Fuzzy Neutrosophic Alpha m-Closed Sets in Fuzzy NeutrosophicTopological Spaces

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Neutrosophic Sets and Systems, Vol. 21, 2018

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University of New Mexico

Fuzzy Neutrosophic Alpham-Closed Sets in Fuzzy NeutrosophicTopological Spaces Fatimah M. Mohammed1,* and Shaymaa F. Matar2 1

Department of Mathematics, College of Education for Pure Sciences, Tikrit University, Tikrit, IRAQ *Corresponding author, Email address: nafea_y2011@yahoo.com

2

Department of Mathematics, College of Education for Pure Sciences, Tikrit University, Tikrit, IRAQ E-mail: shaimaa.1986@yahoo.com

Abstract. In this paper, we state a new class of sets and called them fuzzy neutrosophic Alpham-closed sets, and we prove some theorem related to this definition. Then, we investigate the relation between fuzzy neutrosophic Alpham-closed sets, fuzzy neutrosophic Îą closed sets, fuzzy neutrosophic closed sets, fuzzy neutrosophic semi closed sets and fuzzy neutrosophic pre closed sets. On the other hand, some properties of the fuzzy neutrosophic Alpham-closed set are given. Keywords: fuzzy neutrosophic closed sets, fuzzy neutrosophic Alpham-closed sets, fuzzy neutrosophic topology.

1. Introduction: The concept of fuzzy sets was introduced by Zadeh in 1965 [14]. Then the fuzzy set theory are extension by many researchers. The concept of intutionistic fuzzy sets (IFS) was one of the extension sets by K. Atanassov in 1983 [2, 3, 4], when fuzzy set give the digree of membership of an element in the sets, the intuitionistic fuzzy sets give a degree of membership and a degree of non-membership. Then, several researches were conducted on the generalizations of the notion of intuitionistic fuzzy sets, one of them was Floretin Smarandache in 2010 [7] when he developed another membership in addition to the two memberships which was defined in intuitionistic fuzzy sets and called it neutrosophic set. The concept of neutrosophic sets was defined with membership, non-membership and indeterminacy degrees. In the last year, (2017) Veereswari [13] introduced fuzzy neutrosophic topological spaces. This concept is the solution and representation of the problems with various fields. Neutrosophic topological spaces and many applications have been investigated by Salama et al. in [ 912] In this paper, the concept of Alpham-closed sets in double fuzzy topological spaces [6] were developed. We discussed some

new class of sets and called them fuzzy neutrosophic Alpham-closed sets in fuzzy

neutrosophic topological spaces, and we also discussed some new properties and examples based of this defined concept. 2. Basic definitions and terminologies Definition 2.1 [8]: A neutrosophic topology (đ?‘ đ?‘‡, for short) on a non-empty set đ?‘‹ is a family đ?œ? of neutrosophic subsets of đ?‘‹ satisfying the following axioms: i) ∅đ?‘ , đ?‘‹đ?‘ ∈ đ?œ?. m

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ii) đ??´1â‹‚đ??´2∈ đ?œ? for any đ??´ 1 and đ??´ 2∈ đ?œ?. iii) ⋃đ??´ j∈ đ?œ? for any {đ??´j : j ∈ J}⊆ đ?œ?. In this case the pair (đ?‘‹, đ?œ?) is called a neutrosophic topological space (đ?‘ đ?‘‡đ?‘†, for short) in đ?‘‹. The elements in đ?œ? are called neutrosophic open sets (đ?‘ -open sets for short) in đ?‘‹. A đ?‘ -set is said to be neutrosophic closed set (đ?‘ closed set, for short) if and only if its complement is a đ?‘ -open set. Definition 2.2 [1, 13]: Let X be a non-empty fixed set. A fuzzy neutrosophic set ( FNS, for short), ÎťN is an object having the form ÎťN  ď ť< x, ď ­ÎťN (x), ď łÎťN (x), đ?œˆ ÎťN (x) >: xďƒŽ Xď ˝ where the functions ď ­ÎťN, ď łÎťN, đ?œˆÎťN : X

[0,

1] denote the degree of membership function (namely ď ­ÎťNx), the degree of indeterminacy function (namely ď łÎťN (x )) and the degree of non-membership function (namely đ?œˆÎťN x) respectively, of each set ÎťN we have, 0 ≤ ď ­ÎťN(x) + ď łÎť(x) + đ?œˆÎťN (x) ≤ 3, for each xďƒŽ X. Remark 2.3 [13]: FNS ÎťN = {< x, đ?œ‡ ÎťN (x), đ?œŽ ÎťN (x), đ?œˆ ÎťN (x) >: x ∈ X} can be identified to an ordered triple <x, đ?œ‡ ÎťN,

đ?œŽ ÎťN, đ?œˆ ÎťN > in [0, 1] on X.

