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Neutrosophic topology.

1 Introduction

The concept of neutrosophic sets was first introduced by Smarandache [13, 14] as a generalized of intuitionistic fuzzy sets [1] where besides the degree of of membership, the degree of indeterminacy and the degree of non-membership of each element in X. After the introduction of the neutrosophic sets, neutrosophic set operations were have been investigated. Many researchers have studied topology on neutrosophic sets, such as Smarandache [14] Lupianez [7-10] and Salama [12]. Various topologies have been defined on the neutrosophic sets. For some of them the De Morgan's Laws were not valid.

Thus, in this study we redefine the neutrosophic set operations and investigate some properties related to these definitions. Also, we introduce firstly neutrosophic interior, neutrosophic closure, neutrospohic exterior, neutrosophic boundary and neutrosophic subspace. In this paper we propose to define basic topological structures on neutrosophic sets such that interior, closure, exterior, boundary and subspace.

2 Preliminaries

In this section, we will recall the notions of neutrosophic sets [13]. Moreover, we will give a new approach to neutrosophic set operations.

Definition 1 [13] A neutrosophic set A on the universe of discourse X is defined as

A = {<x,[[mu].sub.A](x), [[sigma].sub.A](x),[[gamma].sub.A](x)>: x [member of] X}

where [[mu].sub.A], [[sigma].sub.A], [[gamma].sub.A] : X [left arrow] ][sup.-]0,1 + [ and [sup.-]0 [less than or equal to] [[mu].sub.A](x) + [[sigma].sub.A](x) + [[gamma].sub.A](x) [less than or equal to] [3.sup.+] From philosophical point of view, the neutrosophic set takes the value from real standard or non-standard subsets of ][sup.-]0,[1.sup.+][. But in real life application in scientific and engineering problems it is difficult to use neutrosophic set with value from real standard or non-standard subset of ][sup.-]0,[1.sup.+][. Hence we consider the neutrosophic set which takes the value from the subset of [0,1]. Set of all neutrosophic set over X is denoted by N (X).

Definition 2 Let A, B [member of] N(X). Then,

i. (Inclusion) If [[mu].sub.A](x) [less than or equal to] [[mu].sub.B](x), [[simga].sub.A](x) [greater than or equal to] [[sigma].sub.B](x) and [v.sub.A](x) [greater than or equal to] [v.sub.B](x) for all x [member of] X, then A is neutrosophic subset of B and denoted by A [??] B. (Or we can say that B is a neutrosophic super set of A.)

ii. (Equality) If A [??] B and B [??] A, then A = B.

iii. (Intersection) Neutrosophic intersection of A and B, denoted by A [??] B, and defined by

A [??] B = {<x,[[mu].sub.a](x) [conjunction] [[mu].sub.B](x), [[sigma].sub.A](x) [disjunction] [[sigma].sub.B](x), [v.sub.A](x) [disjunction] [v.sub.B](x>} : x [member of] X}.

iv. (Union) Neutrosophic union of A and B, denoted by A LI B, and defined by

A [??] B = {<x,[[mu].sub.a](x) [disjunction] [[mu].sub.B](x), [[sigma].sub.A](x) [conjunction] [[sigma].sub.B](x), [v.sub.A](x) [conjunction] [v.sub.B](x>} : x [member of] X}.

v. (Complement) Neutrosophic complement of A is denoted by [A.sup.c] and defined by

[A.sup.c] = {<x, [v.sub.A](x), 1 - [[sigma].sub.a](x),[[mu].sub.a](x>} : x [member of] X}.

vi. (Universal Set) If [[mu].sub.A](x) = 1, [[sigma]a.sub.A](x) = 0 and [v.sub.A](x) = 0 for all x [member of] X, A is said to be neutrosophic universal set, denoted by [??].

vii. (Empty Set) If [[mu].sub.A](x) = 0, [[sigma].sup.A](x) = 1 and [v.sub.A](x) = 1 for all x [member of] X, A is said to be neutrosophic empty set, denoted by [??].

Remark 3 According to Definition 2, [??] should contain complete knowledge. Hence, its indeterminacy degree and nonmembership degree are [??] and its membership degree is 1. Similarly, 0 should contain complete uncertainty. So, its indeterminacy degree and non-membership degree are 1 and its membership degree is 0.

Example 4 Let X = {x, y} and A, B, C [member of] N(X) such that

A = {<x,0.1, 0.4,0.3), (y,0.5, 0.7,0.6>}

B = {<x,0.9, 0.2,0.3), (y,0.6, 0.4,0.5>}

C = {<x,0.5, 0.1,0.4), (y,0.4, 0.3,0.8>}.

