called Neutrosophic [alpha] generalized Semi closed set (Neu-[alpha]GSCS in short) if Neu-[alpha]cl([lambda])[??]U whenever [lambda][??]U and U is a Neu-SOS in E

The complements of Neutrosophic [alpha]GS closed sets is called Neutrosophic [alpha]GS open sets.

3. Neutrosophic [alpha]gs-Continuity maps

In this section we Introduce Neutrosophic [alpha]-generalized semi continuity maps and study some of its properties.

Definition 3.1.

A maps f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N]) is called a Neutrosophic [alpha]-generalized semi continuity(Neu-[alpha]GS continuity in short) [f.sup.-1]([micro]) is a Neu-[alpha]GSCS in ([E.sub.1], [[tau].sub.N]) for every Neu-CS [micro] of ([E.sub.2], [[sigma].sub.N])

Example 3.2.

Let [E.sub.1]={[a.sub.1],[a.sub.2]}, [E.sub.2]={[b.sub.1],[b.sub.2]}, U=<[e.sub.1],(.7,.5,.8),(.5,.5,.4)> and V=<[e.sub.2],(1,.5,.9),(.2,.5,.3)>. Then IN={[0.sub.N],U,[1.sub.N]} and [[sigma].sub.N]={[0.sub.N],V,[1.sub.N]} are Neutrosophic Topologies on [E.sub.1]and E2 respectively.

Define a maps f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])by f([a.sub.1])=[b.sub.1] and f([a.sub.2])=[b.sub.2]. Then f is a Neu-[alpha]GS continuity maps.

Theorem 3.3.

Every Neu-continuity maps is a Neu-[alpha]GS continuity maps.

Proof.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N]) be a Neu-continuity maps. Let [lambda] be a Neu-CS in [E.sub.2]. Since f is a Neu-continuity maps, [f.sup.-1] ([lambda]) is a Neu-CS in [E.sub.1]. Since every Neu-CS is a Neu-[alpha]GSCS,[f.sup.-1]([lambda]) is a Neu-[alpha]GSCS in [E.sub.1]. Hence f is a Neu-[alpha]GS continuity maps.

Example 3.4.

Neu-[alpha]GS continuity maps is not Neu-continuity maps

Let [E.sub.1]={[a.sub.1], [a.sub.2]}, [E.sub.2]={[b.sub.1], [b.sub.2]},U=< [e.sub.1], (.5,.5,.3), (.7,.5, .8)> and V=< [e.sub.2],(.4,.5,.3), (.8,.5, .9)>. Then [[tau].sub.N]={[0.sub.N],U,[1.sub.N]} and [[sigma].sub.N]={[0.sub.N],V,[1.sub.N]} are Neutrosophic sets on [E.sub.1] and [E.sub.2] respectively. Define a maps f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N]) by f([a.sub.1])=[b.sub.1] and f([a.sub.2])=[b.sub.2]. Since the Neutrosophic set [lambda]=<y,(.3,.5, .4),(.9,.5, .8)> is Neu-CS in [E.sub.2], [f.sup.-1]([lambda]) is a Neu-[alpha]GSCS but not Neu-CS in [E.sub.1]. Therefore f is a Neu-[alpha]GS continuity maps but not a Neu-continuity maps.

Theorem 3.5.

Every Neu-[alpha] continuity maps is a Neu-[alpha]GS continuity maps.

Proof.

Let f:([E.sub.1], IN)[right arrow]([E.sub.2], [[sigma].sub.N])be a Neu- [alpha] continuity maps. Let [lambda] be a Neu-CS in [E.sub.2]. Then by hypothesis [f.sup.-1]([lambda]) is a Neu-[alpha]CS in [E.sub.1]. Since every Neu-[alpha]CS is a Neu-[alpha]GSCS,[f.sup.-1]([lambda])is a Neu-[alpha]GSCS in [E.sub.1]. Hence f is a Neu-[alpha]GS continuity maps.

Example 3.6.

Neu-[alpha]GS continuity maps is not Neu-[alpha] continuity maps

Let [E.sub.1]={[a.sub.1],[a.sub.2]}, [E.sub.2]={[b.sub.1],[b.sub.2]},U=< [e.sub.1],(.5,.5, .6), (.7,.5,.6)> and V=< [e.sub.2],(.3,.5,.9), (.5,.5, .7)>. Then [[tau].sub.N]={[0.sub.N],U,[1.sub.N]} and [[sigma].sub.N]={[0.sub.N], V, [1.sub.N]} are Neutrosophic Topologies on [E.sub.1] and [E.sub.2] respectively. Define a maps f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N]) by f([a.sub.1]=[b.sub.2] and f([a.sub.2])=[b.sub.2]. Since the Neutrosophic set [lambda]=< [e.sub.2], (.9,.5, .3), (.7,.5, .5)> is Neu-CS in [E.sub.2], [f.sup.-1]([lambda]) is a Neu-[alpha]GSCS continuity maps.

Remark 3.7.

Neu-G continuity maps and Neu-[alpha]GS continuity maps are independent of each other.

Example 3.8.

Neu-[alpha]GS continuity maps is not Neu-G continuity maps.

