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University of New Mexico
Neutrosophic Crisp Îą-Topological Spaces 1,2,4
A.A.Salama1, I.M.Hanafy2,Hewayda Elghawalby3 and M.S.Dabash4
Department of Mathematics and Computer Science, Faculty of Sciences, Port Said University, Egypt. E-mail: Drsalama44@gmail.com, ihanafy@hotmil.com, majdedabash@yahoo.com
Faculty of Engineering, port-said University, Egypt. E-mail: hewayda2011@eng.psu.edu.eg
3
Abstract. In this paper, a generalization of the neutrosophic topological space is presented. The basic definitions of the neutrosophic crisp Îą-topological space and the neutrosophic crisp Îą-compact space with some of their
characterizations are deduced. Furthermore, we aim to construct a netrosophic crisp Îą-continuous function, with a study of a number its properties.
Keywords: Neutrosophic Crisp Set, Neutrosophic Crisp Topological space, Neutrosophic Crisp Open Set. 1 Introduction In 1965, Zadeh introduced the degree of membership5 and defined the concept of fuzzy set [15]. A degree of nonmembership was added by Atanassov [2], to give another dimension for Zadah's fuzzy set. Afterwards in late 1990's, Smarandache introduced a new degree of indeterminacy or neutrality as an independent third component to define the neutrosophic set as a triple structure [14]. Since then, laid the foundation for a whole family of new mathematical theories to generalize both the crisp and the fuzzy counterparts [4-10]. In this paper, we generalize the neutrosophic topological space to the concept of neutrosophic crisp Îą-topological space. Moreover, we present the netrosophic crisp Îą-continuous function as well as a study of several properties and some characterization of the neutrosophic crisp Îą-compact space. 2 Terminologies We recollect some relevant basic preliminaries, and in particular, the work of Smarandache in [12,13,14], and Salama et al. [4, 5,6,7,8,9,10,11]. Smarandache introduced the neutrosophic components T, I, F which respectively represent the membership, indeterminacy, and nonmembership characteristic mappings of the space X into
ď ?
-

ď ›
the non-standard unit interval 0 ,1 . Hanafy and Salama et al. [3,10] considered some possible definitions for basic concepts of the neutrosophic crisp set and its operations. We now improve some results by the following.
2.1 Neutrosophic Crisp Sets 2.1.1 Definition For any non-empty fixed set X, a neutrosophic crisp set A (NCS for short), is an object having the form A  A1, A2 , A3 where A1 , A2 and A3 are subsets of X satisfying A1 ďƒ‡ A2  ď Ś , A1 ďƒ‡ A3  ď Ś and A2 ďƒ‡ A3  ď Ś . 2.1.2 Remark Every crisp set A formed by three disjoint subsets of a non-empty set X is obviously a NCS having the form A  A1 , A2 , A3 . Several relations and operations between NCSs were defined in [11]. For the purpose of constructing the tools for developing neutrosophic crisp sets, different types of NCSs ď ŚN , X N , Ac
in X were introduced in [9] to be as follows: 2.1.3 Definition ď Ś N may be defined in many ways as a NCS, as follows: i) ď Ś N  ď Ś , ď Ś , X , or ii) ď Ś N  ď Ś , X , X , or iii) ď Ś N  ď Ś , X , ď Ś , or iv) ď Ś N  ď Ś , ď Ś , ď Ś . 2.1.4 Definition X N may also be defined in many ways as a NCS: i) X N  X , ď Ś , ď Ś , or ii) X N  X , X , ď Ś , or iii) X N  X , ď Ś , X , or iv) X N  X , X , X .
