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Generalized Neutrosophic Exponential map.

1 Introduction

Ever since the introduction of fuzzy sets by Zadeh [12] and fuzzy topological space by Chang [5], several authors have tried successfully to generalize numerous pivot concepts of general topology to the fuzzy setting. The concept of intuitionistic fuzzy set was introduced are studied by Atanassov [1] and many works by the same author and his colleagues appeared in the literature [[2],[3],[4]]. The concepts of generalized intuitionistic fuzzy closed set was introduced by Dhavaseelan et al[6]. The concepts of Intuitionistic Fuzzy Exponential Map Via Generalized Open Set by Dhavaseelan et al[8]. After the introduction of the neutrosophic set concept [[10], [11]]. The concepts of Neutrosophic Set and Neutrosophic Topological Spaces was introduced by A.A.Salama and S.A.Alblowi[9].

In this paper the concept of g[ALEPH] compact open topology are introduced. Some interesting properties are discussed. In this paper the concepts of g[ALEPH] local compactness and generalized [ALEPH]--product topology are developed. We have Throughout this paper neutrosophic topological spaces (briefly NTS) ([S.sub.1],[[xi].sub.1]),([S.sub.2], [[xi].sub.2]) and ([S.sub.3], [[xi].sub.3]) will be replaced by [S.sub.1],[S.sub.2] and [S.sub.3], respectively.

2 Preliminiaries

Definition 2.1. [10,11] Let T,I,F be real standard or non standard subsets of][0.sup.-], [1.sup.+][,with [sup.sub.T] = [t.sub.sup],in [f.sub.T] = [t.sub.inf]

[sup.sub.I] [i.sub.sup], in[f.sub.I] = [i.sub.inf]

[sup.sub.F] = [f.sub.sup], in[f.sub.F] = [f.sub.inf]

n - sup = [t.sub.Sup] + [i.sub.sup] + [f.sub.sup]

n - inf = [t.sub.inf] + [i.sub.inf] + [f.sub.inf]. T,I,F are [ALEPH]--components.

Definition 2.2. [10, 11] Let [S.sub.1] be a non-empty fixed set. A [ALEPH]--set (briefly N-set) [LAMBDA] is an object such that [LAMBDA] = {<x[[mu].sub.[and]](x),[[sigma].sub.[and]](x),[[gamma].sub.[and]](x)> : x [member of] [S.sb.1]} where [[mu].sub.[and]](x),[[sigma].sub.[and]](x) and [[gamma].sub.[and]](x) which represents the degree of membership function (namely [[mu].sub.[LAMBDA]](x)), the degree of indeterminacy (namely [[sigma].sub.[and]] (x)) and the degree of non-membership (namely [[gamma].sub.[LAMBDA]] (x)) respectively of each element x [member of] [S.sub.1] to the set [LAMBDA].

Remark 2.1. [10, 11]

(1) An N-set [LAMBDA] = {<x, [[mu].sub.[LAMBDA]](x), [[sigma].sub.[LAMBDA]](x), [[GAMMA].sub.[LAMBDA]](x)> : x [member of] [S.sub.1]} can be identified to an ordered triple <[[mu].sub.[LAMBDA]], [[sigma].sub.[LAMBDA]], [[GAMMA].sub.[LAMBDA]]> in ][0.sup.-], [1.sub.+][on [S.sub.1].

(2) In this paper, we use the symbol [LAMBDA] = <[[mu].sub.[LAMBDA]], [[sigma].sub.[and]], [[GAMMA].Sub.[LAMBDA]]> for the N-set [LAMBDA] = {<x, [[mu].sub.[LAMBDA]](x), [[sigma].sub.[LAMBDA]](x), [[GAMMA].sub.[LAMBDA]](x)> : x [member of] [S.sub.1]}.