Let [E.sub.1]={[a.sub.1], [a.sub.2]}, [E.sub.2]={[b.sub.1], [b.sub.2]}, U=< [e.sub.1],(.5,.5, .6), (.8,.5,.4)> and V=< [e.sub.2],(.7,.5,.4), (.9,.5, .3)>. Then [[tau].sub.N]={[0.sub.N],U,[1.sub.N]} and [[sigma].sub.N]={[0.sub.N],V,[1.sub.N]} are Neutrosophic Topologies on [E.sub.1] and [E.sub.2] respectively. Define a maps f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N]) by f([a.sub.1])=[b.sub.1] and f([a.sub.2])=[b.sub.2]. Then f is Neu-[alpha]GS continuity maps but not Neu-G continuity maps.

Since [lambda]=< [e.sub.1],(.4,.5, .7), (.3,.5, .9)> is Neu-CS in [E.sub.2], [f.sup.-1]([lambda])=< [e.sub.2], (.4,.5, .7), (.7,.5, .3)> is not Neu-GCS in [E.sub.1].

Example 3.9.

Neu-G continuity maps is not Neu-[alpha]GS continuity maps.

Let [E.sub.1]={[a.sub.1], [a.sub.2]}, [E.sub.2]={[b.sub.1],[b.sub.2]}, U=<[e.sub.1],(.6,.5,.4), (.8,.5,.2)> and V=<[e.sub.2],(.3,.5,.7), (.1,.5, .9)>. Then IN={[0.sub.N],U,[1.sub.N]} and [[sigma].sub.N]={[0.sub.N],V,[1.sub.N]} are Neutrosophic Topologies on [E.sub.1] and [E.sub.2] respectively. Define a maps f:([E.sub.1], [[tau].sub.N]) [right arrow]([E.sub.2], [[sigma].sub.N]) by f([a.sub.1]=[b.sub.1] and f([a.sub.2])=[b.sub.2]. Then f is Neu-G continuity maps but not a Neu-[alpha]GS continuity maps.

Since [lambda]=< [e.sub.2],(.7,.5, .3), (.9,.5, .1)> is Neu-CS in [E.sub.2], [f.sup.-1]([lambda])=< [E.sub.1] (.7,.5,.3), (.9,.5,.1)> is not Neu-[alpha]GSCS in [E.sub.1].

Theorem 3.10.

Every Neu-[alpha]GS continuity maps is a Neu-GS continuity maps.

Proof.

Let f f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N] be a Neu-[alpha]GS continuity maps. Let [lambda] be a Neu-CS in [E.sub.2]. Then by hypothesis [f.sup.-1]([lambda]) Neu-[alpha]GSCS in [E.sub.1]. Since every Neu-[alpha]GSCS is a Neu-GSCS, [f.sup.-1]([lambda]) is a Neu-GSCS in [E.sub.1]. Hence f is a Neu-GS continuity maps.

Example 3.11.

Neu-GS continuity maps is not Neu-[alpha]GS continuity maps.

Let [E.sub.1]={[a.sub.1], [a.sub.2]}, [E.sub.2]={[b.sub.1], [b.sub.2]}, U=<[e.sub.1],(.8,.5,.4), (.9,.5,.2)> and V=< [e.sub.2],(.3,.5,.9), (0.l,.5, .9)>. Then [[tau].sub.N]={[0.sub.N],U,[1.sub.N]} and [[sigma].sub.N]={[0.sub.N],V,[1.sub.N]} are Neutrosophic Topologies on [E.sub.1] and [E.sub.2] respectively. Define a maps f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])by f([a.sub.1])=[b.sub.1] and f([a.sub.2])=[b.sub.2].Since the Neutrosophic set [lambda]=< [e.sub.2],(.9,.5,.3),(.9, .5,.l)> is Neu-CS in [E.sub.2], [f.sup.-1]([lambda]) is Neu-GSCS in [E.sub.1] but not Neu-[alpha]GSCS in [E.sub.1]. Therefore f is a Neu-GS continuity maps but not a Neu-[alpha]GS continuity maps.

Remark 3.12.

Neu-P continuity maps and Neu-[alpha]GS continuity maps are independent of each other.

Example 3.13.

Neu-P continuity maps is not Neu-[alpha]GS continuity maps Let [E.sub.1]={[a.sub.1], [a.sub.2]}, [E.sub.2]={[b.sub.1], [b.sub.2]},U= <[e.sub.1], (.3,.5,.7),(.4,.5,.6)> and V=< [e.sub.2],(.8,.5,.3), (.9,.5, .2)>. Then [[tau].sub.N]={[0.sub.N],U,[1.sub.N]} and [[sigma].sub.N]={[0.sub.N], V, [1.sub.N]} are Neutrosophic Topologies on [E.sub.1] and [E.sub.2] respectively. Define a maps f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])by f([a.sub.1])=[b.sub.1] and f([a.sub.2])=[b.sub.2].Since the Neutrosophic set [lambda]=< [e.sub.2],(.3,.5, .8), (.2,.5, .9)> is Neu-CS in [E.sub.2], [f.sup.-1]([lambda]) is Neu-PCS in [E.sub.1] but not Neu-[alpha]GSCS in [E.sub.1]. Therefore f is a Neu-P continuity maps but not Neu-[alpha]GS continuity maps.

