# Basic structure of some classes of neutrosophic crisp nearly open sets & possible application to GIS topology.

1 IntroductionThe fundamental concepts of neutrosophic set, introduced by Smarandache [30, 31, 32] and Salama et al. in [4-29]., provides a natural foundation for treating mathematically the neutrosophic phenomena which exist pervasively in our real world and for building new branches of neutrosophic mathematics. Neutrosophy has laid the foundation for a whole family of new mathematical theories generalizing both their classical and fuzzy counterparts [1, 2, 3, 20, 21, 22, 23, 34 ] such as a neutrosophic set theory. In this paper the structure of some classes of neutrosophic crisp sets are investigated, and some applications are given. Finally we generalize the crisp topological and intuitioistic studies to the notion of neutrosophic crisp set.

2 2 Terminologies

We recollect some relevant basic preliminaries, and in particular, the work of Smarandache in [30, 31, 32] and Salama et al. [4-29]. Smarandache introduced the neutrosophic components T, I, F which represent the membership, indeterminacy, and non-membership values respectively, where J -0 ,1+[is non-standard unit interval. Salama et al. [9, 10, 13, 14, 16, 17] considered some possible definitions for basic concepts of the neutrosophic crisp set and its operations. We now improve some results by the following. Salama extended the concepts of topological space and in tuitionistic topological space to the case of neutrosophic crisp sets.

Definitions 2 [13]

A neutrosophic crisp topology (NCT for short) on a non-empty set X is a family T of neutrosophic crisp subsets in X satisfying the following axioms

i) [[phi].sub.N], [X.sub.N] [member of] T.

ii) [A.sub.1] [intersection] [A.sub.2] [member of] [GAMMA] for any [A.sub.1] and [A.sub.2] [member of] [GAMMA].

iii) [union] [A.sub.j] [member of] [GAMMA] [for all] {[A.sub.j] : j [member of] J} [subset or equal to] [GAMMA].

In this case the pair (X, t) is called a neutrosophic crisp topological space (NCTS for short) in X. The elements in T are called neutrosophic crisp open sets (NCOSs for short) in X. A neutrosophic crisp set F is closed if and only if its complement [F.sup.C] is an open neutrosophic crisp set.

Let (X, [GAMMA]) be a NCTS (identified with its class of neutrosophic crisp open sets), and NCint and NCcl denote neutrosophic interior crisp set and neutrosophic crisp closure with respect to neutrosophic crisp topology

3 Nearly Neutrosophic Crisp Open Sets Definitions. 1

Let (X, [GAMMA]) be a NCTS and A = <[A.sub.1], [A.sub.2], [A.sub.3]> be a NCS in

X, then A is called

Neutrosophic crisp [alpha] - open set iff

A [subset or equal to] NC int(NCcl(NC int(A))),

Neutrosophic crisp [beta] - open set iff

A [subset or equal to] NCcl(NC int(A)).

Neutrosophic crisp semi-open set iff

A [subset or equal to] NC int(NCcl(A)).

We shall denote the class of all neutrosophic crisp [alpha]-open sets NC[[GAMMA].sup.[alpha]], the calls all neutrosophic crisp [beta]-open sets NC[[GAMMA].sup.[beta]], and the class of all neutrosophic crisp semi-open sets NC[[GAMMA].sup.s].

Remark 3.1

A class consisting of exactly all a neutrosophic crisp [alpha]-structure (resp. NC [beta]-structure). Evidently NC[GAMMA] [subset or equal to] NC[[GAMMA].sup.[alpha]] [subset or equal to] NC[[GAMMA].sup.[beta]].

We notice that every non- empty neutrosophic crisp [beta]-open has [NC.sup.[alpha]]-nonempty interior. If all neutrosophic crisp sets the following [{[B.sub.i]}.sub.i[member of]I] are NC [beta]-open sets, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], that is A NC [beta]-structure is a neutrosophic closed with respect to arbitrary neutrosophic crisp unions.

We shall now characterize NC[[GAMMA].sup.[alpha]] in terms NC[[GAMMA].sup.[beta]]

Theorem 3.1

Let (X, [GAMMA]) be a NCTS. NC[[GAMMA].sup.[alpha]] Consists of exactly those neutrosophic crisp set A for which A [intersection] B [member of] NC[[GAMMA].sup.[beta]] for B [member of] NC[[GAMMA].sup.[beta]

Proof

Let A [member of] NC[[GAMMA].sup.[alpha]], B [member of] NC[[GAMMA].sup.[beta]], p [member of] A [intersection] B and U be a neutrosophic crisp neighbourhood (for short NCnbd)of p. Clearly U [intersection] NC int(NCcl(NC int(A))), too is a neutrosophic crisp open neighbourhood of p, so V = (U [intersection] NC int(NCcl (NC int(A)))) n NC int(B) is non-empty. Since V [subset] NCcl(NCint(A)) this implies (U [intersection] NC int(A) [intersection] NC int(B)) = V [intersection] NC int(A) = [[phi].sub.N].