Definition 2.3. [7]Let [S.sub.1] = [empty set] and the N-sets [LAMBDA] and [GAMMA] be defined as

[LAMBDA] = {<x,[[mu].sub.[LAMBDA]](x),[[sigma].sub.[LAMBDA]](x), [[GAMMA].sub.[LAMBDA]] (x)> : x [member of] [S.sub.1]}, [GAMMA] = {<x,[[mu].sub.[GAMMA]](x),[[sigma].sub.[GAMMA]](x), [[GAMMA].sub.[GAMMA]](x)> : x [member of] [S.sub.1]}. Then

(a) [LAMBDA] [??] [GAMMA] iff [[micro].sib.[LAMBDA]] (x) [less than or equal to] [[mu].sub.[GAMMA]] (x), [[sigma].sub.[LAMBDA]] (x) [less than or equal to] [[sigma].sub.[GAMMA]] (x) and [[GAMMA].sub.[LAMBDA]] (x) [greater than or equal to] [[GAMMA].sub.[GAMMA]] (x) for all x [member of] [S.sub.1];

(b) [LAMBDA] = [GAMMA] iff [LAMBDA] [??] [GAMMA] and [GAMMA] [??] [LAMBDA];

(c) [bar.[LAMBDA]] = {<x, [[GAMMA].sub.[LAMBDA]] (x),[[sigma].sub.[LAMBDA]] (x),[[mu].sub.[LAMBDA]] (x)> : x [member of] [S.sub.1]}; [Complement of [LAMBDA]]

(d) [LAMBDA] [intersection] [GAMMA] = {< x, [[mu].sub.[LAMBDA]] (x) [LAMBDA] [[mu].sub.[GAMMA]] (x), [[mu].sub.[LAMBDA]] (x) [LAMBDA] [[mu].sub.[GAMMA]] (x), [[GAMMA].sub.[LAMBDA]] (x) V [[GAMMA].sub.[GAMMA]] (x)> : x [member of] [S.sub.1]};

(e) [LAMBDA] U [GAMMA] = {<x, [[mu].sub.[LAMBDA]] (x) V [[mu].sub.[GAMMA]] (x), [[sigma].sub.[LAMBDA]] (x) V [[sigma].sub.[GAMMA]] (x), [[GAMMA].sub.[LAMBDA]](x) [LAMBDA] [[gamma].sub.[GAMMA]](x)> : x [member of] [S.sub.1]};

(f) [][LAMBDA] = {< x,[[micro].sub.[LAMBDA]] (X),[[sigma].sub.[LAMBDA]] (x), 1-[[mu].sub.[LAMBDA]] (x)> : x [member of] [S.sub.1]};

(g) <>[LAMBDA] = {<x, 1 - [[GAMMA].sub.[LAMBDA]] (x), [[sigma].sub.[LAMBDA]] (x), [[GAMMA].sub.[LAMBDA]] (x)> : x [member of] [S.sub.1].

Definition 2.4. [7] Let {[[LAMBDA].sub.i] : i [member of] J} be an arbitrary family of N-sets in [S.sub.1]. Then

(a) [mathematical expression not reproducible]

(b) [mathematical expression not reproducible]

Since our main purpose is to construct the tools for developing NTS, we must introduce the [ALEPH]--sets [0.sub.N] and [1.sub.N] in X as follows:

Definition 2.5. [7] [0.sub.N] = {<x, 0,0,1> : x [member of] X} and [1.sub.N] = {<x, 1,1,0> : x [member of] X}.

Definition 2.6. [7]A [ALEPH]--topology (briefly N-topology) on [S.sub.1] [not equal to] [empty set] is a family [[xi].sub.1] of N-sets in [S.sub.1] satisfying the following axioms:

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

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

(iii) [union][G.sub.i] [member of] [[xi].sub.1] for arbitrary family {[G.sub.i] | i [member of] [LAMBDA]} [??] [[xi].sub.1].