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2.1.5 Definition Let A  A1 , A2 , A3 a NCS on X , then the complement

of the set A , ( A c for short ferent ways:
C 1 
A  A , A2 , A3 c 1
c
c
c
may be defined in three dif,
C 2  Ac  A3 , A2 , A1 C 3  A c  A3 , A2 c , A1 Several relations and operations between NCSs were introduced in [9] as follows: 2.1.6 Definition Let X be a non-empty set, and the NCSs A and B in the form A  A1, A2 , A3 , B  B1, B2 , B3 , then we may consider two possible definitions for subsets  A ďƒ? B 
 A ďƒ? B  may be defined in two ways: 1) A ďƒ? B ďƒ› A1 ďƒ? B1 , A2 ďƒ? B2 and A3 ďƒŠ B3 or 2) A ďƒ? B ďƒ› A1 ďƒ? B1 , A2 ďƒŠ B2 and A3 ďƒŠ B3 2.1.7 Proposition For any neutrosophic crisp set A , and the suitable choice of ď Ś N , X N , the following are hold: i) ď Ś N ďƒ? A, ď Ś N ďƒ? ď Ś N . ii) A ďƒ? X N , X N ďƒ? X N . 2.1.8 Definition Let X is a non-empty set, and the NCSs A and B in the
form A  A1, A2 , A3 , B  B1, B2 , B3 . Then: 1) A B may be defined in two ways: i) A ďƒ‡ B  A1 ďƒ‡ B1 , A2 ďƒ‡ B 2 , A3 ďƒˆ B3 or ii) A ďƒ‡ B  A1 ďƒ‡ B1 , A2 ďƒˆ B 2 , A3 ďƒˆ B3
2) A B may also be defined in two ways: i) A ďƒˆ B  A1 ďƒˆ B1 , A2 ďƒ‡ B2 , A3 ďƒ‡ B3 or ii) A ďƒˆ B  A1 ďƒˆ B1 , A2 ďƒˆ B2 , A3 ďƒ‡ B3 2.1.9 Proposition For any two neutrosophic crisp sets A and B on X, then the followings are true: 1)  A ďƒ‡ B c  A c ďƒˆ B c . 2)  A ďƒˆ B c  A c ďƒ‡ B c . The generalization of the operations of intersection and union given in definition 2.1.8, to arbitrary family of neutrosophic crisp subsets are as follows: 2.1.10 Proposition Let A j : j ďƒŽ J be arbitrary family of neutrosophic crisp
ď ť
ď ˝
subsets in X, then 1) ďƒ‡ Aj may be defined as the following types : j
i) ďƒ‡ A j  ďƒ‡ A j1 ,ďƒ‡ A j ,ďƒˆ A j ,or 2 3 j
ii)
ďƒ‡ Aj  ďƒ‡ Aj1 ,ďƒˆ Aj 2 ,ďƒˆ Aj3 . j
2)
ďƒˆ Aj may be defined as the following types : j
i) ďƒˆ Aj  ďƒˆ Aj1 ,ďƒˆ Aj ,ďƒ‡ Aj or 2 3 j
ii)
ďƒˆ Aj  ďƒˆ Aj1 ,ďƒ‡ Aj 2 ,ďƒ‡ Aj3 . j
2.1.11 Definition The Cartesian product of two neutrosophic crisp sets A and B is a neutrosophic crisp set A  B given by A  B  A1  B1 , A2  B 2 , A3  B3 . 2.1.12 Definition
Let (đ?‘‹,đ?›¤) be đ?‘ đ??śđ?‘‡đ?‘† and đ??´=⌊đ??´1,đ??´2,đ??´3âŒŞ be a đ?‘ đ??śđ?‘† in đ?‘‹ . Then the neutrosophic crisp closure of đ??´ (đ?‘ đ??śđ?‘?đ?‘™(đ??´) for short) and neutrosophic crisp interior (đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´) for short) of đ??´ are defined by đ?‘ đ??śđ?‘?đ?‘™(đ??´)=∊{đ??ž:đ??ž is a đ?‘ đ??śđ??śđ?‘† in đ?‘‹ and đ??´ ⊆ đ??ž} đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą (đ??´)=âˆŞ{G:G is a đ?‘ đ??śđ?‘‚đ?‘† in đ?‘‹ and G ⊆ đ??´) , Where đ?‘ đ??śđ?‘† is a neutrosophic crisp set and đ?‘ đ??śđ?‘‚đ?‘† is a neutrosophic crisp open set. It can be also shown that đ?‘ đ??śđ?‘?đ?‘™(đ??´) is a đ?‘ đ??śđ??śđ?‘† (neutrosophic crisp closed set) and đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´) is a đ?‘ đ??śđ?‘‚đ?‘† (neutrosophic crisp open set) in đ?‘‹ .