Then,

i. We have that A [member of] B.

ii. Neurosophic union of B and C is

B [??] C = {<x, (0.9 [disjunction] 0.5), (0.2 [conjunction] 0.1), (0.3 [conjunction] 0.4)>, (y, (0.6 [disjunction] 0.4), (0.4 [conjunction] 0.3), (0.5 [conjunction] 0.8)>} = {<x, 0.9, 0.1,0.3), (y, 0.6, 0.3,0.5>}.

iii. Neurosophic intersection of A and C is

A [??] C = {<x, (0.1 [conjunction] 0.5), (0.4 [disjunction] 0.1), (0.3 [disjunction] 0.4)>, <y, (0.5 [conjunction] 0.4), (0.7 [disjunction] 0.3), (0.6 [disjunction] 0.8)>} = {<x,0.1, 0.4,0.3,>, <y, 0.5,0.7, 0.6>}.

iv. Neutrosophic complement of C is

[C.sup.c] = [{<x, 0.5,0.1, 0.4), (y, 0.4,0.3, 0.8>}.sup.c] = {<x, 0.4,1 - 0.1,0.5), (y, 0.8,1 - 0.3, 0.4>} = {<x, 0.4,0.9, 0.5), (y, 0.8,0.7, 0.4>}.

Theorem 5 Let A, B [member of] N(X). Then, followings hold.

i. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ii. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

iii. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

iv. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

v. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

vi. [([A.sup.c]).sup.c] = A

Proof. It is clear.

Theorem 6 Let A,B [member of] N(X). Then, De Morgan's law is valid.

i. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ii. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof.

i. From Definition 2 v

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ii. It can proved by similar way to i.

Theorem 7 Let B [member of] N(X) and {[A.sub.i] : i [member of] I} [subset or equal to] N(X). Then,

i. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ii. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. It can be proved easily from Definition 2.

3 Neutrosophic topological spaces

In this section, we will introduce neutrosophic topological space and give their some properties.

Definition 8 Let [tau] [subset or equal to] N(X), then [tau] is called a neutrosophic topology on X if

i. [??] and [??] belong to [tau],

ii. The union of any number of neutrosophic sets in [tau] belongs to [tau],

iii. The intersection of any two neutrosophic sets in [tau] belongs to [tau].

The pair (X, [tau]) is called a neutrosophic topological space over X. Moreover, the members of [tau] are said to be neutrosophic open sets in X. If [A.sup.c] [member of] [tau], then A [member of] N(X) is said to be neutrosophic closed set in X

Theorem 9 Let (X, [tau]) be a neutrosophic topological space over X. Then

i. [??] and [??] are neutrosophic closed sets over X.

ii. The intersection of any number of neutrosophic closed sets is a neutrosophic closed set over X.

iii. The union of any two neutrosophic closed sets is a neutrosophic closed set over X.

Proof. Proof is clear.

Example 10 Let t = {[??], [??]} and [sigma] = N(X). Then, (X, [tau]) and (X, [sigma]) are two neutrosophic topological spaces over X. Moreover, they are called neutrosophic discrete topological space and neutrosophic indiscrete topological space over X, respectively.

Example 11 Let X = {a,b} and A [member of] N(X) such that A = {<a, 0.2,0.4, 0.6), (b, 0.1,0.3, 0.5>}.

Then, t = {[??], [??], A} is a neutrosophic topology on X.

Theorem 12 Let (X, [[tau].sub.1]) and (X, [[tau].sub.2]) be two neutrosophic topological spaces over X, then (X, [[tau].sub.1] [intersection] [[tau].sub.2]) is a neutrosophic topological space over X.

Proof. Let (X, [[tau].sub.i]) and (X, [[tau].sub.2]) be two neutrosophic topological spaces over X. It can be seen clearly that 0, X [member of] [[tau].sub.1] [intersection] [[tau].sub.2]. If A, B [member of] [[tau].sub.1] [intersection] [[tau].sub.2] then, A, B [member of] [[tau].sub.1] and A, B [member of] [[tau].sub.2]. It is given that A [intersection] B [member of] [[tau].sub.1] and A [intersection] B [member of] [[tau].sub.2]. Thus, A [intersection] B [member of] [[tau].sub.1] [intersection] [[tau].sub.2]. Let {[A.sub.i] : i [member of] I} C[[tau].sub.1][intersection] [[tau].sub.2]. Then, A* [member of][[tau].sub.1][intersection] [[tau].sub.2] for all i [member of] I. Thus, [A.sub.i] [member of] [[tau].sub.1] and [A.sub.i] [member of] [[tau].sub.2] for all i [member of] I. So, we have [[??].sub.i [member of] I] [A.sub.i] [member of] [[tau].sub.1] [intersection] [[tau].sub.2].