Definition 2.4[13]: Let X be a non-empty set and the FNSs ÎťN and βN be in the form: ÎťN = {< x, đ?œ‡ÎťN (x), đ?œŽÎťN (x), đ?œˆÎťN (x) >: x ∈ X} and, βN ={< x, đ?œ‡βN (x), đ?œŽβN (x), đ?œˆβN (x) >: x ∈X} on X then: i.

ÎťN ⊆ βN iff đ?œ‡ÎťN (x) ≤ đ?œ‡ βN (x), đ?œŽ ÎťN (x) ≤ đ?œŽ βN (x) and đ?œˆ ÎťN (x) ≼ đ?œˆ βN (x) for all x ∈ X,

ii. ÎťN = βN iff ÎťN ⊆ βN and βN ⊆ ÎťN, iii. 1N-ÎťN = {<x, đ?œˆÎťN (x), 1 − đ?œŽÎťN (x), đ?œ‡ÎťN (x) >: x ∈X} iv. ÎťN âˆŞ βN = {< x, Max(đ?œ‡ÎťN (x), đ?œ‡βN (x)), Max(đ?œŽÎťN(x), đ?œŽβN (x)), Min(đ?œˆÎťN (x), đ?œˆβN (x))

>: x ∈ X},

v. ÎťN ∊ βN = {< x, Min( đ?œ‡ÎťN (x), đ?œ‡βN (x)), Min(đ?œŽÎťN (x), vi.

đ?œŽβN

(x)),

Max(đ?œˆÎťN

(x),

đ?œˆ

βN

(x))

>:

x

∈

X},

0đ?‘ = < x, 0, 0, 1> and 1đ?‘ = <x, 1, 1, 0>.

Definition 2.5 [13]: A Fuzzy neutrosophic topology (FNT, for short) on a non-empty set X is a family đ?œ?N of fuzzy neutrosophic subsets in X satisfying the following axioms. i.

0đ?‘ , 1đ?‘ ∈ đ?œ?N,

ii.

ÎťN1 ∊ ÎťN2 ∈ đ?œ?N for any ÎťN1, ÎťN2 ∈ đ?œ?N,

iii.

âˆŞ ÎťNj ∈ đ?œ?N, ∀{ÎťNj: j ∈ J} ⊆ đ?œ?N. In this case the pair (X, đ?œ?N) is called fuzzy neutrosophic topological space (FNTS, for short). The

elements of đ?œ? are called fuzzy neutrosophic open sets (FN-open set, for short). The complement of FN-open sets in the FNTS (X, đ?œ?N) are called fuzzy neutrosophic closed sets (FN-closed set, for short). Definition 2.6 [13]: Let (X, đ?œ?N) be FNTS and Îť

đ?‘

= <x, đ?œ‡Îť đ?‘ , đ?œŽÎť đ?‘ , đ?œˆÎťđ?‘ > be FNS in X. Then, the fuzzy

neutrosophic closure of Îťđ?‘ (FNCl, for short) and fuzzy neutrosophic interior of Îťđ?‘ (FNInt, for short) are defined by: m