3 Neutrosophic Crisp đ?›ź-Topological Spaces We introduce and study the concepts of neutrosophic crisp đ?›ź-topological space 3.1 Definition Let (đ?‘‹,đ?›¤) be a neutrosophic crisp topological space (NCTS) and đ??´ =⌊đ??´1,đ??´2,đ??´3âŒŞ be a đ?‘ đ??śđ?‘† in đ?‘‹, then đ??´ is said to be neutrosophic crisp đ?›ź-open set of X if and only if the following is true: đ??´âŠ† đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)). 3.2 Definition A neutrosophic crisp đ?›ź-topology (NCđ?›źT for short) on a non-empty set X is a family ď ‡ď Ą of neutrosophic crisp subsets of X satisfying the following axioms i) ď Ś N , X N ďƒŽ ď ‡ď Ą . ii) A1 ďƒ‡ A2 ďƒŽ ď ‡ď Ą for any A1 and A2 ďƒŽ ď ‡ď Ą .
ď ť
ď ˝
iii) ďƒˆ A j ďƒŽ ď ‡ď Ą  A j : j ďƒŽ J ďƒ? ď ‡ď Ą . j
In this case the pair X , ď ‡ď Ą  is called a neutrosophic crisp đ?›ź-topological space ( NCď ĄTS for short) in X . The eleď Ą
ments in ď ‡ are called neutrosophic crisp đ?›ź-open sets (NCđ?›źOSs for short) in X . A neutrosophic crisp set F is đ?›źclosed if and only if its complement F C is an đ?›ź-open neutrosophic crisp set. 3.3 Remark Neutrosophic crisp đ?›ź-topological spaces are very natural generalizations of neutrosophic crisp topological spaces, as one can prof that every open set in a NCTS is an đ?›ź-open set in a NCđ?›źTS 3.4 Example Let X  ď ťa, b, c, d ď ˝ , ď Ś N , X N be any types of the universal and empty sets on X, and A, B are two neutrosophic crisp sets on X defined by A  ď ťaď ˝, ď ťb, d ď ˝, ď ťcď ˝ ,
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B  ď ťaď ˝, ď ťbď ˝, ď ťcď ˝ , ď ‡  ď ťď ŚN , X N , Aď ˝ then the family ď ‡ď Ą  ď ťď Ś N , X N , A, Bď ˝ is a neutrosophic crisp đ?›ź-topology
on X. 3.5 Definition Let X , ď ‡1ď Ą , X , ď ‡2ď Ą  be two neutrosophic crisp đ?›ź-
topological spaces on X . Then ď ‡1ď Ą is said be contained in ď ‡2ď Ą (in symbols ď ‡1ď Ą ďƒ? ď ‡2ď Ą ) if G ďƒŽ ď ‡2ď Ą for each G ďƒŽ ď ‡1ď Ą . In this case, we also say that ď ‡1ď Ą is coarser than ď ‡2ď Ą . 3.6 Proposition Let ď ť ď ‡ď Ąj : j ďƒŽ J ď ˝ be a family of NCđ?›źTS on X . Then
ďƒ‡ ď ‡ ď Ąj is a neutrosophic crisp đ?›ź-topology on X . Further-
more, ďƒˆ ď ‡ď Ąj is the coarsest NCđ?›źT on X containing all đ?›ź-topologies. Proof Obvious. Now, we can define the neutrosophic crisp đ?›ź-closure and neutrosophic crisp đ?›ź-interior operations on neutrosophic crisp đ?›ź-topological spaces: 3.7 Definition