In this case the ordered pair ([S.sub.1], [[xi].sub.1]) or simply [S.sub.1] is called an NTS and each N-set in [[xi].sub.1] is called a [ALEPH]--open set (briefly N-open set). The complement [bar.[LAMBDA]] of an N-open set [LAMBDA] in [S.sub.1] is called a [ALEPH]--closed set (briefly N-closed set) in [S.sub.1].

Definition 2.7. [7] Let [LAMBDA] be an N-set in an NTS [S.sub.1]. Then

Nint([LAMBDA]) = [union]{G | G is an N-open set in [S.sub.1] and G [??] [LAMBDA]} is called the [ALEPH]--interior (briefly N-interior) of [LAMBDA]; Ncl([LAMBDA]) = [intersection]{G | G is an N-closed set in [S.sub.1] and G [??] [LAMBDA]} is called the [ALEPH]--closure (briefly N-cl) of [LAMBDA].

Definition 2.8. [7] Let X be a nonempty set. If r, t, s be real standard or non standard subsets of][0.sup.-], [1.sup.+][then the [ALEPH]--set [x.sub.r,t,s] is called a [ALEPH]--point(in short NP)in X given by

[mathematical expression not reproducible]

for [x.sub.p] [member of] X is called the support of [x.sub.r,t,S].where r denotes the degree of membership value, t denotes the degree of indeterminacy and s is the degree of non-membership value of [x.sub.r,t,S].

Definition 2.9. [7] Let ([S.sub.1], [[xi].sub.1]) be a NTS. A [ALEPH]--set [LAMBDA] in ([S.sub.1], [[xi].sub.1]) is said to be a g[ALEPH] closed set if N cl ([LAMBDA]) [??] [GAMMA] whenever [LAMBDA] [??] [GAMMA] and [GAMMA] is a [ALEPH]--open set. The complement of a g[ALEPH] closed set is called a g[ALEPH] open set.

Definition 2.10. [7] Let (X, T) be a [ALEPH]--topological space and [LAMBDA] be a [ALEPH]--set in X. Then the [ALEPH]--generalized closure and [ALEPH]--generalized interior of [LAMBDA] are defined by,

(i) NGcl([LAMBDA]) = [intersection]{G:G is a generalized [ALEPH]--closed set in [S.sub.1] and [LAMBDA] [??] G}.

(ii) NGint([LAMBDA]) = [union]{G: G is a generalized [ALEPH]--open set in [S.sub.1] and [LAMBDA] I G}.

3 Neutrosophic Compact Open Topology

Definition 3.1. Let [S.sub.1] and [S.sub.2] be any two NTS. A mapping f : [S.sub.1] [right arrow] [S.sub.2] is generalized neutrosophic[briefly g[ALEPH]] continuous iff for every g[ALEPH] open set V in [S.sub.2], there exists a g[ALEPH] open set U in [S.sub.1] such that f (U) [??] V.

Definition 3.2. A mapping f : [S.sub.1] [right arrow] [S.sub.2] is said to be g[ALEPH] homeomorphism if f is bijective,g[ALEPH] continuous and g[ALEPH] open.

Definition 3.3. Let [S.sub.1] be a NTS. [S.sub.1] is said to be g[ALEPH] Hausdorff space or [T.sub.2] space if for any two [ALEPH]--sets A and B with A [intersection] B = [0.sub.~],there exist g[ALEPH] open sets U and V, such that A [??] U,B [??] V and U [intersection] V = [0.sub.~].

Definition 3.4. A NTS [S.sub.1] is said to be g[ALEPH] locally compact iff for any [ALEPH] set A, there exists a g[ALEPH] open set G, such that A [??] G and G is g[ALEPH] compact. That is each g[ALEPH] open cover of G has a finite subcover.