Example 3.14.

Neu-[alpha]GS continuity maps is not Neu-P continuity maps

Let [E.sub.1]={[a.sub.1], [a.sub.2]}, [E.sub.2]={[b.sub.1], [b.sub.2]}, U=<[e.sub.1],(.4,.5,.8),(.5,.5,.7)> and V=<[e.sub.1],(.5,.5,.7), (.6,.5, .6)> and W=< [e.sub.2],(.8,.5,.4), (.5,.5,.7)>. Then [[tau].sub.N]={[0.sub.N],U,V,[1.sub.N]} and [[sigma].sub.N]={[0.sub.N],W,[1.sub.N]} are Neutrosophic Topologies on [E.sub.1] and [E.sub.2] respectively.

Define a maps f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])by f([a.sub.1]) = [b.sub.1] and f([a.sub.2])=[b.sub.2]. Since the Neutrosophic set [lambda]=<y,(.4,.5, .8), (.7,.5, .5)> is Neu-[alpha]GSCS but not Neu-PCS in [E.sub.2], [f.sup.-1]([lambda])is Neu-[alpha]GSCS in [E.sub.1] but not Neu-PCS in [E.sub.1]. Therefore f is a Neu-[alpha]GS continuity maps but not Neu-P continuity maps.

Theorem 3.15.

Every Neu-[alpha]GS continuity maps is a Neu-[alpha]G continuity maps.

Proof.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])be a Neu-[alpha]GS continuity maps. Let [lambda] be a Neu-CS in [E.sub.2]. Since f is Neu-[alpha]GS continuity maps, [f.sup.-1]([lambda]) is a Neu-[alpha]GSCS in [E.sub.1]. Since every Neu-[alpha]GSCS is a Neu- [alpha]GCS, [f.sup.-1]([lambda]) is a Neu- [alpha]GCS in [E.sub.1]. Hence f is a Neu- [alpha]G continuity maps.

Example 3.16.

Neu- [alpha]G continuity maps is not Neu-[alpha]GS continuity maps

Let [E.sub.1]={[a.sub.1], [a.sub.2]}, [E.sub.2]={[b.sub.1], [b.sub.2]}, U=<[e.sub.1],(.l,.5,.7),(.3,.5, .6)> and V=< [e.sub.2],(.7,.5,.4), (.6,.5, .5)>. Then [[tau].sub.N]={[0.sub.N],U,[1.sub.N]} and [[sigma].sub.N]={[0.sub.N],V,[1.sub.N]} are Neutrosophic Topologies on [E.sub.1] and [E.sub.2] respectively. Define a maps f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])by f([a.sub.1])=[b.sub.1] and f([a.sub.2])=[b.sub.2].Since the Neutrosophic set [lambda]=< [e.sub.2],(.4,.5,.7),(.5,.5, .6)> is Neu-CS in [E.sub.2], [f.sup.-1]([lambda]) is Neu-[alpha]GCS in [E.sub.1] but not Neu-[alpha]GSCS in [E.sub.1]. Therefore f is a Neu-[alpha]G continuity maps but not a Neu-[alpha]GS continuity maps.

Theorem 3.17.

Every Neu-[alpha]GS continuity maps is a Neu-G[alpha] continuity maps.

Proof.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])be a Neu-[alpha]GS continuity maps. Let [lambda] be a Neu-CS in [E.sub.2]. Since f is Neu-[alpha]GS continuity maps, [f.sup.-1]([lambda])is a Neu-[alpha]GSCS in [E.sub.1]. Since every Neu-[alpha]GSCS is a Neu-G[alpha]CS, [f.sup.-1]([lambda]) is a Neu-G[alpha]CS in [E.sub.1]. Hence f is a Neu-G[alpha] continuity maps.

Example 3.18.

Neu-G[alpha] continuity maps is not Neu-[alpha]GS continuity maps Let [E.sub.1]={[a.sub.1], [a.sub.2]}, [E.sub.2]={[b.sub.1], [b.sub.2]}, U=<[e.sub.1], (.5,.5,.7), (.3,.5, .9)> and V=< [e.sub.2],(.6,.5,.6), (.5,.5,.7)>. Then [[tau].sub.N]={[0.sub.N],U,[1.sub.N]} and [[sigma].sub.N]={[0.sub.N],V,[1.sub.N]} are Neutrosophic Topologies on [E.sub.1] and [E.sub.2] respectively. Define a maps f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])by f([a.sub.1])=[b.sub.1] and f([a.sub.2])=[b.sub.2]. Since the Neutrosophic set [lambda]=<y,(.6,.5,.6), (.7,.5, .5)> is Neu-CS in [E.sub.2], [f.sup.-1]([lambda])is Neu-G[alpha] CS in [E.sub.1] but not Neu-[alpha]GSCS in [E.sub.1]. Therefore f is a Neu-G[alpha] continuity maps but not a Neu-[alpha]GS continuity maps.

Remark 3.19.

We obtain the following diagram from the results we discussed above.

Theorem 3.20.

A maps f:([E.sub.1],[[tau].sub.N])[right arrow]([E.sub.2],[[sigma].sub.N])is Neu-[alpha]GS continuity if and only if the inverse image of each Neutrosophic set in [E.sub.2] is a Neu-[alpha]GSOS in [E.sub.1].