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FNCl(Îťđ?‘ ) = ∊ { βN: βN is FN-closed set in X and Îťđ?‘ ⊆ βN }, FNInt (Îťđ?‘ ) = âˆŞ { βN: βN is FN-open set in X and βN ⊆ Îťđ?‘ }. Note that FNCl(Îťđ?‘ ) be FN-closed set and FNInt (Îť đ?‘ ) be FN-open set in X. Further, i. ÎťN be FN-closed set in X iff FNCl (Îťđ?‘ ) = Îťđ?‘ , ii. ÎťN be FN-open set in X iff FNInt (Îťđ?‘ ) = Îťđ?‘ . Proposition 2.7 [13]: Let (X, đ?œ?N) be FNTS and ÎťN, βN are FNSs in X. Then, the following properties hold: i. FNInt(ÎťN) ⊆ ÎťN and ÎťN ⊆ FNCl(ÎťN), ii. ÎťN ⊆ βN â&#x;š FNInt (ÎťN) ⊆ FNInt (βN) and ÎťN ⊆ βN â&#x;š FNCl(ÎťN) ⊆ FNCl(βN), iii. Int(FNInt(ÎťN)) = FNInt(ÎťN) and FNCl(FNCl(ÎťN)) = FNCl(ÎťN), iv. FNInt (ÎťN ∊ βN) = FNInt(ÎťN) ∊ FNInt(βN) and FNCl(ÎťN âˆŞ βN) = FNCl(ÎťN) âˆŞ FNCl(βN), v. FNInt(1đ?‘ ) =1đ?‘ and FNCl(1đ?‘ ) = 1N, vi. FNInt(0đ?‘ ) = 0đ?‘ and FNCl(0đ?‘ ) = 0N. Definition 2.8 [2]: FNS ÎťN in FNTS (X, đ?œ?N) is called: i. fuzzy neutrosophic semi-open set (FNS-open, for short) if ÎťN ⊆ FNCl(FNInt(ÎťN), ii. fuzzy neutrosophic semi-closed set (FNS-closed, for short) if FNInt(FNcl(ÎťN)) ⊆ ÎťN, iii. fuzzy neutrosophic pre-open set (FNP-open, for short) if ÎťN ⊆ FNInt (FNCl(ÎťN)). iv. fuzzy neutrosophic pre-closed set (FNP-closed, for short) if FNCl(FNInt(ÎťN)) ⊆ ÎťN, v. fuzzy neutrosophic Îą-open set (FNÎą-open, for short ) if ÎťN ⊆ FNInt(FNCl(FNInt(ÎťN))), vi. fuzzy neutrosophic Îą-closed set (FNÎą-closed, for short) if FNCl(FNInt(FNCl(ÎťN))) ⊆ ÎťN. m

3. Fuzzy Neutrosophic Alpha - Closed Sets in Fuzzy Neutrosophic Topological Spaces. Now, the concept of fuzzy neutrosophic Alpham-closed set in fuzzy neutrosophic topological space is introduced, as follows: Definition 3.1: Fuzzy neutrosophic subset ÎťN of FNTS (X, đ?œ?N) is called fuzzy neutrosophic Alpham-closed set (FNÎąm- closed set, for short ) if FNint(FNcl (ÎťN)) ⊆ UN, wherever ÎťN ⊆ UN and UN be FNÎą-open set. And ÎťN is said to be fuzzy neutrosophic Alpham-open set (FNÎąm-open set, for short) in (X, đ?œ?N) if the complement 1N-ÎťN be FNÎąm-closed set in (X, đ?œ?N). Proposition 3.2: For any FNS, the following statements satisfy: i. Every FN-open set is FNÎą-open set. ii. Every FNÎą-closed set is FNÎąm-closed set. iii. Every FN-closed set is FNÎąm-closed set. iv. Every FNS-closed set is FNÎąm-closed set. Proof:

m

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i. Let Îťđ?‘ ={<x, ď ­ÎťN (x), đ?œŽÎťN (x), đ?œˆÎťN (x) >: x

X} be FN-open set in FNTS (X, đ?œ?N).Then, by Definition 2.6 (ii)

we get, ÎťN = FNInt(ÎťN)‌‌(1) And, by Proposition 2.7 (i) we get, ÎťN ⊆ FNCl(ÎťN). But, ÎťN ⊆ FNCl(FNInt(ÎťN)). Then, FNInt(ÎťN) ⊆ FNInt(FNCl(FNInt(ÎťN))). Therefore, by (1) we get, ÎťN ⊆ FNInt(FNCl(FNInt(ÎťN))). Hence, ÎťN be FNÎą-open set in (X, đ?œ?N). ii. Let ÎťN ={<x, ď ­ÎťN(x), đ?œŽÎťN(x), đ?œˆÎťN(x) >: x

X } be FNÎą-closed set in FNTS (X, đ?œ?N).