Let X , ď ‡ď Ą  be NCđ?›źTS and A  A1, A2 , A3 be a NCS in X, then the neutrosophic crisp đ?›ź- closure of A (NCđ?›źCl(A) for short) and neutrosophic crisp đ?›ź-interior crisp (NCđ?›źInt (A) for short) of A are defined by
ď ť
ď ˝
{ Aj } jďƒŽJ   Aj1 , Aj2 , Aj3 : j ďƒŽ J . Then we see
that we have two types defined as follows:
Type1: NCď ĄInt( A)  ďƒˆ Aj ,ďƒˆ Aj ,ďƒ‡ Aj  1
2
3
( NCď ĄInt ( A)) c  ďƒ‡ Acj1 ,ďƒ‡ Acj2 ,ďƒˆ Acj3 
Hence NCď ĄCl ( Ac )  ( NCď ĄInt ( A)) c Type 2: NCď ĄInt ( A)  ďƒˆ Aj ,ďƒ‡ Aj ,ďƒ‡ Aj  ( NCď ĄInt ( A)) c  ďƒ‡ Acj ,ďƒˆ Acj ,ďƒˆ Acj3  . 1
1
2
3
2
Hence NCď ĄCl ( Ac )  ( NCď ĄInt ( A)) c b) Similar to the proof of part (a). 3.9 Proposition Let X , ď ‡ď Ą  be a NCđ?›źTS and A , B are two neutrosophic crisp đ?›ź-open sets in X . Then the following properties hold: (a) NCď ĄInt( A) ďƒ? A, (b) A ďƒ? NCď ĄCl ( A), (c) A ďƒ? B ďƒž NCď ĄInt ( A) ďƒ? NCď ĄInt ( B), (d) A ďƒ? B ďƒž NCď ĄCl ( A) ďƒ? NCď ĄCl ( B), (e) NCď ĄInt ( A ďƒ‡ B)  NCď ĄInt( A) ďƒ‡ NCď ĄInt( B), (f) NCď ĄCl( A ďƒˆ B)  NCď ĄCl( A) ďƒˆ NCď ĄCl( B), (g) NCď ĄInt ( X N )  X N , (h) NCď ĄCl (ď ŚN )  ď ŚN . Proof. Obvious
NC ď ĄCl ( A)  ďƒ‡ď ťK : K is an NCS in X and A ďƒ? Kď ˝ 4 Neutrosophic Crisp đ?›ź-Continuity
NC ď ĄInt ( A)  ďƒˆď ťG : G is an NCOS in X and G ďƒ? Aď ˝
where NCS is a neutrosophic crisp set, and NCOS is a neutrosophic crisp open set. It can be also shown that NCď ĄCl ( A) is a NCđ?›źCS (neutrosophic crisp đ?›ź-closed set) and NCď ĄInt ( A) is a NCđ?›źOS (neutrosophic crisp đ?›ź-open set) in X . a) A
is a NCđ?›ź-closed in X A  NCď ĄCl ( A) .
b)
A
is a NC�-open
A  NCď ĄInt( A) .
in
X
if and only if if and only if
3.8 Proposition
For any neutrosophic crisp đ?›ź-open set A in X , ď ‡ď Ą  we have (a) NCď ĄCl ( Ac )  ( NCď ĄInt ( A)) c , (b) NCď ĄInt ( Ac )  ( NCď ĄCl ( A))c . Proof a) Let A  A1, A2 , A3 and suppose that the family of all neutrosophic crisp subsets contained in A are indexed by the family
In this section, we consider đ?‘“:đ?‘‹â&#x;śđ?‘Œ to be a map between any two fixed sets X and Y. 4.1 Definition (a) If A  A1, A2 , A3 is a NCS in X, then the neutrosophic crisp image of A under f , denoted by f ( A), is the a NCS in Y defined by f ( A)  f ( A1 ), f ( A2 ), f ( A3 ) .