Proof.

first part Let [lambda] be a Neutrosophic set in [E.sub.2]. This implies [[lambda].sup.C] is Neu-CS in [E.sub.2]. Since f is Neu-[alpha]GS continuity, f (-) [.sup.1]([[lambda].sup.C]) is Neu-[alpha]GSCS in [E.sub.1]. Since [f.sup.-1]([[lambda].sup.C])=[([f.sup.-1]([lambda])).sup.C], [f.sup.-1]([lambda]) is a Neu-[alpha]GSOS in [E.sub.1].

Converse part Let [lambda] be a Neu-CS in [E.sub.2]. Then [[lambda].sup.C] is a Neutrosophic set in [E.sub.2]. By hypothesis [f.sup.-1]([[lambda].sup.C]) is Neu-[alpha]GSOS in [E.sub.1]. Since [f.sup.-1]([[lambda].sup.C])=[([f.sup.-1]([lambda])).sup.C], [([f.sup.-1]([lambda])).sup.C] is a Neu-[alpha]GSOS in [E.sub.1]. Therefore [f.sup.-1]([lambda]) is a Neu-[alpha]GSCS in [E.sub.1]. Hence f is Neu-[alpha]GS continuity.

Theorem 3.21.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])be a maps and [f.sup.-1]([lambda]) be a Neu-RCS in [E.sub.1]for every Neu-CS [lambda] in [E.sub.2]. Then f is a Neu-[alpha]GS continuity maps.

Proof.

Let [lambda] be a Neu-CS in [E.sub.2] and [f.sup.-1]([lambda]) be a Neu-RCS in [E.sub.1]. Since every Neu-RCS is a Neu-[alpha]GSCS, [f.sup.-1]([lambda]) is a Neu-[alpha]GSCS in [E.sub.1]. Hence f is a Neu-[alpha]GS continuity maps.

Definition 3.22.

A Neutrosophic Topology (E, [[tau].sub.N]) is said to be an

(i)[Neu-.sub.[alpha]ga][U.sub.1/2](in short [Neu-.sub.[alpha]ga][U.sub.1/2]) space,if every Neu-[alpha]GSCS in E is a Neu-CS in E,

(ii)[Neu-.sub.[alpha]gb][U.sub.1/2](in short [Neu-.sub.[alpha]gb][U.sub.1/2]) space,if every Neu-[alpha]GSCS in E is a Neu-GCS in E,

(iii)[Neu-.sub.[alpha]gc][U.sub.1/2](in short [Neu-.sub.[alpha]gc][U.sub.1/2]) space, if every Neu-[alpha]GSCS in E is a Neu-GSCS in E.

Theorem 3.23.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])be a Neu-[alpha]GS continuity maps, then f is a Neu-continuity maps if [E.sub.1] is a [Neu-.sub.[alpha]ga][U.sub.1/2] space.

Proof.

Let [lambda] be a Neu-CS in [E.sub.2]. Then [f.sup.-1]([lambda])is a Neu-[alpha]GSCS in [E.sub.1], by hypothesis.Since [E.sub.1] is a [Neu-.sub.[alpha]ga][U.sub.1/2],[f.sup.-1]([lambda]) is a Neu-CS in [E.sub.1]. Hence f is a Neu-continuity maps.

Theorem 3.24.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])be a Neu-[alpha]GS continuity maps, then f is a Neu-G continuity maps if [E.sub.1] is a [Neu-.sub.[alpha]gb][U.sub.1/2] space.

Proof.

Let [lambda] be a Neu-CS in [E.sub.2]. Then [f.sup.-1]([lambda]) is a Neu-[alpha]GSCS in [E.sub.1], by hypothesis. Since [E.sub.1] is a [Neu-.sub.[alpha]gb][U.sub.1/2],[f.sup.-1]([lambda]) is a Neu-GCS in [E.sub.1]. Hence f is a Neu-G continuity maps.

Theorem 3.25.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])be a Neu-[alpha]GS continuity maps, then f is a Neu-GS continuity maps if [E.sub.1] is a [Neu-.sub.[alpha]gc][U.sub.1/2] space.

Proof.

Let [lambda] be a Neu-CS in [E.sub.2]. Then [f.sup.-1]([lambda]) is a Neu-[alpha]GSCS in [E.sub.1], by hypothesis. Since [E.sub.1] is a [Neu-.sub.[alpha]gc][U.sub.1/2],[f.sup.-1]([lambda]) is a Neu-GSCS in [E.sub.1]. Hence f is a Neu-GS continuity maps.

Theorem 3.26.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])be a Neu-[alpha]GS continuity maps and g:([E.sub.2], [[sigma].sub.N])[right arrow]([E.sub.3], [[rho].sub.N]) be an Neutrosophic continuity, then gof :([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.3], [[rho].sub.N]) is a Neu-[alpha]GS continuity.

Proof.

Let [lambda] be a Neu-CS in [E.sub.3]. Then [g.sup.-1]([lambda])is a Neu-CS in [E.sub.2], by hypothesis. Since f is a Neu-[alpha]GS continuity maps, [f.sup.-1]([g.sup.-1]([lambda])) is a Neu-[alpha]GSCS in [E.sub.1]. Hence gof is a Neu-[alpha]GS continuity maps.