Remark 3.1. Let [S.sub.1] and [S.sub.2] be two NTS with [S.sub.2] [ALEPH]--compact. Let [x.sub.r,t,S] be any [ALEPH]--point in [S.sub.1]. The [ALEPH]--product space [S.sub.1] x [S.sub.2] containing {[x.sub.r,t,S]} x [S.sub.1]. It is cleat that {[x.sub.r,t,S]} x [S.sub.2] is [ALEPH]--homeomorphic to [S.sub.2]

Remark 3.2. Let [S.sub.1] and [S.sub.2] be two NTS with [S.sub.2] [ALEPH]--compact. Let [x.sub.r,t,S] be any [ALEPH]--point in [S.sub.1]. The [ALEPH]--product space [S.sub.1] x [S.sub.2] containing {[x.sub.r,t,S]} x [S.sub.2]. {[x.sub.r,t,S]} x [S.sub.1] is [ALEPH]--compact.

Remark 3.3. A [ALEPH]--compact subspace of a [ALEPH]--Hausdorff space is [ALEPH]--closed.

Proposition 3.1. A g[ALEPH] Hausdorff topological space [S.sub.1],the following conditions are equivalent.

(a) [S.sub.1] is g[ALEPH] locally compact

(b) for each [ALEPH] set A, there exists a g[ALEPH] open set G in [S.sub.1] such that A [??] G and N Gcl (G) is g[ALEPH] compact

Proof. ([alpha]) [??] (b) By hypothesis for each [ALEPH]--set A in [S.sub.1], there exists a g[ALEPH] open set G, such that A [??] G and G is g[ALEPH] compact.Since [S.sub.1] is g[ALEPH] Hausdorff, by Remark 3.3(g[ALEPH] compact subspace of g[ALEPH] Hausdorff space is g[ALEPH] closed),G is g[ALEPH] closed,thus G = NGcl(G). Hence A [??] G = NGcl (G) and NGcl(G) is g[ALEPH] compact. (b) [??] (a) Proof is simple.

Proposition 3.2. Let [S.sub.1] be a g[ALEPH] Hausdorff topological space. Then [S.sub.1] is g[ALEPH] locally compact on an [ALEPH]--set A in [S.sub.1] iff for every g[ALEPH] open set G containing A, there exists a g[ALEPH] open set V, such that A [??] V,NGcl (V) is g[ALEPH] compact and NGcl(V) [??]

Proof. Suppose that [S.sub.1] is g[ALEPH] locally compact on an [ALEPH]--set A. By Definition 3.4,there exists a g[ALEPH] open set G, such that A [??]d G is g[ALEPH] compact. Since [S.sub.1] is g[ALEPH] Hausdorff space, by Remark 3.3(g[ALEPH] compact subspace of g[ALEPH] Hausdorff space is g[ALEPH] closed),G is g[ALEPH] closed,thus G = NGcl G).Consider an [ALEPH]--set A [??Since [S.sub.1] is g[ALEPH] Hausdorff space, by Definition 3.3, for any two [ALEPH]--sets A and B with A [intersection] B = [0.sub.~], there exist a g[ALEPH] open sets C and D,such that A [??] C, B [??] D and C [intersection] D = [0.sub.~]. Let V = C [intersection] G. Hence V [??] G implies NGcl(V) [??] NGcl(G) = G. Since NGcl (V) is g[ALEPH] closed and G is g[ALEPH] compact,by Remark 3.3(every g[ALEPH] closed subset of a g[ALEPH] compact space is g[ALEPH] compact) it follows that NGcl(V) is [ALEPH]--compact. Thus A [??] NGcl (V) [??] G and NGcl(G) is g[ALEPH] compact.

The converse follows from Proposition 3.1(b).

Definition 3.5. Let [S.sub.1] and [S.sub.2] be two NTS. The function T : [S.sub.1] x [S.sub.2] [right arrow] [S.sub.2] x [S.sub.1] defined by T(x, y) = (y, x) for each (x, y) [member of] [S.sub.1] x [S.sub.2] is called a [ALEPH]--switching map.

Proposition 3.3. The [ALEPH]--switching map T : [S.sub.1] x [S.sub.2] [right arrow] [S.sub.2] x [S.sub.1] defined as above is g[ALEPH] continuous.

