Basic Structure of Some Classes of Neutrosophic Crisp Nearly Open Sets & Possible Application to GIS Topology

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1 eutrosophic Sets and Systems, Vol 7, Basic Structure of Some lasses of eutrosophic risp early Open Sets & Possible Application to GIS A A Salama Department of Mathematics and omputer Science, Faculty of Sciences, Port Said University, Egypt drsalama44@gmailcom Abstract Since the world is full of indeterminacy, the neutrosophics found their place into contemporary research The fundamental concepts of neutrosophic set, introduced by Smarandache in [30, 31, 32] and Salama et al in [4-29] In Geographical information systems (GIS) there is a need to model spatial regions with indeterminate boundary and under indeterminacy In this paper the structure of some classes of neutrosophic crisp nearly open sets are investigated and some applications are given Finally we generalize the crisp topological and intuitioistic studies to the notion of neutrosophic crisp set Possible applications to GIS topological rules are touched upon Keywords: eutrosophic risp Set; eutrosophic risp ; eutrosophic risp Open Set; eutrosophic risp early Open Set; eutrosophic GIS 1 Introduction The 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 eutrosophy 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 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 intuitionistic topological space to the case of neutrosophic crisp sets Definition1 2 [13] A neutrosophic crisp topology (T for short) on a non-empty set X is a family of neutrosophic crisp subsets in X satisfying the following axioms i) ii) iii), X A 1 A 2 for any 1 A and A A j : j J j A In this case the pair X, is called a neutrosophic crisp topological space ( TS for short) in X The elements in are called neutrosophic crisp open sets (OSs for short) in X A neutrosophic crisp set F is closed if and only if its complement F is an open neutrosophic crisp set Let X, be a TS ( identified with its class of neutrosophic crisp open sets ), and int and cl denote neutrosophic interior crisp set and neutrosophic crisp closure with respect to neutrosophic crisp topology 2 A A Salama, Basic Structure of Some lasses of eutrosophic risp early Open Sets & Possible Application to GIS

2 eutrosophic Sets and Systems, Vol 7, early eutrosophic risp Open Sets Definition3 1 Let X, be a TS and A A1, A2, A3 be a S in X, then A is called eutrosophic crisp open set iff A int( cl( int( ), eutrosophic crisp open set iff A cl ( int( eutrosophic crisp semi-open set iff A int( cl ( We shall denote the class of all neutrosophic crisp open sets, the calls all neutrosophic crisp open sets, and the class of all neutrosophic crisp semi-open sets s Remark 31 A class consisting of exactly all a neutrosophic crisp structure (resp structure) Evidently We notice that every non- empty neutrosophic crisp open has nonempty interior If all neutrosophic crisp sets the following { B } are open sets, i i I then { Bi} ii cl ( int( Bi )) cl ( int( B i )), ii that is A structure is a neutrosophic closed with respect to arbitrary neutrosophic crisp unions We shall now characterize in terms Theorem 31 Let X, be a TS onsists of exactly those neutrosophic crisp set A for which A B for B Let A, B, p A B and U be a neutrosophic crisp neighbourhood (for short nbd )of p learly U int( cl( int( ), too is a neutrosophic crisp open neighbourhood of p, so V ( U int( cl( int( )) int( B) is non-empty Since V cl( int( this implies ( U int( A) int( B)) = V int( A) It follows that A B cl( c int( A) int( B)) = cl( int( A B)) ie A B onversely, let A B for all A Assume that B then in particular p A ( int( cl ( A) int( ) p cl(b), { p} B c Then where ( cl ( int( ) learly, and consequently A {{ p} B} But A {{ p} B} { p} Hence {p} is a neutrosophic crisp open As p ( cl ( int( this implies p int( cl( int( ), contrary to assumption Thus p A implies p cl( int( and A This completes the proof Thus we have found that is complete determined by ie all neutrosophic crisp topologies with the same -structure also determined the same -structure, explicitly given Theorem 31 We shall that conversely all neutrosophic crisp topologies with the same -structure, so that is com- pletely determined by Theorem 32 Every neutrosophic crisp -structure is a neutrosophic crisp topology ontains 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 B for all B constitutes a neutrosophic crisp topology, hence the theorem Hence forth we shall also use the term -topology for -structure two neutrosophic crisp topologies deterring the same - structure shall be called -equivalent, and the equivalence classes shall be called -classes We may now characterize the following way c in terms of in Proposition 31 Let X, be a TS Then = and hence -equivalent topologies determine the same -structure Let cl and int denote neutrosophic closure and eutrosophic crisp interior with respect to A A Salama, Basic Structure of Some lasses of eutrosophic risp early Open Sets & Possible Application to GIS