It follows that

A [intersection] B [subset] NCcl (Nc int(A) [intersection] NC int(B)) = NCcl(NC int(A [intersection] B)) i.e. A [intersection] B [member of] NC[[GAMMA].sup.[beta]]. Conversely, let A [intersection] B [member of] NC[[GAMMA].sup.[beta]] for all B [member of] NC[[GAMMA].sup.[beta]] then in particular A [member of] NC[[GAMMA].sup.[beta]]. Assume that p [member of] A [intersection] [(NC int(NCcl (A) [intersection] NC int(A))).sup.c]. Then p [member of] NCcl (B), where [(NCcl (NC int(A))).sup.c]. Clearly {p} [union] B [member of] NC[[GAMMA].sup.[beta]], and consequently A [intersection] {{p} [union] B} [member of] NC[[GAMMA].sup.[beta]]. But A [intersection] {{p} [union] B} = {p} . Hence {p} is a neutrosophic crisp open. As p [member of] (NCcl (NC int(A)) this implies p [member of] NC int(NCcl (NC int(A))), contrary to assumption. Thus p [member of] A implies p e NCcl (NC int(A)) and A e NCra. This completes the proof. Thus we have found that NC[[GAMMA].sup.[alpha]] is complete determined by NC[GAMMA] [beta] i.e. all neutrosophic crisp topologies with the same NC [beta]-structure also determined the same NC a -structure, explicitly given Theorem 3.1.

We shall that conversely all neutrosophic crisp topologies with the same NC [alpha]-structure, so that NCr p is completely determined by NCr a .

Theorem 3.2

Every neutrosophic crisp NC [alpha]-structure is a neutrosophic crisp topology.

Proof

NC[[GAMMA].sup.[beta]] Contains the neutrosophic crisp empty set and is closed with respect to arbitrary unions. A standard result gives the class of those neutrosophic crisp sets A for which A [intersection] B [member of] NC[[GAMMA].sup.[beta]] for all B [member of] NC[[GAMMA].sup.[beta]] constitutes a neutrosophic crisp topology, hence the theorem. Hence forth we shall also use the term [NC.sup.[alpha]]-topology for NC a -structure two neutrosophic crisp topologies deterring the same NC a structure shall be called [NC.sup.[alpha]]-equivalent , and the equivalence classes shall be called [NC.sup.[alpha]]-classes.

We may now characterize NCrp in terms of NCra in the following way.

Proposition 3.1

Let (X, [GAMMA]) be a NCTS. Then NC[[GAMMA].sup.[beta]] = NC[[GAMMA].sup.[alpha][beta]] and hence NC [alpha]-equivalent topologies determine the same NC [beta]-structure.

Proof

Let NC [alpha] - cl and NC [alpha]-int denote neutrosophic closure and Neutrosophic crisp interior with respect to NC[[GAMMA].sup.[alpha]]. If p [member of] B [member of] NC[[GAMMA].sup.[beta]] and p [member of] B [member of] NC [[GAMMA].sup.[alpha]], then (NC int(NCcl (NC int(A))) n NC int(B)) [not equal to] [[phi].sub.N]. Since NC int(NCcl (NC int(A))) is a crisp neutrosophic neighbourhood of point p. So certainly NC int(B) meets NCcl (NC int(A)) and therefore (bing neutrosophic open) meets NC int(A) , proving A n NC int(B) [not equal to] [[phi].sub.N] this means B [subset] NCacl(NC int(B)) i.e. B e NCTa on the other hand let A [member of] NC[[GAMMA].sup.[alpha]], p [member of] A and p [member of] V [member of] NC[GAMMA]. As V [member of] NC[[GAMMA].sup.[alpha]] and p [member of] NCcl(NC int(A)) we have V [intersection] NC int(A) [not equal to] [[phi].sub.N] and there exist a nutrosophic crisp set W [member of] [GAMMA] such that W [subset] V [intersection] NC [alpha] int(A) [subset] A.

In other words V [intersection] NC int(A) [not equal to] [[phi].sub.N] and p [member of] NCcl (NC int(A)). Thus we have verified NC[[GAMMA].sup.[alpha][beta]] [subset] NC[[GAMMA].sup.[alpha]], and the proof is complete combining Theorem 1 and Proposition 1. We get NC[[GAMMA].sup.[alpha][alpha]] = NC[[GAMMA].sup.[alpha]].

Corollary 3.2

A neutrosophic crisp topology NC[GAMMA] a NC[alpha]-topology iff NCT = NC[[GAMMA].sup.[alpha]]. Thus an NC[alpha]-topology belongs to the NC[alpha]-class if all its determining a Neutrosophic crisp topologies, and is the finest topology of finest neutrosophic topology of this class. Evidently NC [[GAMMA].sup.[beta]] is a neutrosophic crisp topology iff NC[[GAMMA].sup.[alpha]] = NC [[GAMMA].sup.[beta]]. In this case [NCT.sup.[beta][beta]] = NC[[GAMMA].sup.[alpha][beta]] = NC [[GAMMA].sup.[beta]].