(b) If đ?‘“ is a bijective map then đ?‘“-1: đ?‘Œ â&#x;ś đ?‘‹ is a map defined such that: for any NCS B  B1 , B2 , B3 in Y, the neutrosophic crisp preimage of B, denoted by f ď€1 ( B), is a NCS in X defined by f ď€1 ( B)  f ď€1 ( B1 ), f ď€1 ( B2 ), f ď€1 ( B3 ) . Here we introduce the properties of neutrosophic images and neutrosophic crisp preimages, some of which we shall frequently use in the following sections. 4.2 Corollary Let A = ď ťAi : i ďƒŽ J ď ˝ , be NCđ?›źOSs in X, and B = ď ťB j : j ďƒŽ K ď ˝ be NCđ?›źOSs in Y, and f : X ď‚Ž Y a function. Then (a) A1 ďƒ? A2 ďƒ› f ( A1) ďƒ? f ( A2 ), B1 ďƒ? B2 ďƒ› f ď€1( B1) ďƒ? f ď€1( B2 ),
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(b) A ďƒ? f ď€1( f ( A)) and if f is injective, then A  f ď€1( f ( A) ) . (c) f ď€1( f ( B)) ďƒ? B and if f is surjective, then f ď€1 ( f ( B) )  B . (d) f ď€1 (ďƒˆBi ) )  ďƒˆ f ď€1 ( Bi ), f ď€1 (ďƒ‡Bi )) ďƒ? ďƒ‡ f ď€1 ( Bi ), (e) f (ďƒˆ Ai )  ďƒˆ f ( Ai ) , f (ďƒ‡ Ai ) ďƒ? ďƒ‡ f ( Ai ) ďƒ— f ď€1 (YN )  X N , f ď€1 (ď Ś N )  ď Ś N .
(f)
(g) f (ď ŚN )  ď ŚN , f ( X N )  YN , if f is subjective. Proof Obvious. 4.3 Definition Let X , ď ‡1ď Ą  and Y , ď ‡2ď Ą  be two NCđ?›źTSs, and let f : X ď‚Ž Y be a function. Then f is said to be
�-continuous iff the neutrosophic crisp preimage of each
NCS in ď ‡2ď Ą is a NCS in ď ‡1ď Ą . 4.4 Definition Let X , ď ‡1ď Ą  and Y , ď ‡2ď Ą  be two NCđ?›źTSs and let f : X ď‚Ž Y be a function. Then f is said to be open iff
the neutrosophic crisp image of each NCS in ď ‡1ď Ą is a NCS in ď ‡2ď Ą . 4.5 Proposition Let X , ď ‡oď Ą  and Y ,ď š oď Ą be two NCđ?›źTSs.


If f : X ď‚Ž Y is đ?›ź-continuous in the usual sense, then in this case, f is đ?›ź-continuous in the sense of Definition 4.3 too. Proof Here we consider the NCđ?›źTs on X and Y, respectively, as follows : ď ‡1ď Ą  G, ď Ś , G c : G ďƒŽ ď ‡oď Ą and
ď ť H ,ď Ś, H
ď ť
ď ˝
ď ˝
for each CNC B in Y. 4.8 Corollary Consider ( X , ď ‡1ď Ą ) and Y , ď ‡2ď Ą to be two NCđ?›źTSs, and let f : X ď‚Ž Y be a function. if ď ‡1ď Ą  ď ťf ď€1 ( H ) : H ďƒŽ ď ‡2ď Ą ď ˝ . Then ď ‡1ď Ą will be the coarsest NCđ?›źT on X which makes the function f : X ď‚Ž Y đ?›ź-continuous. One may call it the initial neutrosophic crisp đ?›ź-topology with respect to f . 5 Neutrosophic Crisp đ?›ź-Compact Space First we present the basic concepts: 5.1 Definition Let X , ď ‡ď Ą  be an NCđ?›źTS. (a)