3 20 eutrosophic Sets and Systems, Vol 7, 2015 If p B and p B, then ( int( cl( int( ) int( B)) Since int( cl( int( ) is a crisp neutrosophic neighbourhood of point p So certainly int( B) meets cl( int( and therefore (bing neutrosophic open ) meets int( A), proving A int( B) this means B cl( int( B)) ie on the other hand let A, A B p and p V As V and p cl( int( we have V int( A) and there exist a nutrosophic crisp W V int( A) A set W such that In other words V int( A) and p cl( int( Thus we have verified, and the proof is complete combining Theorem 1 and Proposition 1 We get orollary 32 A neutrosophic crisp topology a topology iff Thus an topology belongs to the class if all its determining a eutrosophic crisp topologies, and is the finest topology of finest neutrosophic topology of this class Evidently is a neutrosophic crisp topology iff In this case = orollary 33 Structure is a neutrosophic crisp topology, then We proceed to give some results an the neutrosophic structure of neutrosophic crisp topology Proposition 34 The 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 we have A ( int( cl ( int( ( int( cl ( int( A ), where ( int( cl ( int( A ) clearly is neutrosophic crisp nowhere dense set, we easily see that B cl ( int( and consequently A B int( cl( int( so the proof is complete orollary 34 A neutrosophic crisp topology is a topology iff all neutrosophic crisp nowhere dense sets are neutrosophic crisp closed For a neutrosophic crisp 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 35 eutrosophic crisp topologies which are equivalent determine the same class of neutrosophic crisp nowhere dense sets Definition 32 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 36 If Structure is a neutrosophic crisp topology, all a neutrosophic crisp topologies for which are neutrosophic crisp extremely disconnected In particular: Either all or none of the neutrosophic crisp topologies of a class are extremely disconnected Let that and suppose there is a A cl(a) p cl ( A) ( int( cl ( A A Salama, Basic Structure of Some lasses of eutrosophic risp early Open Sets & Possible Application to GIS such Let with { p} int( cl(, M int( cl( we have { p } M int( cl( cl( int( M ), { p} cl( A) cl( int( cl( cl ( int( B)) Hence both and M are in The intersection M {p} is not neutrosophic crisp open since p cl ( A) M, hence not open so is not a neutrosophic crisp topology ow suppose is not a topology, and There is a such that Assume that cl ( int( ) Then cl ( int( ) int( cl ( int( )) ie, 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

4 eutrosophic Sets and Systems, Vol 7, orollary 35 A neutrosophic crisp topology is a neutrosophic crisp extremely disconnected if and only if is a neutrosophic crisp topology 4 onclusion and future work eutrosophic set is well equipped to deal with missing data By employing Ss 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 S 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 References [1] K Atanassov, intuitionistic fuzzy sets, in VSgurev, ed, Vii ITKRS Session, Sofia (June 1983 central Sci and Techn Library, Bulg Academy of Sciences (1984) [2] K Atanassov, intuitionistic fuzzy sets, Fuzzy Sets and Systems 20, 87-96,(1986) [3] K Atanassov, Review and new result on intuitionistic fuzzy sets, preprint IM-MFAIS-1-88, Sofia, (1988) [4] S A Alblowi, AASalama and Mohmed Eisa, ew oncepts of eutrosophic Sets, International Journal of Mathematics and omputer Applications Research (IJMAR),Vol 4, Issue 1, (2014)pp59-66 [5] IM Hanafy, AA Salama and K Mahfouz, orrelation of eutrosophic Data, International Refereed Journal of Engineering and Science (IRJES), Vol(1), Issue 2 (2012) PP39-33 [6] IM Hanafy, AA Salama and KM Mahfouz, eutrosophic lassical Events and Its Probability, International Journal of Mathematics and omputer Applications Research(IJMAR) Vol(3),Issue 1, (2013)pp [7] A A Salama and SA Alblowi, Generalized eutrosophic Set and Generalized eutrosophic Topological Spaces, Journal omputer Sci Engineering, Vol (2) o (7), (2012)pp [8] A A Salama and S A Alblowi, eutrosophic Set and eutrosophic Topological Spaces, ISOR J Mathematics, Vol(3), Issue(3), (2012) pp31-35 [9] A A Salama, eutrosophic risp Point & eutrosophic risp Ideals, eutrosophic Sets and Systems, Vol1, (2013) pp [10] A A Salama and F Smarandache, Filters via eutrosophic risp Sets, eutrosophic Sets and Systems, Vol1, (2013) pp [11] AA Salama and SA Alblowi, Intuitionistic Fuzzy Ideals Topological Spaces, Advances in Fuzzy Mathematics, Vol(7), umber 1, (2012) pp [12] AA Salama, and H Elagamy, eutrosophic Filters, International Journal of omputer Science Engineering and Information Technology Reseearch (IJSEITR), Vol3, Issue(1), (2013) pp [13] A A Salama, FSmarandache and Valeri Kroumov, eutrosophic crisp Sets & eutrosophic crisp Topological Spaces, eutrosophic Sets and Systems, Vlo(2),(2014),pp25-30 [14] A A Salama, Florentin Smarandache and Valeri Kroumov, eutrosophic losed Set and eutrosophic ontinuous Functions eutrosophic Sets and Systems, 2014, Vol (4) pp4-8 [15] A A Salama, Mohamed Eisa and M M Abdelmoghny, eutrosophic Relations Database, International Journal of Information Science and Intelligent System, 3(1) (2014)pp33-46 [16] A A Salama, Florentin Smarandache and S A Alblowi, ew eutrosophic risp Topological oncepts, eutrosophic Sets and Systems, Vol(4), (2014)pp50-54 [17] A A Salama, Said Broumi and Florentin Smarandache, eutrosophic risp Open Set and eutrosophic risp ontinuity via eutrosophic risp Ideals, IJ Information Engineering and Electronic Business, Vol3, (2014)pp1-8 [18] AA Salama, Florentin Smarandache and SA Alblowi The haracteristic Function of a eutrosophic Set, eutrosophic Sets and Systems, 2014, Vol (3), pp14-18 [19] A A Salama, eutrosophic risp Points & eutrosophic risp Ideals, eutrosophic Sets and Systems, 2013,Vol(1) pp50-53 [20] A A Salama, Mohamed Abdelfattah and S A Alblowi, Some Intuitionistic Topological otions of Intuitionistic Region, Possible Application to GIS Topological Rules, International Journal of Enhanced Research in A A Salama, Basic Structure of Some lasses of eutrosophic risp early Open Sets & Possible Application to GIS