Theorem 3.27.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])be a maps from Neutrosophic Topology in [E.sub.1] in to a Neutrosophic Topology [E.sub.2]. Then the following conditions set are equivalent if [E.sub.1] is a [Neu-.sub.[alpha]ga][U.sub.1/2] space.

(i) f is a Neu-[alpha]GS continuity maps.

(ii) if [micro] is a Neutrosophic set in [E.sub.2] then [f.sup.-1]([micro]) is a Neu-[alpha]GSOS in [E.sub.1].

(iii) [f.sup.-1] (Neu-int([micro])) [??]Neu-int(Neu-Cl(Neu-int([f.sup.-1] ([micro])))) for every Neutrosophic set [micro] in [E.sub.2].

Proof.

(i)[right arrow] (ii): is obviously true.

(ii)[right arrow] (iii): Let [micro] be any Neutrosophic set in [E.sub.2]. Then Neu-int([micro]) is a Neutrosophic set in [E.sub.2]. Then [f.sup.-1](Neu-int([micro])) is a Neu-[alpha]GSOS in [E.sub.1]. Since [E.sub.1] is a [Neu-.sub.[alpha]ga][U.sub.1/2] space, [f.sup.-1](Neu-int([micro]))is a Neutrosophic set in [E.sub.1]. Therefore [f.sup.-1] (Neu-int([micro]))=Neu-int([f.sup.-1] (Neu-int([micro])))[??]Neu-int(Neu-Cl(Neu-int([f.sup.-1] ([micro])))).

(iii)[right arrow](i) Let [micro] be a Neu-CS in [E.sub.2]. Then its complement [[mu].sup.C] is a Neutrosophic set in [E.sub.2]. By Hypothesis [f.sup.-1](Neu-int([[mu].sup.C])) [??]Neu-int(Neu-Cl(Neu-int([f.sup.-1](Neu-int([[mu].sup.C]))))). This implies that [f.sup.-1]([[mu].sup.C])[??]Neu-int(Neu-Cl(Neu-int([f.sup.-1](Neu-int([[mu].sup.C]))))). Hence [f.sup.-1]([[mu].sup.C]) is a Neu-[alpha]OS in [E.sub.1]. Since every Neu-[alpha]OS is a Neu-[alpha]GSOS, [f.sup.-1][[mu].sup.C]is a Neu-[alpha]GSOS in [E.sub.1]. Therefore [f.sup.-1]([micro]) is a Neu-[alpha]GSCS in [E.sub.1]. Hence f is a Neu-[alpha]GS continuity maps.

Theorem 3.28.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])be a maps. Then the following conditions set are equivalent if [E.sub.1] is a [Neu-.sub.[alpha]ga][U.sub.1/2] space.

(i) f is a Neu-[alpha]GS continuity maps.

(ii) [f.sup.-1]([lambda]) is a Neu-[alpha]GSCS in [E.sub.1] for every Neu-CS [lambda] in [E.sub.2].

(iii) Neu-Cl(Neu-int(Neu-Cl([f.sup.-1] ([lambda])))) [??][f.sup.-1] (Neu-Cl([lambda])) for every Neutrosophic set [lambda] in [E.sub.2].

Proof.

(i)[right arrow] (ii): is obviously true.

(ii)[right arrow] (iii): Let [lambda] be a Neutrosophic set in [E.sub.2]. Then Neu-Cl([lambda]) is a Neu-CS in [E.sub.2]. By hypothesis,[f.sup.-1] (Neu-Cl([lambda]))is a Neu-[alpha]GSCS in [E.sub.1]. Since [E.sub.1] is a [Neu-.sub.[alpha]ga][U.sub.1/2] space, [f.sup.-1](Neu- Cl([lambda])) is a Neu-CS in [E.sub.1]. Therefore Neu-Cl([f.sup.-1](Neu-Cl([lambda])))=[f.sup.-1] (Neu-Cl([lambda])).NowNeu-Cl(Neu-int(Neu-Cl([f.sup.-1]([lambda]))))cNeu-Cl (Neu-int(Neu-Cl([f.sup.-1](Neu-Cl([lambda]))))) [??][f.sup.-1] (Neu-Cl([lambda])).

(iii)[right arrow](i): Let [lambda] be a Neu-CS in [E.sub.2]. By hypothesis Neu-Cl(Neu-int(Neu-Cl([f.sup.-1]([lambda]))))[??][f.sup.-1](Neu-Cl([lambda]))=[f.sup.-1]([lambda]). This implies [f.sup.-1]([lambda]) is a Neu-[alpha]CS in [E.sub.1] and hence it is a Neu-[alpha]GSCS in [E.sub.1]. Therefore f is a Neu-[alpha]GS continuity maps.

Definition 3.29.

Let (E, [[tau].sub.N]) be a Neutrospohic topology. The Neutrospohic alpha generalized semi closure (Neu-[alpha]GSCl([lambda])in short) for any Neutrosophic set [lambda] is Defined as follows. Neu-[alpha]GSCl([lambda])=n{K|K is a Neu-[alpha]GSCS in [E.sub.1] and [lambda] [??]K). If [lambda] is Neu-[alpha]GSCS, then Neu-[alpha]GSCl([lambda])=[lambda].