We now introduce the concept of g[ALEPH] compact open topology in the set of all g[ALEPH] continuous functions from a NTS [S.sub.1] to a NTS [S.sub.2].

Definition 3.6. Let [S.sub.1] and [S.sub.2] be two NTS and let [mathematical expression not reproducible] such that f is g[ALEPH] continuous}. We give this class [mathematical expression not reproducible] a topology called the g[ALEPH] compact open topology as follows:Let K = {K [member of] [I.sup.s.sub.1] : K is g[ALEPH] compact [S.sub.1]} and V = {V [member of] [I.sup.s.sub.1] : V is g[ALEPH] open in [S.sub.2]}. For any K [member of] K and V [member of] V, let [S.sub.K,V] = [mathematical expression not reproducible]

The collection of all such {[S.sub.K,V] : K [member of] K, V [member of] V} generates an [ALEPH]--structure on the class [mathematical expression not reproducible]

4 Generalized Neutrosophic Evaluation Map and Generalized Neutrosophic Exponential Map

We now consider the g[ALEPH] product topological space [mathematical expression not reproducible] and define a g[ALEPH] continuous map from [mathematical expression not reproducible] into [S.sub.2].

Definition 4.1. The mapping [mathematical expression not reproducible] defined by e(f, A) = f (A) for each [ALEPH]--set A in [S.sub.1] and [mathematical expression not reproducible] is called the g[ALEPH] evaluation map.

Definition 4.2. Let [S.sub.1],[S.sub.2] and [S.sub.3] be three NTS and f : [S.sub.3] x [S.sub.1] [right arrow] [S.sub.2] be any function. Then the induced map [mathematical expression not reproducible] is defined by [mathematical expression not reproducible] for [ALEPH]--sets [A.sub.1] of [S.sub.1] and [A.sub.1] of [S.sub.3].

Conversely,given a function [mathematical expression not reproducible] a corresponding function f can be also be defined be the same rule.

Proposition 4.1. Let [S.sub.1] be a g[ALEPH] locally compact Hausdorff space. Then the g[ALEPH] evaluation map [mathematical expression not reproducible] is g[ALEPH] continuous.

Proof. Consider (f, [A.sub.1]) [mathematical expression not reproducible],where [mathematical expression not reproducible] and [ALEPH]--set [A.sub.1] of [S.sub.1]. Let V be a g[ALEPH] open set containing f ([A.sub.1]) = e(f, [A.sub.1]) in [S.sub.2]. Since [S.sub.1] is g[ALEPH] locally compact and f is g[ALEPH] continuous,by Proposition 3.2, there exists an g[ALEPH] open set U in [S.sub.1], such that [A.sub.1] [??] NGcl(U) and NGcl(U) is g[ALEPH] compact and f (NGcl(U)) [??] V.

Consider the g[ALEPH] open set [S.sub.NGcl(U),V] x U in [mathematical expression not reproducible] is such that f [member of] [S.sub.NGcl(U),V] and [A.sub.1] [??] U. Let (g, [A.sub.2]) be such that g [member of] [S.sub.NGcl(U),V] and [A.sub.2] [??] U be arbitrary, thus g(NGcl(U)) [??] V. Since [A.sub.2] [??] U,we have g([A.sub.2]) [??] V and e(g, [A.sub.2]) = g([A.sub.2]) [??] V. Thus e([S.sub.NGcl(U),V] x U) [??] V. Hence e is g[ALEPH] continuous.

Proposition 4.2. Let [S.sub.1] and [S.sub.2] be two NTS with [S.sub.2] is g[ALEPH] compact. Let [A.sub.1] be any [ALEPH]--set in [S.sub.1] and N be a g[ALEPH] open set in the g[ALEPH] product space [S.sub.1] x [S.sub.2] containing {[A.sub.1]} x [S.sub.2]. Then there exists some g[ALEPH] open W with [A.sub.1] [??] W in [S.sub.1], such that {[A.sub.1]} x [S.sub.1] C W x [S.sub.2] [??] N.