5 22 eutrosophic Sets and Systems, Vol 7, 2015 Management&omputerApplications, (2014),Vol(3), no (6)pp1-13 [21] A A Salama, Mohamed Abdelfattah and Mohamed Eisa, A ovel Model for Implementing Security over Mobile Ad-hoc etworks using Intuitionistic Fuzzy Function, International Journal of Emerging Technologies in omputational and Applied Sciences (IJETAS), 2014,Vol(6)pp1-7 [22] A A Salama and Said Broumi, Roughness of eutrosophic Sets, Elixir Appl Math 74 (2014)pp [23] A A Salama, Mohamed Abdelfattah and Mohamed Eisa, Distances, Hesitancy Degree and Flexible Querying via eutrosophic Sets, International Journal of omputer Applications, Volume 101 o10, (2014)pp [24] AA Salama, Haithem A El-Ghareeb, Ayman M Maine and Florentin Smarandache Introduction to Develop Some Software Programs for dealing with eutrosophic Sets, eutrosophic Sets and Systems, 2014,Vol(4), pp51-52 [25] A A Salama, F Smarandache, and M Eisa Introduction to Image Processing via eutrosophic Technique, eutrosophic Sets and Systems, 2014, Vol (5) pp59-63 [26] A A Salama, Haitham A El-Ghareeb, Ayman M Manie and M M Lotfy, Utilizing eutrosophic Set in Social etwork Analysis e- Learning Systems, International Journal of Information Science and Intelligent System, 3(2), (2014)pp61-72 [27] A A Salama, O M Khaled, and K M Mahfouz eutrosophic orrelation and Simple Linear Regres-sion, eutrosophic Sets and Systems, 2014, Vol (5) pp3-8 [28] A A Salama, and F Smarandache eutrosophic risp Set Theory, eutrosophic Sets and Systems, 2014, Vol (5) pp27-35 [29] A A Salama, Florentin Smarandache and S A ALblowi, ew eutrosophic risp Topological oncepts, eutrosophic Sets and Systems, 2014, Vol(4)pp50-54 [30] Florentin Smarandache, eutrosophy and eutrosophic Logic, First International onference on eutrosophy, eutrosophic Logic, Set, Probability, and Statistics University of ew Mexico, Gallup, M 87301, USA(2002) [31] Florentin Smarandache, A Unifying Field in Logics: eutrosophic Logic eutrosophy, eutrosophic Set, eutrosophic Probability American Research Press, Rehoboth, M, (1999) [32] Florentin Smarandache, eutrosophic set, a generialization of the intuituionistics fuzzy sets, Inter J Pure Appl Math, 24 (2005), [33] Debasis Sarker, Fuzzy ideal theory, Fuzzy local function and generated fuzzy topology, Fuzzy Sets and Systems 87, (1997) [34] LA Zadeh, Fuzzy Sets, Inform and ontrol 8, (1965) Received: October 3, 2014 Accepted: December 20, 2014 A A Salama, Basic Structure of Some lasses of eutrosophic risp early Open Sets & Possible Application to GIS