Corollary 13 Let {<X, [t.sub.i]) : i [member of] I} be a family of neutrosophic topological spaces over X. Then, (X, [[intersection].sub.i [member of] I] [t.sub.i]) is a neutrosophic topological space over X.

Proof. It can proved similar way Theorem 12.

Remark 14 If we get the union operation instead of the intersection operation in Theorem 12, the claim may not be correct. This situation can be seen following example.

Example 15 Let X = {a, b} and A, B [member of] N(X) such that

A = {<a, 0.2,0.4, 0.6), (b, 0.1,0.3, 0.5>}

B = {<a, 0.4,0.6, 0.8), (b, 0.3,0.5, 0.7>}.

Then, [[tau].sub.1] = {[??], [??], A} and [[tau].sub.2] = {[??], [??], B} are two neutrosophic topology on X. But, [[tau].sub.1] [union] [[tau].sub.2] = {[??], [??], A, B} is not neutrosophic topology on X. Because, A [??] B [not member of] [[tau].sub.1] [union] [[tau].sub.2]. So, [[tau].sub.1] [union] [[tau].sub.2] is not neutrosophic topological space over X.

Definition 16 Let (X, [tau]) be a neutrosophic topological space over X and A [member of] N(X). Then, the neutrosophic interior of A, denoted by int(A) is the union of all neutrosophic open subsets of A. Clearly int(A) is the biggest neutrosophic open set over X which containing A.

Theorem 17 Let (X, [tau]) be a neutrosophic topological space over X and A, B [member of] N(X). Then

i. int([??]) = 0 and int([??]) = [??].

ii. int(A) [??] A.

iii. A is a neutrosophic open set if and only if A = int(A).

iv. int(int(A)) = int(A).

v. A [??] B implies int(A) int(B).

vi. int(A) [??] int(B) [??] int(A [??] B).

vii. int(A [??] B) = int(A) [??] int(B).

Proof. i. and ii. are obvious.

iii. If A is a neutrosophic open set over X, then A is itself a neutrosophic open set over X which contains A. So, A is the largest neutrosophic open set contained in A and int(A) = A. Conversely, suppose that int(A) = A. Then, A [member of] t.

iv. Let int(A) = B. Then, int(B) = B from iii. and then, int(int(A)) = int(A).

v. Suppose that A [??] B. As int(A) [??] A [??] B. int(A) is a neutrosophic open subset of B, so from Definition 16, we have that int(A) [??] int(B).

vi. It is clear that A [??] A [??] B and B [??] A [??] B. Thus, int(A) [??] int(A [??] B) and int(B) [??] int(A [??] B). So, we have that int(A) [??] int(B) [??] int(A [??] B) by v.

vii. It is known that int(A [??] B) [??] int(A) and int(A [??] B) [??] int(B) by v so that int(A [??] B) [??] int(A) [??] int(B). Also, from int(A) [??] A and int(B) [??] B, we have int(A) [??] int(B) [??] A [??] B. These imply that int(A [??] B) = int(A) [??] int(B).

Example 18 Let X = {a, b} and A, B, C [member of] N(X) such that

A = {<a, 0.5, 0.5,0.5), (b, 0.3, 0.3,0.3>}

B = {<a, 0.4, 0.4,0.4), (b, 0.6, 0.6,0.6>}

C = {<a, 0.7, 0.7,0.7), (b, 0.2, 0.2,0.2>}.

Then, t = {[??], [??], A} is a neutrosophic soft topological space over X. Therefore, int(B) = {0,X, int(C) = {0,X and int(B [??] C) = A. So, int(B) [??] int(C) [not equal to] int(B [??] C).

Definition 19 Let (X, [tau]) be a neutrosophic topological space over X and A [member of] N(X). Then, the neutrosophic closure of A, denoted by cl(A) is the intersection of all neutrosophic closed super sets of A. Clearly cl(A) is the smallest neutrosophic closed set over X which contains A.

Example 20 In the Example 10, according to the neutrosophic topological space (X, [sigma]), neutrosophic interior and neutrosophic closure of each element of N(X) is equal to itself.

Theorem 21 Let (X, [tau]) be a neutrosophic topological space over X and A,B [member of] N(X). Then

i. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

ii. A [??] cl(A).

iii. A is a neutrosophic closed set if and only if A = cl(A).

iv. cl(cl(A)) = cl(A).

v. A [??] B implies cl(A) [??] cl(B).

vi. cl(A [??] B) = cl(A) [??] cl(B).

vii. cl(A [??] B) [??] cl(A) [??] cl(B).