Theorem 3.30.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])be a Neu-[alpha]GS continuity maps. Then the following conditions set are hold.

(i) f(Neu-[alpha]GSCl([lambda]))[??]Neu-Cl(f([lambda])), for every Neutrosophic set [lambda] in [E.sub.1].

(ii) Neu-[alpha]GSCl([f.sup.-1]([micro]))[??][f.sup.-1](Neu-Cl([micro])),for every Neutrosophic set [micro] in [E.sub.2].

Proof.

(i) Since Neu-Cl(f([lambda]))is a Neu-CS in [E.sub.2] and f is a Neu-[alpha]GS continuity maps, [f.sup.-1] (Neu-Cl(f([lambda])))is Neu-[alpha]GSCS in [E.sub.1]. That is Neu-[alpha]GSCl([lambda])[??][f.sup.-1](Neu-Cl(f([lambda]))). Therefore f(Neu-[alpha]GSCl([lambda])) [??]Neu-Cl(f([lambda])),for every Neutrosophic set [lambda] in [E.sub.1].

(ii) Replacing [lambda] by [f.sup.-1]([micro]) in (i) we get f(Neu-[alpha]GSCl([f.sup.-1]([micro])))[??]Neu-Cl(f([f.sup.-1]([micro])))cNeu-Cl([micro]).Hence Neu-[alpha]GSCl([f.sup.-1] ([micro])) [??][f.sup.-1] (Neu-Cl([micro])), for every Neutrosophic set [micro] in [E.sub.2].

4. Neutrosophic [alpha]-Generalized Semi Irresolute Maps

In this section we Introduce Neutrosophic [alpha]-generalized semi irresolute maps and study some of its characterizations.

Definition 4.1.

A maps f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])is called a Neutrosophic alpha-generalized semi irresolute (Neu-[alpha]GS irresolute) maps if [f.sup.-1]([lambda]) is a Neu-[alpha]GSCSin ([E.sub.1],[[TAU].sub.N]) for every Neu-[alpha]GSCS [lambda] of (E2, [[sigma].sub.N])

Theorem 4.2.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])be a Neu-[alpha]GS irresolute, then f is a Neu-[alpha]GS continuity maps.

Proof.

Let f be a Neu-[alpha]GS irresolute maps. Let [lambda] be any Neu-CS in [E.sub.2]. Since every Neu-CS is a Neu-[alpha]GSCS, [lambda] is a Neu-[alpha]GSCS in [E.sub.2]. By hypothesis [f.sup.-1]([lambda]) is a Neu-[alpha]GSCS in [E.sub.2]. Hence f is a Neu-[alpha]GS continuity maps.

Example 4.3.

Neu-[alpha]GS continuity maps is not Neu-[alpha]GS irresolute maps.

Let [E.sub.1]={[a.sub.1], [a.sub.2]), [E.sub.2]={[b.sub.1], [b.sub.2]), U=< [e.sub.1],(.4,.5, .7), (.5,.5,.6)> and V=< e2,(.8,.5,.3), (.4,.6, .7)>. Then [[tau].sub.N]={[0.sub.N],U,[1.sub.N]) and [[sigma].sub.N] = {[0.sub.N],V,[1.sub.N]) are Neutrosophic Topologies on [E.sub.1] and [E.sub.2] respectively. Define a maps f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])by f([a.sub.1])=[b.sub.1] and f([a.sub.2])=[b.sub.2]. Then f is a Neu-[alpha]GS continuity. We have [micro]=< [e.sub.2],(.2,.5, .9), (.6,.5, .5)> is a Neu-[alpha]GSCS in [E.sub.2] but [f.sup.-1]([micro]) is not a Neu-[alpha]GSCS in [E.sub.1]. Therefore f is not a Neu-[alpha]GS irresolute maps.

Theorem 4.4.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])be a Neu-[alpha]GS irresolute, then f is a Neutrosophic irresolute maps if [E.sub.1] is a [Neu-.sub.[alpha]ga][U.sub.1/2] space.

Proof.

Let [lambda] be a Neu-CS in [E.sub.2]. Then [lambda] is a Neu-[alpha]GSCS in [E.sub.2]. Therefore [f.sup.-1]([lambda]) is a Neu-[alpha]GSCS in [E.sub.1], by hypothesis.

Since [E.sub.1] is a [Neu.sub.-[alpha]ga][U.sub.1/2] space, [f.sup.-1]([lambda]) is a Neu-CS in [E.sub.1]. Hence f is a Neutrosophic irresolute maps.

Theorem 4.5.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])and g:([E.sub.2], [[sigma].sub.N])[right arrow]([E.sub.3], [[rho].sub.N]) be Neu-[alpha]GS irresolute maps, then gof:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.3], [[rho].sub.N])is a Neu-[alpha]GS irresolute maps.

Proof.

Let [lambda] be a Neu-[alpha]GSCS in [E.sub.3]. Then [g.sup.-1]([lambda]) is a Neu-[alpha]GSCS in [E.sub.2]. Since f is a Neu-[alpha]GS irresolute maps. [f.sup.-1](([g.sup.-1]([lambda]))) is a Neu-[alpha]GSCS in [E.sub.1]. Hence gof is a Neu-[alpha]GS irresolute maps.