Then, FNCl(FNInt(FNCl(ÎťN))) ⊆ ÎťN. Now, let βN be FN-open set such that, ÎťN⊆ βN. Since, βN be FN-open set then, is FNÎą-open set by (i).Then, FNInt(FNCl(ÎťN)) ⊆ FNInt(ÎťN) ⊆ ÎťN ⊆ βN. Therefore, FNInt(FNCl(ÎťN)) ⊆ βN. Hence, ÎťN be FNÎąm- closed set in (X, đ?œ?N). iii. Let ÎťN = {< x, ď ­ÎťN(x), đ?œŽÎťN(x), đ?œˆÎťN(x) >: x

X } is FN-closed set in

FNTS (X, đ?œ?N).Then, by Definition 2.6 (i). We get, ÎťN = FNcl(ÎťN).... (1). And, by Proposition 2.7 (i). We get, FNint(ÎťN) ⊆ ÎťN‌‌(2). But, FNint(FNcl(ÎťN)) ⊆ FNcl(ÎťN).Then, by (1). We get, FNint(FNcl(ÎťN) ⊆ ÎťN. Now, let βN is FN-open set such that, ÎťN ⊆ UN. By, Proposition 3.2 (i). If, βN is FNopen set. Then, is FNÎą-open set. Then, FNint(FNcl(ÎťN) ⊆ ÎťN⊆βN. Therefore, FNint(FNcl(ÎťN)) ⊆ βN. Hence, ÎťN is FNÎąm-closed set in (X, đ?œ?N). iv. Let ÎťN ={<x, ď ­ÎťN(x), đ?œŽÎťN(x), đ?œˆÎťN(x) >:x

X } be FNS-closed set in FNTS (X, đ?œ?N).

Then, FNInt(FNCl(ÎťN)) ⊆ΝN. Now, let βN be FN-open set such that, ÎťN⊆ βN Since, βN be FN-open set then, is FNÎą-open set, by (i) Then, FNInt(FNCl (ÎťN)) ⊆ ÎťN ⊆ βN. Therefore, FNInt(FNCl (ÎťN)) ⊆ βN. Hence, ÎťN be FNÎąm- closed set in (X, đ?œ?N). Remark 3.3: The converse of Proposition 3.2 is not true in general and we can show it by the following examples: Example 3.4: i. Let X={x} define FNSs ÎťN and βN in X as follows: ÎťN ={<x, 0.7, 0.6, 0.5>: x X }, βN ={<x, 0.8, 0.9, 0.4>: x

X }.

And the family đ?œ?N = {0N, 1N, ÎťN, βN } be FNT such that, 1N-đ?œ?N ={1N, 0N, <x, 0.5, 0.4, 0.7>, <x, 0.4, 0.1, 0.8>}. Now if ,

m

Fatimah M. Mohammed and Shaymaa F. Matar, Fuzzy Neutrosophic Alpha -Closed Sets in Fuzzy Neutrosophic Topological Spaces

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N

Neutrosophic Sets and Systems, Vol. 21, 2018

= {<x, 0.8, 0.6, 0.5 >: x X }.

Then, FNInt( Therefore, Hence,

N) N

N

={<x, 0.7, 0.6, 0.5>: x

⊆ FNInt(FNCl(FNInt(

X }, FNCl(FNInt(

N))

= 1N,and FNInt(FNCl(FNInt(

N)))

=1N.

N))).

be FNÎą-open set. But, not FN-open set.

ii. Let X={x} define FNSs ÎťN, βN, ĆžN and ΨN in X as follows: ÎťN = {<x, 1, 0.5, 0.7>: x

X}, βN = {<x, 0, 0.9, 0.2>: x

X },

ĆžN = {<x, 1, 0.9, 0.2>: x

X } and ΨN = {<x, 0, 0.5, 0.7>: x

X}

And the family đ?œ?N ={0N, 1N, ÎťN, βN, ĆžN, ΨN } be FNT.such that, 1N-đ?œ?N ={1N, 0N, <x, 0.7, 0.5, 1 >, <x, 0.2, 0.1, 0 >, <x, 0.2, 0.1,1 >, <x, 0.7, 0.5, 0 >} Now if,

N

= {<x, 0, 0.4, 0.8>: x

X } and UN = {<x, 0, 0.5, 0.7>: x

Where, UN be FN-open set such that,

N

X}.

⊆ UN.

Since,UN be FN-open set then, is FNÎą-open set by Proposition 3.2 (i). Then, FNCl

N)

={<x, 0.7, 0.5, 0>: x

Therefore, FNInt(FNCl( Hence,

N))

X }and FNInt(FNCl(

N))

={<x, 0, 0.5, 0.7>: x X }.

⊆ UN..

m

be FNÎą - closed set.

N

But, FNCl(

N)

FNInt(FNCl(

={<x, 0.7, 0.5, 0>: x

N))

X },

={<x, 0, 0.5, 0.7>: x

FNCl(FNInt(FNCl(

N)))

X } and

={<x, 0.7, 0.5, 0>: x

Therefore, FNCl(FNInt(FNCl(

N)))

N.