If a family
ď ťG
i1 , Gi2 , Gi3
X satisfies the condition ďƒˆ
ď ťG
i1
ď ˝
: i ďƒŽ J of NCđ?›źOSs in
ď ˝
: i ďƒŽ J  X N , then it is called
, Gi2 , Gi3
an neutrosophic đ?›ź-open cover of X. (b) A finite subfamily of an đ?›ź-open cover Gi1 , Gi2 , Gi3 : i ďƒŽ J on X, which is also a neutrosophic đ?›ź-
ď ť
ď ˝
open cover of X , is called a neutrosophic crisp finite �open subcover. 5.2 Definition A neutrosophic crisp set A  A1 , A2 , A3 in a NC�TS
X , ď ‡  is called neutrosophic crisp đ?›ź-compact iff every ď Ą
neutrosophic crisp open cover of A has a finite neutrosophic crisp open subcover. 5.3 Definition A family Ki , Ki , Ki : i ďƒŽ J of neutrosophic crisp đ?›ź-
ď ť
1
2
ď ˝
3
In this case we have, for each H , ď Ś , H c ďƒŽ ď ‡2ď Ą ,
compact sets in X satisfies the finite intersection property (FIP for short) iff every finite subfamily K i , K i , K i : i  1,2,..., n of the family satisfies the
H ďƒŽ ď ™o ,
condition ďƒ‡ K i , K i , K i : i  1,2,..., n ď‚š ď Ś N .
ď ‡2ď Ą 
c
: H ďƒŽ ď ™oď Ą ,
ď Ą
f
ď€1
 f
H ,ď Ś, H ď€1
c
 f
H ,ď Ś , ( f
ď€1
ď€1
( H ), f
( H ))
c
ď€1
(ď Ś ), f
ď€1
(H ) c
ďƒŽ ď ‡1 . ď Ą
4.6 Proposition Let f : ( X , ď ‡1ď Ą ) ď‚Ž (Y , ď ‡2ď Ą ) . f is continuous iff the neutrosophic crisp preimage of each CNđ?›źCS (crisp neutrosophic đ?›ź-closed set) in ď ‡1ď Ą is a CNđ?›źCS in ď ‡2ď Ą . Proof Similar to the proof of Proposition 4.5. 4.7 Proposition The following are equivalent to each other: (a) f : ( X , ď ‡1ď Ą ) ď‚Ž (Y , ď ‡2ď Ą ) is continuous . (b) f ď€1 (CNď ĄInt ( B) ďƒ? CNď ĄInt ( f ď€1 ( B)) for each CNS B in Y. (c) CNď ĄCl ( f ď€1 ( B)) ďƒ? f ď€1 (CNď ĄCl ( B))
ď ť
1
2
ď ˝
3
ď ť
1
2
ď ˝
3
5.4 Definition A NCđ?›źTS X , ď ‡ď Ą  is called neutrosophic crisp đ?›źcompact iff each neutrosophic crisp đ?›ź-open cover of X has a finite đ?›ź-open subcover. 5.5 Corollary A NCđ?›źTS X , ď ‡ď Ą  is a neutrosophic crisp đ?›ź-compact iff every family Gi , Gi , Gi : i ďƒŽ J of neutrosophic
ď ť
1
2
3
ď ˝
crisp đ?›ź-compact sets in X having the the finite intersection properties has nonempty intersection. 5.6 Corollary Let X , ď ‡1ď Ą  , Y , ď ‡2ď Ą  be NCđ?›źTSs and f : X ď‚Ž Y be a continuous surjection. If X , ď ‡1ď Ą  is a neutrosophic crisp


đ?›ź-compact, then so is Y , ď ‡2ď Ą .