Proof. i. and ii. are clear. Moreover, proofs of vi. and vii. are similar to Theorem 17 vi. and vii..

iii. If A is a neutrosophic closed set over X then A is itself a neutrosophic closed set over X which contains A. Therefore, A is the smallest neutrosophic closed set containing A and A = cl(A). Conversely, suppose that A = cl(A). As A is a neutrosophic closed set, so A is a neutrosophic closed set over X.

iv. A is a neutrosophic closed set so by iii., then we have A = cl(A).

v. Suppose that A [??] B. Then every neutrosophic closed super set of B will also contain A. This means that every neutrosophic closed super set of B is also a neutrosophic closed super set of A. Hence the neutrosophic intersection of neutrosophic closed super sets of A is contained in the neutrosophic intersection of neutrosophic closed super sets of B. Thus cl(A) [??] cl(B).

Example 22 Lei X = {a, b} and A,B [member of] N(X) such that

A = {<a, 0.5, 0.5,0.5), (b, 0.4,0.4,0.4>}

B = {<a, 0.6, 0.6,0.6), (b, 0.3,0.3,0.3>}.

Then,

[tau] = {[??], [??], A, B, A [??] B, A [??] B}

is a neutrosophic topology on X. Moreover, set of neutrosophic closed sets over X is

{[??], [??], [A.sup.c], [B.sup.c], [(A [??] B).sup.c], [(A [??] B).sup.c]}.

Therefore

[A.sup.c] = {<a, 0.5,0.5, 0.5), (b, 0.4,0.6, 0.4>}

[B.sup.c] = {<a, 0.6,0.4, 0.6), (b, 0.3,0.7, 0.3>}

[(A [??] B).sup.c] = {<a, 0.6,0.4, 0.5), (b, 0.4,0.6, 0.4>}

[(A [??] B).sup.c] = {<a, 0.5,0.5, 0.6), (b, 0.3,0.7, 0.4>}.

Thus, we have that

A [??] B = {<a, 0.5, 0.5,0.6), (b, 0.3, 0.7,0.4>}

cl(A) = [??]

cl(B) = [??]

cl(A [??] B) = [(A [??] B).sup.c]

cl(A [??] B) [??] cl(A) [??] cl(B).

Remark 23 Example 18 and Example 22 show that there is not equality in Theorem 17 vi. and Theorem 21 vii.

Theorem 24 Let (X, [tau]) be a neutrosophic topological space over X and A, B [member of] N(X). Then

i. int([A.sup.c]) = [(cl(A)).sup.c],

ii. cl([A.sup.c]) = [(int(A)).sup.c].

Proof. Let A, B [member of] N(X). Then,

i. It is known that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Right hand of above equality is int([A.sup.c]), thus int([A.sup.c]) = [(cl(A)).sup.c].

ii. If it is taken [A.sup.c] instead of A in i., then it can be seen clearly that [(cl([A.sup.c])).sup.c] = int[(([A.sup.c]).sup.c]) = int(A). So, cl([A.sup.c]) = [(int(A)).sup.c].

Definition 25 Let (X, [tau]) be a neutrosophic topological space over X then the neutrosophic exterior of a neutrosophic set A over X is denotedby ext(A) and is defined as ext(A) = int([A.sup.c]).

Theorem 26 Let (X, [tau]) be a neutrosophic topological space over X and A, B [member of] N(X). Then

i. ext(A [??] B) = ext(A) [??] ext(B)

ii. ext(A) [??] ext(B) [??] ext(A [??] B)

Proof. Let A, B [member of] N(X). Then,

i. By Definition 25, Theorem 6 and Theorem 17 vii.

ext(A [??] B) = int([(A [??] B).sup.c])

= int([A.sup.c] [??] [B.sup.c])

= int([A.sup.c]) [??] int([B.sup.c])

= ext(A) [??] ext(B)

ii. It is similar to i.

Definition 27 Let (X, [tau]) be a neutrosophic topological space over X and A [member of] N(X). Then, the neutrosophic boundary of a neutrosophic set A over X is denoted by fr(A) and is defined as fr(A) = cl(A) [??] cl([A.sup.c]). It must be noted that fr(A) = fr([A.sup.c]).

Example 28 Let consider the neutrosophic sets A and B in the Example 22. According to the neutrosophic topology in Example 11 we have fr(A) = [??] and fr(C) = [(A [??] B).sup.c].