X}

Hence,

N

be not FNÎą- closed set.

iii. Take, the example which defined in ii. Then, we can see Take again, the example which defined in ii. Then,

N be

N be

FNÎąm-closed set. But, not FN-closed set.

FNÎąm-closed set. But, not FNS-closed set.

Remark 3.5: The relation between FNP-closed sets and FNιm- closed sets are independent and we can show it by the following examples. Example 3.6: (1) Let X={x} define FNSs ΝN and βN in X as follows: ΝN = {<x, 0.1, 0.2, 0.4>: x

X }, βN = {<x, 0.7, 0.5, 0.2>: x

X },

And, the family đ?œ?N ={0N, 1N, ÎťN, βN } be FNT such that, 1N-đ?œ?N ={1N, 0N, <x, 0.4, 0.8, 0.1>, <x, 0.2, 0.5, 0.7>}. Now if,

N

= {<x, 0.1, 0.3, 0.4>: x

UN = {<x, 0.7, 0.5, 0.2>: x

X }and

X }where, UN be FN-open set such that,

N

⊆ U N.

Since, UN be FN-open set then, is FNÎą-open set by Proposition 3.2 (i) Then, FNCl (

N)

FNInt(FNCl(

N))={<x,

Hence,

N

0.1, 0.2, 0.4>: x X }.Therefore, FNInt(FNCl(

N))

⊆ U N.

be FNÎą - closed set. N)

FNCl(FNInt ( N

X }and

m

But, FNInt( Hence,

={<x, 0.4, 0.8, 0.1>: x

={<x, 0.1, 0.2, 0.4>: x N))

X },

= {<x, 0.4, 0.8, 0.1>: x X }.Therefore, FNCl(FNInt (

N))

N.

be not FNP-closed set.

(2) Let X={a, b} define FNS ÎťN in X as follows: ÎťN = <x, (a\0.5, b\0.5) , (a\0.5, b\0.5) , (a\0.4, b\0.5) >. And the family đ?œ?N ={0N, 1N, ÎťN } be FNT. m

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Neutrosophic Sets and Systems, Vol. 21, 2018

Such that, 1N-đ?œ?N = {1N, 0N, <x, , (a\0.4, b\0.5), (a\0.5, b\0.5) , (a\0.5, b\0.5) >}. Now if,

N

= <x, (a\0.5, b\0.4) , (a\0.5, b\0.5) , (a\0.6, b\0.5) > And, UN = ÎťN be FN-open set such that,

N

U N.

Since, UN be FN-open set then, is FNÎą-open set by Proposition 3.2 (i) Then, FNInt (

N)

= 0N and FNCl(FNInt(

Therefore, FNCl(FNInt ( But, FNCl (

N)

N))

N.

N))

= 0N.

Hence,

= 1N and FNInt(FNCl(

Therefore, FNInt(FNCl(

N))

N))

N

be FNP-closed set.

= 1 N.

UN. Hence,

N

be not FNÎąm- closed set.

Proposition 3.7: If ÎťN be FNÎąm- closed set and ÎťN ⊆ ĆžN ⊆ FNInt(FNCl(ÎťN)), Then, ĆžN be FNÎąm- closed set. Proof: Let ÎťN ={<x, ď ­ÎťN (x), đ?œŽÎťN (x), đ?œˆÎťN (x) >: x

X} be FNιm- closed set such that,ΝN ⊆ ƞN ⊆ FNInt(FNCl(ΝN)).

Now let βN be FNÎą- open set such that, ĆžN⊆ βN. Since, ÎťN be FNÎąm-closed set then, we have FNInt(FNCl(ÎťN)) ⊆ βN, where ÎťN ⊆ βN. Since, ÎťN ⊆ ĆžN and ĆžN ⊆ FNInt(FNCl(ÎťN)) we get, FNInt(FNCl(ĆžN)) ⊆ FNInt(FNCl(FNInt(FNCl(ÎťN)))) ⊆ FNInt(FNCl(ÎťN)) ⊆ βN. Therefore, FNInt(FNCl(ĆžN)) ⊆ βN. Hence, ĆžN be FNÎąm- closed set in (X, đ?œ?N). Proposition 3.8: Let (X, đ?œ?N) be FNTS. So, the intersection of two FNÎąm-closed sets be FNÎąm-closed set. Proof: Let ÎťN and βN are FNS-closed sets on FNTS (X, đ?œ?N) Then, FNInt(FNCl(ÎťN)) ⊆ ÎťN‌‌(1) And, FNInt(FNCl(βN)) ⊆ βN‌‌(2) Consider ÎťN

βN

FNInt(FNCl(ÎťN))

FNInt(FNCl(βN))

FNInt(FNCl(ÎťN)

FNCl(βN))

FNInt(FNCl(ÎťN Therefore, FNInt(FNCl(ÎťN

βN)).