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5.7 Definition (a) If a family Gi1 , Gi2 , Gi3 : i ďƒŽ J of neutrosophic
ď ť
ď ˝
crisp đ?›ź-compact sets in X satisfies the condition A ďƒ? ďƒˆ Gi1 , Gi2 , Gi3 : i ďƒŽ J , then it is called a
ď ť
ď ˝
neutrosophic crisp open cover of A. (b) Let’s consider a finite subfamily of a neutrosophic crisp open subcover of Gi1 , Gi2 , Gi3 : i ďƒŽ J .
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5.8 Corollary Let X , ď ‡1ď Ą  , Y , ď ‡2ď Ą  be NCđ?›źTSs and f : X ď‚Ž Y be a continuous surjection. If A is a neutrosophic crisp đ?›źcompact in X , ď ‡1ď Ą  , then so is f ( A) in Y , ď ‡2ď Ą . 6. Conclusion In this paper, we presented a generalization of the neutrosophic topological space. The basic definitions of the neutrosophic crisp Îą-topological space and the neutrosophic crisp Îą-compact space with some of their characterizations were deduced. Furthermore, we constructed a netrosophic crisp Îą-continuous function, with a study of a number its properties. References [1] S.A. Alblowi, A. A. Salama and Mohmed Eisa, New Concepts of Neutrosophic Sets, International Journal of Mathematics and Computer Applications Research (LIMCAR)Nol. 4, Issue 1, (2014) 59-66 [2] K Atanassov, intuitionistic fuzzy sets, Fuzzy Sets and Systems , vol. 20, (1986), pp. 87-96 [3] I.M. Hanafy, A.A. Salama and K.M. Mahfouz,â€? Neutrosophic Crisp Events and Its Probability" International Journal of Mathematics and Computer Applications Research (IJMCAR) Vol. 3, Issue 1, (2013), pp. 171-178. [4] A.A. Salama and S.A. Alblowi, "Generalized Neutrosophic Set and Generalized Neutrosophic Topological Spaces ", Journal computer Sci. Engineering, Vol. 2, No. 7, (2012), pp. 29-32 [5] A.A. Salama and S.A. Alblowi, Neutrosophic set and neutrosophic topological space, ISORJ. Mathematics, Vol. 3, Issue 4, (2012), pp. 31-35. [6] A.A. Salama and S.A. Alblowi, Intuitionistic Fuzzy Ideals Topological Spaces, Advances in Fuzzy Mathematics , Vol. 7, No. 1, (2012), pp. 51- 60. [7] A.A. Salama, and H. Elagamy, "Neutrosophic Filters," International Journal of Computer Science Engineering and Information Technology Research (IJCSEITR), Vol.3, Issue (1), (2013), pp. 307-312. [8] A. A. Salama, S. A. Alblowi & Mohmed Eisa, New Concepts of Neutrosophic Sets, International Journal of Mathematics and Computer Applications Research (IJMCAR),Vol.3, Issue 4, Oct 2013, (2013), pp. 95102.
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A.A. Salama and Florentin Smarandache, Neutrosophic crisp set theory, Educational Publisher, Columbus, (2015).USA A. A. Salama and F. Smarandache, "Filters via Neutrosophic Crisp Sets", Neutrosophic Sets and Systems, Vol.1, No. 1, (2013), pp. 34-38. A. A. Salama,"Neutrosophic Crisp Points & Neutrosophic Crisp Ideals", Neutrosophic Sets and Systems, Vol.1, No. 1, (2013) pp. 50-54. Florentin Smarandache, An introduction to the Neutrosophy probability applied in Quantum Physics , International Conference on Neutrosophic Physics , Neutrosophic Logic, Set, Probability, and Statistics University of New Mexico, Gallup, NM 87301, USA, 2-4 December (2011) . Florentin 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) . F. Smarandache. A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability. American Research Press, Rehoboth, NM, (1999). L.A.Zadeh, Fuzzy Sets. Inform. Control, vol 8, (1965), pp 338-353. Received: April 15, 2016. Accepted: June 22, 2016.
A.A.Salama, I.M.Hanafy, Hewayda Elghawalby and M.S.Dabash, Neutrosophic Crisp � -Topological Spaces