New Trends in Neutrosophic Theory and Applications
A.A.SALAMA1, I.M.HANAFY1, HEWAYDA ELGHAWALBY3, M.S.DABASH4 1,2,4 Department of Mathematics and Computer Science, Faculty of Sciences, Port Said University, Egypt. Emails: drsalama44@gmail.com, ihanafy@hotmail.com, majdedabash@yahoo.com 3 Faculty of Engineering, port-said University, Egypt. Email: hewayda2011@eng.psu.edu.eg
Some GIS Topological Concepts via Neutrosophic Crisp Set Theory Abstract In this paper we introduce and study the neutrosophic crisp pre-open, semi-open, đ?›˝- open set, neutrosophic crisp continuity and neutrosophic crisp compact spaces are introduced. Furthermore, we investigate some of their properties and characterizations. Possible application to GIS topology rules are touched upon.
Keywords Neutrosophic crisp topological spaces, neutrosophic crisp sets, neutrosophic crisp continuity, neutrosophic crisp compact space.
1. Introduction Smarandache [26, 27] introduced the notion of neutrosophic sets, which is a generalization of Zadeh's fuzzy set [28]. In Zadah's sense, there is no precise definition for the set. Later on, Atanassov presented the idea of the intuitionistic fuzzy set [1], where he goes beyond the degree of membership introducing the degree of non-membership of some element in the set. The new presented concepts attracted several authors to develop the classical mathematics. For instance, Chang [2] and Lowen [6] started the discipline known as "Fuzzy Topology", where they forwarded the concepts from fuzzy sets to the classical topological spaces. Furthermore, Salama et al. [14, 17, 20] established several notations for what they called, "Neutrosophic topological spaces"]. In this paper, we study in more details some weaker and stronger structures constructed from the neutrosophic crisp topology introduced in [7], as well as the concepts neutrosophic crisp interior and the neutrosophic closure. The remaining of this paper is structured as follows: in §2, some basic definitions are presented, while the new concepts of neutrosophic crisp nearly open sets are introduced in §3, in addition to providing a study of some of its properties. The neutrosophic crisp continuous function and neutrosophic crisp compact spaces are presented in §4 and §5, respectively.
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Florentin Smarandache, Surapati Pramanik (Editors)
2. Terminologies We recollect some relevant basic preliminaries, in particular, the work introduced by We recollect some relevant basic preliminaries, in particular, the work introduced by Smarandache and Salama [7], Salama et al. [8] and Smarandache [25,26,27]. The neutrosophic components T, I, F: Xâ&#x;ś]0−, 1+[to represent the membership, indeterminacy, and non-membership values of some universe X, respectively, where ]0− , 1+ [ is the nonstandard unit Interval. Definition 2.1 [7] Let đ?‘‹ be a non-empty fixed sample space. A neutrosophic crisp set (đ?‘ đ??śđ?‘† for short) đ??´ is an object having the form đ??´ = (đ??´1, đ??´2, đ??´3) where đ??´1, đ??´2 and đ??´3 are subsets of đ?‘‹. Where đ??´1 contains all those members of the space X that accept the event A and đ??´3 contains all those members of the space X that rejected the event A, while đ??´2 contains those who stand in a distance from accepting or rejecting A. Definition 2.2 Salama [7] defined the object having the form đ??´ = (đ??´1, đ??´2, đ??´3) to be 1) (Neutrosophic Crisp Set with Type 1),if satisfying đ??´1∊đ??´2=∅ , đ??´1∊đ??´3 =∅ and đ??´2∊đ??´3 = ∅. (đ?‘ đ??śđ?‘† -Type 1 ). 2) (Neutrosophic Crisp Set with Type 2), if satisfying đ??´1∊đ??´2=∅ , đ??´1∊đ??´3 =∅ and đ??´2∊đ??´3 = ∅ and đ??´1âˆŞđ??´2âˆŞđ??´3 = đ?‘‹ (đ?‘ đ??śđ?‘† -Type 2 ). 3) (Neutrosophic Crisp Set with Type 3) if satisfying đ??´1∊đ??´2∊đ??´3 = ∅ and đ??´1âˆŞđ??´2âˆŞđ??´3 = đ?‘‹. (đ?‘ đ??śđ?‘† -Type3 for short) . Every neutrosophic crisp set đ??´ of a non-empty set đ?‘‹ is obviously ađ?‘ đ??śđ?‘† having the form đ??´ = (đ??´1, đ??´2, đ??´3). Definition 2.3 [7] Let đ??´ = (đ??´1, đ??´2, đ??´3) a đ?‘ đ??śđ?‘† on đ?‘‹, then the complement of the set đ??´ , (đ??´c for short ) was presented in [7], to have one of the following forms: (C1) đ??