Theorem 4.6.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])be a Neu-[alpha]GS irresolute and g:([E.sub.2], [[sigma].sub.N]) [right arrow]([E.sub.3], [[rho].sub.N]) be Neu-[alpha]GS continuity maps, then gof:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.3], [[RHO].sub.N]) is a Neu-[alpha]GS continuity maps.

Proof.

Let [lambda] be a Neu-CS in [E.sub.3]. Then [g.sup.-1]([lambda]) is a Neu-[alpha]GSCS in [E.sub.2]. Since f is a Neu-[alpha]GS irresolute, [f.sup.-1](([g.sup.-1]([lambda])) is a Neu-[alpha]GSCS in [E.sub.1]. Hence gof is a Neu-[alpha]GS continuity maps.

Theorem 4.7.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])be a Neu-[alpha]GS irresolute, then f is a Neu-G irresolute maps if [E.sub.1] is a [Neu-.sub.[alpha]gb][U.sub.1/2] space.

Proof.

Let [lambda] be a Neu-[alpha]GSCS in [E.sub.2]. By hypothesis, [f.sup.-1]([lambda]) is a Neu-[alpha]GSCS in [E.sub.1]. Since [E.sub.1] is a [Neu-.sub.[alpha]gb][U.sub.1/2] space, [f.sup.-1]([lambda]) is a Neu-GCS in [E.sub.1]. Hence f is a Neu-G irresolute maps.

Theorem 4.8.

Let f:([E.sub.1], [[tau].sub.N])[right arrow]([E.sub.2], [[sigma].sub.N])be a maps from a Neutrosophic Topology [E.sub.1] Into a Neutrosophic Topology [E.sub.2]. Then the following conditions set are equivalent if [E.sub.1] and [E.sub.2] are [Neu-.sub.[alpha]ga][U.sub.1/2] spaces.

(i) f is a Neu-[alpha]GS irresolute maps.

(ii) [f.sup.-1]([micro]) is a Neu-[alpha]GSOS in [E.sub.1] for each Neu-[alpha]GSOS [micro] in [E.sub.2].

(iii) Neu-Cl([f.sup.-1]([micro]))[??][f.sup.-1](Neu-Cl([micro])) for each Neutrosophic set [micro] of [E.sub.2].

Proof.

(i) [right arrow](ii) : Let [micro] be any Neu-[alpha]GSOS in [E.sub.2]. Then [[mu].sup.C] is a Neu-[alpha]GSCS in [E.sub.2]. Since f is Neu-[alpha]GS irresolute, [f.sup.-1][[mu].sup.C] is a Neu-[alpha]GSCS in [E.sub.1]. But [f.sup.-1][[mu].sup.C]=[([f.sup.-1]([micro])).sup.C]Therefore [f.sup.-1]([micro]) is a Neu-[alpha]GSOS in [E.sub.1].

(ii)[right arrow](iii) : Let [micro] be any Neutrosophic set in [E.sub.2] and [micro][??]Neu-Cl([micro]). Then [f.sup.-1]([micro])[??][f.sup.-1](Neu-Cl([micro])). Since Neu-Cl([micro]) is a Neu-CS in [E.sub.2], Neu-Cl([micro]) is a Neu-[alpha]GSCS in [E.sub.2]. Therefore [(Neu-Cl([micro])).sup.C] is a Neu-[alpha]GSOS in [E.sub.2]. By hypothesis, [f.sup.-1][((Neu-Cl([micro])).sup.C]) is a Neu-[alpha]GSOS in [E.sub.1]. Since [f.sup.-1][((Neu-Cl([micro])).sup.C])=[([f.sup.-1](Neu-Cl([micro]))).sup.C],[f.sup.-1](Neu-Cl([micro])) is a Neu-[alpha]GSCS in [E.sub.1]. Since [E.sub.1] is [Neu-.sub.[alpha]ga][U.sub.1/2] space,[f.sup.-1](Neu-Cl([micro])) is a Neu-CS in [E.sub.1]. Hence Neu-Cl([f.sup.-1]([micro]))[??]Neu-Cl([f.sup.-1](Neu-Cl([micro])))=[f.sup.-1](Neu-Cl([micro])). That is Neu-Cl([f.sup.-1]([micro]))[??][f.sup.-1](Neu-Cl([micro])).

(iii)[right arrow](i) : Let [micro] be any Neu-[alpha]GSCS in [E.sub.2]. Since [E.sub.2] is [Neu.sub.-[alpha]ga][U.sub.1/2] space, [micro] is a Neu-CS in [E.sub.2] and Neu-Cl([micro])=[micro].Hence [f.sup.-1]([micro])=[f.sup.-1](Neu-Cl([micro]) [??] Neu-Cl([f.sup.-1]([micro])). But clearly [f.sup.-1]([micro])[??]Neu-Cl([f.sup.-1]([micro])). Therefore Neu-Cl([f.sup.-1] ([micro]))=[f.sup.-1]([micro]). This implies [f.sup.-1]([micro]) is a Neu-CS and hence it is a Neu-[alpha]GSCS in [E.sub.1]. Thus f is a Neu-[alpha]GS irresolute maps.