Proof. It is clear that by Remark 3.1, {[A.sub.1]} x [S.sub.2] is g[ALEPH] homeomorphism to [S.sub.2] and hence by Remark 3.2, {[A.sub.1]} x [S.sub.2] is g[ALEPH] compact. We cover {[A.sub.1]} x [S.sub.1] by the basis elements {U x V}(for the g[ALEPH] product topology) lying in N.Since {[A.sub.1]} x [S.sub.2] is g[ALEPH] compact,{U x V} has a finite subcover, say a finite number of basis elements [U.sub.1] x [V.sub.1],...,[U.sub.n] x [V.sub.n]. Without loss of generality we assume that {[A.sub.1]} [??] [U.sub.i] for each i = 1, 2,..., n.Since otherwise the basis elements would be superfluous.

Let W = [[intersection].sup.n.sub.i=1] [U.sub.i]. Clearly W is g[ALEPH] open and [A.sub.1] [??] W. We show that W x [S.sub.2] [??] [[intersection].sup.n.sub.i=1]([U.sub.i] x [V.sub.i]). Let ([A.sub.1],B) be an [ALEPH]--set in W x [S.sub.2]. Now ([A.sub.1],B) [??] [U.sub.i] x [V.sub.i] for some i, thus B [??] [V.sub.i]. But [A.sub.1] [??] [U.sub.i] for every i = 1, 2,..., n(because [A.sub.1] [??] W). Therefore, ([A.sub.1], B) [??] [U.sub.i] x [V.sub.i] as desired. But [U.sub.i] x [V.sub.i] [??] N for all i = 1, 2,..., n and W x [S.sub.2] [??] [U.sup.n.i=1]([U.sub.i] x [V.sub.i]), therefore W x [S.sub.2] [??]N.

Proposition 4.3. Let [S.sub.3] be a g[ALEPH] locally compact Hausdorff space and [S.sub.1], [S.sub.2] be arbitrary NTS. Then a map f : [S.sub.3] x [S.sub.1] [right arrow] [S.sub.2] is g[ALEPH] continuous iff [mathematical expression not reproducible] is g[ALEPH] continuous,where f is defined by the rule (f([A.sub.1]))([A.sub.2]) = f ([A.sub.1],[A.sub.1]

Proof. Suppose that f is g[ALEPH] continuous. Consider the functions [mathematical expression not reproducible] where [i.sub.Z] denote the [ALEPH]--identity function on Z,t denote the [ALEPH]--switching map and e denote the g[ALEPH] evaluation map. Since [mathematical expression not reproducible] it follows that f = et ([i.sub.Z] x f) and f being the composition of g[ALEPH] continuous functions is itself g[ALEPH].

Conversely,suppose that f is g[ALEPH] continuous,let [A.sub.1] be any arbitrary [ALEPH]--set in [S.sub.1]. We have f([A.sub.1]) [member of] [mathematical expression not reproducible] Consider [mathematical expression not reproducible] is g[ALEPH] compact and [mathematical expression not reproducible] is g[ALEPH] open},containing

[??] ([A.sub.1]). We need to find a g[ALEPH] open W with [A.sub.1] [??] W,such that f ([A.sub.1]) [??] [S.sub.K,U]; this will suffice to prove f to be a g[ALEPH] continuous map.