´c = (đ??´1đ?‘? , đ??´đ?‘?2 , đ??´đ?‘?3 ) or (C2) đ??´c = (đ??´3, đ??´2, đ??´1) or (C3) đ??´c= (đ??´3, đ??´đ?‘?2 , đ??´1). Several relations and operations between đ?‘ đ??śđ?‘† were defined in [7], which we are introducing in the following: Definition 2.4 [7] Let đ?‘‹ be a non-empty set, and đ?‘ đ??śđ?‘†đ??´ and đ??ľ in the form đ??´ = (đ??´1, đ??´2, đ??´3), đ??ľ= (đ??ľ1, đ??ľ2, đ??ľ3), then we may consider two possible definitions for subsets (đ??´âŠ†đ??ľ). The concept of (đ??´âŠ†đ??ľ) may be defined as two types: Type 1. đ??´âŠ†đ??ľâ&#x;şđ??´1⊆đ??ľ1, đ??´2⊆đ??ľ2 and đ??´3⊇đ??ľ3 or Type 2. đ??´âŠ†đ??ľâ&#x;şđ??´1⊆đ??ľ1, đ??´2⊇đ??ľ2 and đ??´3⊇đ??ľ3 Proposition 2.5[7] For any neutrosophic crisp set đ??´ the following are hold đ?œ™N⊆đ??´, đ?œ™N⊆đ?œ™N đ??´âŠ†đ?–ˇN, đ?–ˇN⊆đ?–ˇN
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New Trends in Neutrosophic Theory and Applications
Definition 2.6[7] Let đ?‘‹be a non-empty set, and the two đ?‘ đ??śđ?‘†đ?‘ đ??´ and đ??ľgiven in the form đ??´ = (đ??´1, đ??´2, đ??´3) , đ??ľ = (đ??ľ1, đ??ľ2, đ??ľ3), then : 1) đ??´âˆŠđ??ľ may be defined as two types: i)Type 1. đ??´âˆŠđ??ľ = ⌊đ??´1∊đ??ľ1, đ??´2∊đ??ľ2, đ??´3âˆŞđ??ľ3âŒŞ ii) Type 2. đ??´âˆŠđ??ľ = ⌊đ??´1∊đ??ľ1, đ??´2âˆŞđ??ľ2, đ??´3âˆŞđ??ľ3âŒŞ 2) đ??´âˆŞđ??ľ may be defined as two types: i) Type 1. đ??´âˆŞđ??ľ = ⌊đ??´1âˆŞđ??ľ1, đ??´2âˆŞđ??ľ2, đ??´3∊đ??ľ3âŒŞ ii) Type 2. đ??´âˆŞđ??ľ = ⌊đ??´1âˆŞđ??ľ1, đ??´2∊đ??ľ2, đ??´3∊đ??ľ3âŒŞ Definition 2.7[7] A neutrosophic crisp topology (đ?‘ đ??śđ?‘‡ ) on a non-empty set đ?‘‹ is a family đ?›¤of neutrosophic crisp subsets ofđ?‘‹ satisfying the following axioms: i) ∅đ?‘ , đ?‘‹đ?‘ ∊đ?›¤. ii) đ??´1∊đ??´2∊ đ?›¤, ∀đ??´ 1, đ??´ 2∊ đ?›¤. iii) âˆŞ đ??´ j∊đ?›¤, ∀{đ??´ j : j∊J} ⊆đ?›¤. In this case, the pair (đ?‘‹, đ?›¤) is called a neutrosophic crisp topological space (đ?‘ đ??śđ?‘‡đ?‘†) in đ?‘‹. The elements of đ?›¤are called neutrosophic crisp open sets (đ?‘ đ??śđ?‘‚đ?‘†đ?‘ ) in đ?‘‹. A neutrosophic crisp set F is closed if and only if its complement Fc is an open neutrosophic crisp set. Definition 2.8[7] 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??´)) and neutrosophic interior crisp (đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´) ) of đ??´ are defined by đ?‘ đ??śđ?‘?đ?‘™(đ??´)=∊{đ??ž:đ??ž is an đ?‘ đ??śđ??śđ?‘† in đ?‘‹ and đ??´âŠ†đ??ž} đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą (đ??´) = âˆŞ{G:G is an 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 Nearly Open Sets
Definition 3.1 Let (đ?‘‹, đ?›¤) be a đ?‘ đ??śđ?‘‡đ?‘† and đ??´ =⌊đ??´1, đ??´2, đ??´3âŒŞ be a đ?‘ đ??śđ?‘† in đ?‘‹, then đ??´ is called: Neutrosophic crisp đ?›ź-open set iffđ??´âŠ† đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)). [24] i) Neutrosophic crisp pre-open set iffđ??´ ⊆ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ??´)) . ii) Neutrosophic crisp semi-open set iffđ??´ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą (đ??´)) . iii) Neutrosophic crisp đ?›˝- open set iffđ??´ ⊆ (đ?‘ đ??śđ?‘?đ?‘™ (đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™ (đ??´)). We shall denote the class of all neutrosophic crisp đ?›ź- open sets asđ?‘ đ??śđ?›¤đ?›ź, and the class of all neutrosophic crisp pre-open sets as đ?‘ đ??śđ?›¤p, and the class of all neutrosophic crisp semiopen sets as đ?‘ đ??śđ?›¤đ?‘†, and the class of all neutrosophic crisp đ?›˝- open sets as đ?‘ đ??śđ?›¤đ?›˝. Definition 3.2 Let (đ?‘‹, đ?›¤) be a đ?‘ đ??śđ?‘‡đ?‘† and đ??ľ = ⌊đ??ľ1, đ??ľ2, đ??ľ3âŒŞ be a đ?‘ đ??śđ?‘† in đ?‘‹, then đ??ľ is called: i) Neutrosophic crisp đ?›ź-closed set iff (đ?‘ đ??śđ?‘?đ?‘™ (đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™ (đ??ľ))⊆ đ??ľ. ii) Neutrosophic crisp pre- closed set iff đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą (đ??ľ)) ⊆ đ??ľ. iii) Neutrosophic crisp semi- closed set iff đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ??ľ)) ⊆ đ??ľ. iv) Neutrosophic crisp đ?›˝- closed set iff đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??ľ)) ⊆đ??ľ. One can easily show that, the complement of a neutrosophic crisp (đ?›ź, pre, semi, đ?›˝)- open set is a neutrosophic crisp (đ?›ź, pre, semi, đ?›˝)- closed set,respectively.