Conclusion

In this research paper using Neu-[alpha]GSCS(Neutrosophic [alpha]gs-closed sets) we are defined Neu-[alpha]GS continuity maps and analyzed its properties.after that we were compared already existing Neutrosophic continuity maps to Neu-[alpha]GSCS continuity maps. Furthermore we were extended to this maps to Neu-[alpha]GS irresolute maps, Finally This concepts can be extended to future Research for some mathematical applications.

References

[1] Abdel-Basset, M., Saleh, M., Gamal, A., & Smarandache, F. (2019). An approach of TOPSIS technique for developing supplier selection with group decision making under type-2 neutrosophic number. Applied Soft Computing, 77, 438-452.

[2] Abdel-Baset, M., Chang, V., Gamal, A., & Smarandache, F. (2019). An integrated neutrosophic ANP and VIKOR method for achieving sustainable supplier selection: A case study in importing field. Computers in Industry, 106, 94-110.

[3] Abdel-Basset, M., Manogaran, G., Gamal, A., & Smarandache, F. (2[0.sub.1]9). A group decision making framework based on neutrosophic TOPSIS approach for smart medical device selection. Journal of medical systems, 43(2), 38.

[4] Abdel-Baset, M., Chang, V., & Gamal, A. (2019). Evaluation of the green supply chain management practices: A novel neutrosophic approach. Computers in Industry, 108, 210-220.

[5] Abdel-Basset, M., Manogaran, G., Gamal, A., & Smarandache, F. (2018). A hybrid approach of neutrosophic sets and DEMATEL method for developing supplier selection criteria. Design Automation for Embedded Systems, 1-22.

[6] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1986), 87-94.

[7] I.Arokiarani, R. Dhavaseelan, S. Jafari, M. Parimala, On Some New Notions and Functions In Neutrosophic Topological Spaces, Neutrosophic Sets and Systems, Vol. 16, 2017, (16-19)

[8] V.Banu priya, S.Chandrasekar, Neutrosophic [alpha] generalized semi closed set(Communicated)

[9] R. Dhavaseelan and S. Jafari, Generalized Neutrosophic closed sets, New trends in Neutrosophic theory and applications Volume II- 26l-273,(20l8).

[10] R. Dhavaseelan, S. Jafari and md. Hanif page, Neutrosophic generalized [alpha]-contra-continuity, creat. math. inform. 27 (2O18), no. 2, 133 - 139

[11] Florentin Smarandache, Neutrosophic and NeutrosophicLogic, First International Conference On Neutrosophic,Neutrosophic Logic, Set, Probability, and Statistics University of New MeEico, Gallup, NM 87301, USA (2002), smarand@unm.edu

[12] Floretin Smaradache, Neutrosophic Set: - A Generalization of Intuitionistic Fuzzy set, Journal of Defense Resourses Management. 1(2010), 107-114.

[13] Ishwarya, P and Bageerathi, K., On Neutrosophic semiopen sets in Neutrosophic topological spaces, International Jour. of Math. Trends and Tech. 2016, 214-223.

[14] D.Jayanthi, [alpha] Generalized Closed Sets in Neutrosophic Topological Spaces, InternationalJournal of Mathematics Trends and Technology (IJMTT)- Special Issue ICRMIT March 2018.

[15] A.A. Salama and S.A. Alblowi, Generalized Neutrosophic Set and Generalized Neutrosophic Topological Spaces, Journal computer Sci. Engineering, Vol.(ii) No.(7)(20l2).

[16] A.A.Salama and S.A. Alblowi, Neutrosophic set and Neutrosophic topological space, ISOR J.mathematics, Vol.(iii), Issue(4), (2012). pp-31-35

[17] V.K.Shanthi, S. Chandrasekar, K.SafinaBegam, Neutrosophic Generalized Semi Closed Sets In Neutrosophic Topological Spaces, International Journal of Research in Advent Technology, Vol.6, No.7, July 2018, 1739-1743

[18] V. Venkateswara Rao, Y. Srinivasa Rao, Neutrosophic Pre-open Sets and Pre-closed Sets in Neutrosophic Topology, International Journal of ChemTech Research, Vol.10 No. 10, pp 449-458, 2017

[19] Nabeeh, N. A., Smarandache, F., Abdel-Basset, M., El-Ghareeb, H. A., & Aboelfetouh, A. (2019). An Integrated Neutrosophic-TOPSIS Approach and Its Application to Personnel Selection: A New Trend in Brain Processing and Analysis. IEEE Access, 7, 29734-29744.

[20] Zadeh, L. A. "Fuzzy sets", Information and Control, 8, 338-353 (1965).

Received: January 7, 2019. Accepted: May 20, 2019

V.Banupriya (1), S.Chandrasekar (2)

(1) Assistant Professor, Department of Mathematics, RMK College of Engineering and Technology, Puduvoyal, Tiruvallur(DT), Tamil Nadu, India.E-mail: spriya.maths@gmail.com.

(2) Assistant Professor, PG and Research Department of Mathematics, Arignar Anna Government Arts College,Namakkal(DT),Tamil Nadu, India. E-mail: chandrumat@gmail.com.
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