βN)) ⊆ ÎťN

βN

Now, let ĆžN be FN-open set such that, ÎťN

βN ⊆ ĆžN.

Since, ĆžN be FN-open set then it is FNÎą-open set, by Proposition 3.2 (i). Then, FNInt(FNCl(ÎťN

βN)) ⊆ ÎťN

Therefore, FNInt(FNCl(ÎťN

βN ⊆ ĆžN.

βN)) ⊆ ĆžN. Hence, ÎťN

βN be FNÎąm-closed set in (X, đ?œ?N).

Remark 3.9: The union of any FNÎąm-closed sets is not necessary to be FNÎąm-closed set and we can show it by the following example. Example 3.10: Take, Example 3.4 (ii) if, N1

={<x, 0.4,0.5, 1>: x X } and

Then, FNCl(

N1)

={<x, 0.7, 0.5,1>: x

Therefore, FNInt(FNCl( Hence,

N1 be

N2

N1)

={<x, 0.2, 0, 0.8>: x X } and FNInt(FNCl(

X}. And UN ={<x, 1,0.5, 0.7>: x X } N1)

= 0N.

⊆ UN.

m

FNÎą - closed set. m

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Neutrosophic Sets and Systems, Vol. 21, 2018

And, FNCl(

N2)

={<x, 0.2, 0.1,0>: x

Therefore, FNInt(FNCl( Therefore,

N1

N2

N2))

X}, FNInt(FNCl(

⊆ UN. Hence,

N2 be

N2))

= 0N.

m

FNÎą - closed set.

be not FNÎąm- closed set.

Definition 3.11: Let (X, đ?œ?N) be FNTS and Îťđ?‘ = <x, ď ­ÎťN (x), đ?œŽÎťN (x), đ?œˆÎťN (x)> be FNS in X. Then, the fuzzy neutrosophic Alpham closure of Îťđ?‘ (FNÎąmCl, for short) and fuzzy neutrosophic Alpham interior of Îťđ?‘ (FNÎąmInt, for short) are defined by: i. FNÎąmCl(Îť đ?‘ ) = ∊{βđ?‘ : βđ?‘ is FNÎąm-closed set in X and Îťđ?‘ ⊆ βđ?‘ }, ii. FNÎąmInt(Îť đ?‘ ) = âˆŞ{βđ?‘ : βđ?‘ is FNÎąm-open set in X and βđ?‘ ⊆ Îťđ?‘ }. Proposition 3.12: Let (X, đ?œ?N) be FNTS and ÎťN, βN are FNSs in X. Then the following properties hold: i.

FNÎąmCl(0đ?‘ ) = 0N and FNÎąmCl (1đ?‘ ) =1đ?‘ ,

ii. ÎťN ⊆ FNÎąmCl(ÎťN), iii. If ÎťN ⊆ βN, then FNÎąmCl(ÎťN) ⊆ FNÎąmCl(βN), iv. ÎťN be FNÎąm-closed set iff Îťđ?‘ = FNÎąmCl (Îťđ?‘ ), v. FNÎąmCl(ÎťN) = FNÎąmCl(FNÎąmCl(ÎťN)). Proof: i.

by Definition 3.11 (i) we get,

FNÎąmCl(0đ?‘ ) = ∊ {βđ?‘ : βđ?‘ is FNÎąm-closed set in X and 0đ?‘ ⊆ βđ?‘ } = 0đ?‘ . And, FNÎąmCl(1đ?‘ ) = ∊ {βđ?‘ : βđ?‘ is FNÎąm-closed set in X and 1đ?‘ ⊆ βđ?‘ } =1đ?‘ . ii. ÎťN ⊆ ∊ {βđ?‘ : βđ?‘ is FNÎąm-closed set in X and ÎťN ⊆ βđ?‘ } = FNÎąmCl (ÎťN). iii.

Suppose that ÎťN ⊆ βđ?‘ then, ∊{βđ?‘ : βđ?‘ is FNÎąm-closed set in X and ÎťN ⊆ βđ?‘ } ⊆ ∊{ ĆžN: ĆžN is FNÎąm-closed set in X and βđ?‘ ⊆ ĆžN }. Therefore, FNÎąmCl(ÎťN) ⊆ FNÎąmCl(βN).

iv.