Theorem 29 Let (X, [tau]) be a neutrosophic topological space over X and A,B [member of] N(X). Then

i. [(fr(A)).sup.c] = ext(A) [??] int(A).

ii. cl(A) = int(A) [??] fr(A).

Proof. Let A, B [member of] N(X). Then,

i. By Theorem 24 i., we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ii. By Theorem 24 i., we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 30 Let (X, [tau]) be a neutrosophic topological space over X and A [member of] N(X). Then

i. A is a neutrosophic open set over X if and only if A [??] fr(A) = [??].

ii. A is a neutrosophic closed set over X if and only if fr(A) [??] A.

Proof. Let A [member of] N(X). Then

i. Assume that A is a neutrosophic open set over X. Thus int(A) = A. By Theorem 24, fr(A) = cl(A) [??] fr([A.sup.c]) = cl(A) [??] [(int(A)).sup.c]. So,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Conversely, let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which implies [A.sup.c] is a neutrosophic set and so A is a neutrosophic open set.

ii. Let A be a neutrosophic closed set. Then, cl(A) = A. By Definition 27, fr(A) = cl(A) [??] fr([A.sup.c]) [??] cl(A) = A. Therefore, fr(A) [??] A. Conversely, fr(A) [??] A. Then fr(A) [??] [A.sup.c] = [??]. From fr([A.sup.c]) = fr([A.sup.c]), fr([A.sup.c]) [??] [A.sup.c] = 0. By i., [A.sup.c] is a neutrosophic open set and so A is a neutrosophic closed set.

Theorem 31 Let (X, [tau]) be a neutrosophic topological space over X and A [member of] N(X). Then

i. fr(A) [??] int(A) = [??]

ii. fr(int(A)) [??] fr(A)

Proof. Let A [member of] N(X). Then,

i. From Theorem 30 i., it is clear.

ii. By Theorem 24 ii.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 32 Let (X, [tau]) be a neutrosophic topological space and Y be a non-empty subset of X. Then, a neutrosophic relative topology on Y is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], otherwise.

Thus, (Y, [[tau].sub.Y]) is called a neutrosophic subspace of (X, [tau]).

Example 33 Let X = {a, b, c}, Y = ja, b} [subset or equal to] X and A, B [member of] N(X) such that

A = {<a, 0.4,0.2, 0.2), (b, 0.5,0.4, 0.6), (c, 0.2,0.5, 0.7>}

B = {<a, 0.4,0.5, 0.3), (b, 0.5,0.6, 0.5), (c, 0.3,0.7, 0.8>}.

Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a neutrosophic topology on X. Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a neutrosophic relative topology on Y such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4 Conclusion

In this work, we have redefined the neutrosophic set operations in accordance with to define neutrosophic topological structures. Then, we have presented some properties of these operations. We have also investigated neutrosophic topological structures of neutrosophic sets. Hence, we hope that the findings in this paper will help researcher enhance and promote the further study on neutrosophic topology.

References

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[6] D. (Joker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, 88(1) (1997), 81-89.

[7] F. G. Lupianez, On neutrosophic topology, The International Journal of Systems and Cybernetics, 37(6) (2008), 797-800.

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[9] F. G. Lupianez, On various neutrosophic topologies, The International Journal of Systems and Cybernetics, 38(6) (2009), 1009-1013.

[10] F. G. Lupianez, On neutrosophic paraconsistent topology, The International Journal of Systems and Cybernetics, 39(4) (2010), 598-601.

[11] P. K. Maji, Neutrosophic soft set, Annals of Fuzzy mathematics and Informatics, 5(1) (2013), 157-168.

[12] A. Salama and S. AL-Blowi, Generalized neutrosophic set and generalized neutrosophic topological spaces, Computer Science and Engineering, 2(7) (2012), 129-132.

[13] F. Smarandache, Neutrosophic set--a generalization of the intuitionistic fuzzy set, International Journal of Pure and Applied Mathematics, 24(3) (2005) 287-297.

[14] F. Smarandache, Neutrosophy and neutrosophic logic, first international conference on neutrosophy, neutrosophic logic, set, probability, and statistics, University of New Mexico, Gallup, NM 87301, USA(2002).

Received: November 10, 2016. Accepted: December 20, 2016

Serkan Karatag (1) and Cemil Kuru (2)

(1) Department of Mathematics, Faculty of Arts and Sciences, Ordu University, 52200 Ordu, Turkey, posbiyikliadam@gmail.com

(2) Department of Mathematics, Faculty of Arts and Sciences, Ordu University, 52200 Ordu, Turkey, cemilkuru@outlook.com
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Date:Oct 1, 2016
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