Corollary 3.3

NC[beta]-Structure B is a neutrosophic crisp topology, then B = [B.sup.[alpha]] = [B.sup.[beta]].

We proceed to give some results an the neutrosophic structure of neutrosophic crisp NC[alpha]-topology

Proposition 3.4

The NC[alpha]-open with respect to a given neutrosophic crisp topology are exactly those sets which may be written as a difference between a neutrosophic crisp open set and neutrosophic crisp nowhere dense set

If A [member of] [NCT.sup.[alpha]] we have A = (NC int(NCcl (NC int(A)) [intersection] [(NC int(NCcl(NC int(A)) [intersection] [A.sup.C]).sup.C], where (NCint(NCcl(NCint(A)) [intersection] [A.sup.C]) clearly is neutrosophic crisp nowhere dense set, we easily see that B [subset] NCcl (NC int(A)) and consequently A [subset] B [subset] NC int(NCcl (NC int(A)) so the proof is complete.

Corollary 3.4

A neutrosophic crisp topology is a NCa - topology iff all neutrosophic crisp nowhere dense sets are neutrosophic crisp closed.

For a neutrosophic crisp NC[alpha]-topology may be characterized as neutrosophic crisp topology where the difference between neutrosophic crisp open and neutrosophic crisp nowhere dense set is again a neutrosophic crisp open, and this evidently is equivalent to the condition stated.

Proposition 3.5

Neutrosophic crisp topologies which are NC[alpha]-equivalent determine the same class of neutrosophic crisp nowhere dense sets.

Definition 3.2

We recall a neutrosophic crisp topology a neutrosophic crisp extremely disconnected if the neutrosophic crisp closure of every neutrosophic crisp open set is a neutrosophic crisp open .

Proposition 3.6

If NC[alpha]--Structure B is a neutrosophic crisp topology, all a neutrosophic crisp topologies [GAMMA] for which [[GAMMA].sup.[beta]] = B are neutrosophic crisp extremely disconnected.

In particular: Either all or none of the neutrosophic crisp topologies of a NC[alpha]--class are extremely disconnected.

Proof

Let [[GAMMA].sup.[beta]] = B and suppose there is a A [member of] T such that NCcl(A) [not member of] [GAMMA] Let p [member of] NCcl (A) [intersection] (NC int[(NCcl (A)).sup.C] with B = {p} [union] NC int(NCcl (A)), M = [(NC int(NCcl(A))).sup.C] we have {p} [subset] M = [(NCint(NCcl(A))).sup.C] = NCcl (NC int(M), {p} [subset] NCcl (A) = NCcl (NC int(NCcl (A)) [subset] NCcl (NC int(B)). Hence both B and M are in [[GAMMA].sup.[beta]]. The intersection BnM = {p} is not neutrosophic crisp open since p e NCcl (A) [intersection] [M.sup.C], hence not NC[beta]- open so. [[GAMMA].sup.[beta]] = B is not a neutrosophic crisp topology. Now suppose B is not a topology, and [[GAMMA].sup.[beta]] = B There is a B [member of] [[GAMMA].sup.[beta]] such that B [not member of] [[GAMMA].sup.[alhpa]]. Assume that NCcl (NC int(B) [member of] T. Then B [subset] NCcl (NC int(B) = NC int(NCcl (NC int(B)) .i.e. B [member of] [[GAMMA].sup.[alpha]], contrary to assumption. Thus we have produced a neutrosophic crisp open set whose neutrosophic crisp closure is not neutronsophic crisp open, which completes the proof.

Corollary 3.5

A neutrosophic crisp topology [GAMMA] is a neutrosophic crisp extremely disconnected if and only if [[GAMMA].sup.[beta]] is a neutrosophic crisp topology.

4 Conclusion and future work

Neutrosophic set is well equipped to deal with missing data. By employing NSs in spatial data models, we can express a hesitation concerning the object of interest. This article has gone a step forward in developing methods that can be used to define neutrosophic spatial regions and their relationships. The main contributions of the paper can be described as the following: Possible applications have been listed after the definition of NS. Links to other models have been shown. We are defining some new operators to describe objects, describing a simple neutrosophic region. This paper has demonstrated that spatial object may profitably be addressed in terms of neutrosophic set. Implementation of the named applications is necessary as a proof of concept.

Received: October 3, 2014. Accepted: December 20, 2014.

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A. A. Salama

Department of Mathematics and Computer Science, Faculty of Sciences, Port Said University, Egypt

drsalama44@gmail.com

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Author: | Salama, A.A. |
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Publication: | Neutrosophic Sets and Systems |

Article Type: | Report |

Date: | Jan 1, 2015 |

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