If, ÎťN be FNÎąm-closed set, then FNÎąmCl (ÎťN) = ∊{βđ?‘ : βđ?‘ is FNÎąm-closed set in X and ÎťN ⊆ βđ?‘ }‌‌(1) And, by (ii) we get, ÎťN ⊆ FNÎąmCl (ÎťN)‌‌(2) but, ÎťN is necessarily to be the smallest set. Thus, ÎťN = ∊{βđ?‘ : βđ?‘ is FNÎąm-closed set in X and ÎťN ⊆ βđ?‘ }, Therefore, Îť đ?‘ = FNÎąmCl (Îťđ?‘ ).

Conversely; Let ÎťN = FNÎąmCl (ÎťN) by using Definition 3.11 (i),we get, Îťđ?‘ be FNÎąm-closed set. v.

Since, by (iv) we get, Îťđ?‘ = FNÎąmCl (Îťđ?‘ ) Then, FNÎąmCl (Îťđ?‘ ) = FNÎąmCl(FNÎąmCl(Îťđ?‘ )).

Proposition 3.13: Let (X, đ?œ?N) be FNTS and ÎťN, βN are FNSs in X. Then the following properties hold: i.

FNÎąmInt(0đ?‘ ) = 0N and FNÎąmInt (1đ?‘ ) =1đ?‘ ,

ii.

FNιmInt(ΝN) ⊆ ΝN,

iii.

If ÎťN ⊆ βN, then FNÎąmInt(ÎťN) ⊆ FNÎąmInt(βN),

iv.

ÎťN be FNÎąm-open set iff Îťđ?‘ = FNÎąmInt (Îťđ?‘ ), m

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FNÎąmInt (ÎťN) = FNÎąmInt (FNÎąmInt (ÎťN)).

v. Proof:

i. by Definition 3.11 (ii) we get, FNÎąmInt(0đ?‘ ) = âˆŞ{βđ?‘ : βđ?‘ is FNÎąm-open set in X and βđ?‘ ⊆ 0đ?‘ } = 0đ?‘ , And, FNÎąmInt(1đ?‘ ) = âˆŞ{βđ?‘ : βđ?‘ is FNÎąm-open set in X and βđ?‘ ⊆ 1đ?‘ } =1đ?‘ . ii. Follows from Definition 3.11 (ii). iii. FNÎąmInt(Îť đ?‘ ) = âˆŞ{βđ?‘ : βđ?‘ is FNÎąm-open set in X and βđ?‘ ⊆ Îťđ?‘ }.Since, Îťđ?‘ ⊆ βđ?‘ then, âˆŞ{βđ?‘ : βđ?‘ is FNÎąm-open set in X and βđ?‘ ⊆ Îťđ?‘ }âŠ†âˆŞ{ ĆžN: ĆžN is FNÎąm-open set in X and ĆžN ⊆ βđ?‘ } Therefore, FNÎąmInt(ÎťN) ⊆ FNÎąmInt(βN). We must proof that, FNÎąmInt(Îťđ?‘ ) ⊆ Îťđ?‘ and Îťđ?‘ ⊆ FNÎąmInt(Îťđ?‘ ).

iv.

Suppose that Îťđ?‘ be FNÎąm-open set in X. Then, FNÎąmInt(Îťđ?‘ ) = âˆŞ{βđ?‘ : βđ?‘ is FNÎąm-open set in X and βđ?‘ ⊆ Îťđ?‘ }. by using (ii) we get, FNÎąmInt(ÎťN) ⊆ ÎťN‌.(1) Now to proof, ÎťN ⊆ FNÎąmInt(ÎťN), we have, For all ÎťN ⊆ ÎťN, the FNÎąmInt(ÎťN) ⊆ ÎťN So, we get ÎťN ⊆ âˆŞ{βđ?‘ : βđ?‘ is FNÎąm-open set in X and βđ?‘ ⊆ Îťđ?‘ }= FNÎąmInt(ÎťN) ‌‌(2) From (1) and (2) we have, Îť đ?‘ = FNÎąmInt (Îťđ?‘ ). Conversely; assume that Îť đ?‘ = FNÎąmInt (Îťđ?‘ ) and by using Definition 3.11 (ii) we get, Îťđ?‘ be FNÎąm-open set in X. By (iv) we get, Îťđ?‘ = FNÎąmInt(Îťđ?‘ )

v.