For any [ALEPH]--set [A.sub.2] in K, we have (f([A.sub.1]))([A.sub.2]) = f ([A.sub.2], [A.sub.1]) [member of] U thus f (K x {[A.sub.1]}) [??] U, that is K x {[A.sub.1]} [??] [f.sup.-1](U). Since f is g[ALEPH] continuous,[f.sup.-1](U) is a g[ALEPH] open set in [S.sub.3] x [S.sub.1]. Thus [f.sup.-1](U) is a g[ALEPH] open set [S.sub.3] x [S.sub.1] containing K x {[A.sub.1]}. Hence by Proposition 4.2, there exists a g[ALEPH] open W with [A.sub.1] [??] W in [S.sub.1], such that K x {[A.sub.1]} [??] K x W [??] [f.sup.-i](U). Therefore f (K x W) [??] U. Now for any [A.sub.1] [??] W and [A.sub.2] [??] K,f ([A.sub.2], [A.sub.1]) = (f([A.sub.1])) ([A.sub.2]) [??] U. Therefore f([A.sub.1])(K) [??] U for all [A.sub.1] [??] W. That is f([A.sub.1]) [member of] [S.sub.k,u] for all [A.sub.1] [??] W.Hence f(W) [??] [S.sub.K,U] as desired.

Proposition 4.4. Let S and [S.sub.3] be two g[ALEPH] locally compact Hausdorff spaces. Then for any NTS [S.sub.2], the function [mathematical expression not reproducible] defined by E(f) = f(that is E(f)([A.sub.1]) ([A.sub.2]) = f ([A.sub.2], [A.sub.1]) = (f([A.sub.1])) ([A.sub.2])) for all f : [S.sub.3] x X [right arrow] [S.sub.2] is a g[ALEPH] homeomorphism.

Proof.

(a) Clearly E is onto.

(b) For E to be injective. Let E(f) = E(g) for f, g : [S.sub.3] x [S.sub.1] [right arrow] [S.sub.2]. Thus f = G,where f and f are the induced maps of f and g respectively. Now for any [ALEPH]--set A in S and any [ALEPH]--set [A.sub.2] in [S.sub.3], f ([A.sub.1], [A.sub.1]) = (f([A.sub.1]))([A.sub.2]) = (G([A.sub.1])) ([A.sub.2]) = g([A.sub.2], [A.sub.1]);thus f = g.

(c) For proving the g[ALEPH] continuity of E,consider any g[ALEPH] subbasis neighbourhood V of f in [mathematical expression not reproducible] that is V is of the form [S.K,W] where K is a g[ALEPH] compact subset of [S.sub.1] and W is g[ALEPH] open in [mathematical expression not reproducible] Without loss of generality we may assume that W = [S.sub.L,U], where L is a g[ALEPH] compact subset of [S.sub.1] and U is a g[ALEPH] open set in [S.sub.2]. Then f(K) [??] [S.sub.L,U] = W and this implies that f(K)(L) [??] U. Thus for any [ALEPH]--set [A.sub.1] [??] K and for all

[ALEPH]--sets [A.sub.2] [??] L. We have (f([A.sub.1])) ([A.sub.2]) [??] U, that is f ([A.sub.2], [A.sub.1]) [??] U and therefore f (L x K) [??] U. Now since L is g[ALEPH] compact in [S.sub.3] and K is g[ALEPH] compact in [S.sub.1],L x K is also g[ALEPH] compact in [S.sub.3] x [S.sub.1] [6] and since U is a g[ALEPH] open set in [S.sub.2], we conclude that [mathematical expression not reproducible] We assert that E([S.sub.LxK,U]) [??] [S.sub.K,W].Let g [member of] [S.sub.LxK,tU] be arbitrary. Thus g(L x K) [??] U, that is g([A.sub.2], [A.sub.1]) = (g([A.sub.1]))([A.sub.2]) [??] U for all [ALEPH]--sets [A.sub.2] [??] L in [S.sub.2] and for all [ALEPH]--sets [A.sub.1] [??] K in [S.sub.1].So (G([A.sub.1])) (L) [??] U for all [ALEPH]--sets [A.sub.1] [??] K in [S.sub.1] , that is G([A.sub.1]) [??] [S.sub.L,U] = W for all [ALEPH]--sets [A.sub.1] [??] K in U.Hence we have G(K) [??] W,that is G = E(g) [member of] [S.sub.K,W] for any g [member of] [S.sub.LxK,U]. Thus E([S.sub.LxK,U]) [??] [S.sub.K,W]. This proves that E is g[ALEPH] continuous.