Then, FNÎąmInt(Îťđ?‘ ) = FNÎąmInt(FNÎąmInt(Îťđ?‘ )). Proposition 3.14: Let (X, đ?œ?N) be FNTS. Then, for any fuzzy neutrosophic subsets ÎťN of X. i.

1N- (FNÎąmInt(ÎťN)) = FNÎąmCl(1N-ÎťN),

ii.

1N- (FNÎąmCl(ÎťN)) = FNÎąmInt(1N-ÎťN).

i.

FNÎąmInt(Îť đ?‘ ) = âˆŞ{βđ?‘ : βđ?‘ is FNÎąm-open set in X and βđ?‘ ⊆ Îťđ?‘ }, by the complement we get,

Proof: 1N-(FNÎąmInt(Îť đ?‘ )) = 1N- (âˆŞ{βđ?‘ : βđ?‘ is FNÎąm-open set in X and βđ?‘ ⊆ Îťđ?‘ }). So, 1N- (FNÎąmInt(Îť đ?‘ )) = ∊{ (1N-βđ?‘ ): (1N-βđ?‘ ) is FNÎąm-closed set in X and (1N-Îťđ?‘ ) ⊆ (1N-βđ?‘ ) }. Now, replacing (1N-βđ?‘ ) by ĆžN we get, 1N- (FNÎąmInt(Îť đ?‘ )) = ∊{ ĆžN: ĆžN is FNÎąm-closed set in X and (1N-Îťđ?‘ ) ⊆ ĆžN} = FNÎąmCl(1N-ÎťN). ii.

FNÎąmCl(Îť đ?‘ ) = ∊{βđ?‘ : βđ?‘ is FNÎąm-closed set in X and Îťđ?‘ ⊆ βđ?‘ }, by the complement we get, 1N-(FNÎąmCl(Îť đ?‘ )) = 1N- (∊{βđ?‘ : βđ?‘ is FNÎąm-closed set in X and Îťđ?‘ ⊆ βđ?‘ }). So, 1N- (FNÎąmCl(Îť đ?‘ )) = âˆŞ{(1N-βđ?‘ ): (1N-βđ?‘ ) is FNÎąm-open set in X and (1N-βđ?‘ ) ⊆(1N- Îťđ?‘ )}. Again replacing (1N-βđ?‘ ) by ĆžN we get, 1N- (FNÎąmCl(Îť đ?‘ )) = âˆŞ{ ĆžN: ĆžN is FNÎąm-open set in X and ĆžN ⊆(1N-Îťđ?‘ )}= FNÎąmint(1N-ÎťN).

Proposition 3.15: Fuzzy neutrosophic interior of FN-closed set be FNÎąm-closed set. m

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proof:

Let ÎťN ={<x, ď ­ÎťN(x), đ?œŽÎťN(x), đ?œˆÎťN(x) >:x X} be FN-closed set in FNTS (X, đ?œ?N).Then, by Definition 2.6

(i) we get, ΝN = FNCl(ΝN). So, FNInt(ΝN) = FNInt(FNCl(ΝN))‌‌(1), And, by Proposition 2.7 (i) we get, FNInt(ΝN)

ΝN ‌‌(2)

From (1) and (2) we get, FNInt(FNCl(ÎťN))

ÎťN

Now, let βN be FN-open set such that, ÎťN⊆ βN. Since, βN be FN-open set, then βN is FNÎą-open set by Proposition 3.2 (i) Therefore, FNInt(FNCl(ÎťN)) ⊆ βN. Hence, ÎťN be FNÎąm-closed set in (X, đ?œ?N). Remark 3.16: The relationship between different sets in FNTS can be showing in the next diagram and the converse is not true in general. FNÎą-closed set

FN-closed set

FNS-closed set

FNÎąm-closed set

FNP-closed set Diagram 1

Conclusion In this paper, the new concept of a new class of sets and called them fuzzy neutrosophic Alpham-closed sets. we investigated the relation between fuzzy neutrosophic Alpham-closed sets, fuzzy neutrosophic Îą closed sets, fuzzy neutrosophic closed sets, fuzzy neutrosophic semi closed sets and fuzzy neutrosophic pre closed sets with some properties.

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Fatimah M. Mohammed and Shaymaa F. Matar, Fuzzy Neutrosophic Alpha -Closed Sets in Fuzzy Neutrosophic Topological Spaces

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