(d) For proving the g[ALEPH] continuity of [E.sup.-1], we consider the following g[ALEPH] evaluation maps: [mathematical expression not reproducible] defined by [e.sub.1](f, [A.sub.1]) = f([A.sub.1]) where [mathematical expression not reproducible] and [A.sub.1] is an [ALEPH]--set in [S.sub.1] and [mathematical expression not reproducible] defined by [e.sub.2](g,[A.sub.2]) = g([A.sub.2]) where [mathematical expression not reproducible] and [A.sub.1] is a [ALEPH]--set in [S.sub.3]. Let [psi] denote the composition of the following g[ALEPH] continuous functions [mathematical expression not reproducible] where i, [i.sub.Z] denote the [ALEPH]--identity maps on [mathematical expression not reproducible] and [S.sub.3] respectively and T, t denote the [ALEPH]--switching maps. Thus [mathematical expression not reproducible] We consider the map [mathematical expression not reproducible] (as defined in the statement of the proposition in fact it is E). So E [mathematical expression not reproducible] Now for any [ALEPH]--sets [A.sub.2] in [S.sub.1],A in [S.sub.1] and [mathematical expression not reproducible] again to check that (E([psi]) o E)(f)([A.sub.2], [A.sub.1]) = f ([A.sub.2], [A.sub.1]); hence E([psi]) o E=identity. Similarly for any [mathematical expression not reproducible] and [ALEPH]--sets [A.sub.1] in [S.sub.1],[A.sub.2] in [S.sub.3], again to check that (E o E([psi]))(g)([A.sub.1], [A.sub.2]) = (g([A.sub.1]))([A.sub.2]);hence E o E([psi])=identity. Thus E is a g[ALEPH] homeomorphism.

Definition 4.3. The map E in Proposition 4.4 is called the g[ALEPH] exponential map.

As easy consequence of Proposition 4.4 is as follows.

Proposition 4.5. Let [S.sub.1], [S.sub.2] and [S.sub.3] be three g[ALEPH] locally compact Hausdorff spaces. Then the map [mathematical expression not reproducible] defined by N (f, g) = g o f is g[ALEPH] continuous.

Proof. Consider the following compositions: [mathematical expression not reproducible] where T,t denote the [ALEPH]--switching maps, [i.sub.X], i denote the [ALEPH]--identity functions on [S.sub.1] and [mathematical expression not reproducible] respectively and [e.sub.2] denote the g[ALEPH] evaluation maps. Let [phi] = [e.sub.2] o (i x [e.sub.2]) o (t x [i.sub.X]) o T. By proposition 4.4, we have an exponential map.[mathematical expression not reproducible] Since [mathematical expression not reproducible] Let N = E([phi]),that is N : [mathematical expression not reproducible] is an g[ALEPH] continuous. For [mathematical expression not reproducible] and for any [ALEPH] set [A.sub.1] in [S.sub.1] , it is easy to see that N(f, g) ([A.sub.1]) = g(f([A.sub.1])).

5 Conclusions

In this paper, we introduced the concept of g[ALEPH] compact open topology and Some characterization of this topology are discussed.

References

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Received: March 18, 2019. Accepted: June 23, 2019

(1*) R. Dhavaseelan, (2) R. Subash Moorthy and (3) S. Jafari

(1) Department of Mathematics, Sona College of Technology, Salem-636005,Tamil Nadu,India. E-mail: dhavaseelan.r@gmail.com

(2) Department of Mathematics, Amrita School of Engineering,Coimbatore-641112,Tamil Nadu,India. E-mail: subashvnp@gmail.com

(3) Department of Mathematics, College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, Denmark. E-mail: jafaripersia@gmail.com

(*) Correspondence: Author (dhavaseelan.r@gmail.com)
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