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Remark 3.3 For the class consisting of exactly all a đ?‘ đ??śđ?›ź- structure and đ?‘ đ??śđ?›˝- structure, evidently, đ?‘ đ??śđ?›¤âŠ† đ?‘ đ??śđ?›¤đ?›źâŠ† đ?‘ đ??śđ?›¤đ?›˝ . We notice that every non-empty đ?‘ đ??śđ?›˝- open has đ?‘ đ??śđ?›ź-open non-empty interior. If all neutrosophic crisp sets the family {đ??ľi}i∊I, are đ?‘ đ??ś đ?›˝- open sets, then Proposition 3.4 Consider, {âˆŞđ??ľi}i∊I, is a family of đ?‘ đ??śđ?›˝- open sets, then {âˆŞđ??ľi}i∊I⊂đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??ľi))⊂đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??ľi)) , that is A đ?‘ đ??śđ?›˝- structure is a neutrosophic closed with respect to arbitrary neutrosophic crisp unions . We shall now characterize đ?‘ đ??śđ?›¤đ?›ź in terms ofđ?‘ đ??śđ?›¤đ?›˝ . Definition 3.5 Let (đ?‘‹, đ?›¤) be a đ?‘ đ??śđ?‘‡đ?‘† and đ??´ = ⌊đ??´1, đ??´2, đ??´3âŒŞ be a đ?‘ đ??śđ?‘† in đ?‘‹, then: đ?‘ đ??śđ?‘?đ?‘™đ?›ź(đ??´) = â‹‚{ G:G⊇đ??´ and G is đ?‘ đ??śđ?›ź-closed} đ?‘ đ??śđ?‘–đ?‘›đ?‘Ąđ?›ź (đ??´) = ⋃{G:G⊆đ??´ and G is đ?‘ đ??śđ?›ź- open} đ?‘ đ??śđ?‘?đ?‘™ pre (đ??´) = â‹‚{ G:G⊇đ??´ and G is đ?‘ đ??śpre-closed} đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą pre (đ??´) = ⋃{G:G⊆đ??´ and G is đ?‘ đ??śpre- open} Definition 3.6 đ?‘ đ??śđ?‘?đ?‘™ semi (đ??´) = â‹‚{ G:G⊇đ??´ and G is đ?‘ đ??śsemi-closed} đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą semi (đ??´) = ⋃{G:G⊆đ??´ and G is đ?‘ đ??śsemi- open} đ?‘ đ??śđ?‘?đ?‘™đ?›˝(đ??´) = â‹‚{ G:G⊇đ??´ and G is đ?‘ đ??śđ?›˝-closed} đ?‘ đ??śđ?‘–đ?‘›đ?‘Ąđ?›˝(đ??´) = ⋃{G:G⊆đ??´ and G is đ?‘ đ??śđ?›˝- open} Theorem 3.7 Let (đ?‘‹, đ?›¤) be a đ?‘ đ??śđ?‘‡đ?‘†. đ?‘ đ??śđ?›¤đ?›ź Consists of exactly those đ?‘ đ??śđ?‘†đ??´ for which đ??´âˆŠđ??ľâˆˆ đ?‘ đ??śđ?›¤đ?›˝ for đ??ľâˆŠđ?‘ đ??śđ?›¤đ?›˝. Proof Let đ??´âˆˆ đ?‘ đ??śđ?›¤đ?›ź, đ??ľâˆˆ đ?‘ đ??śđ?›¤đ?›˝, đ?‘ƒâˆˆđ??´âˆŠđ??ľ and đ?‘ˆ be a neutrosophic crisp neighborhood (for short đ?‘ đ??śđ?‘›đ?‘?đ?‘‘) of p. Clearly đ?‘ˆ ∊ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)), too is a neutrosophic crisp open neighborhood of đ?‘ƒ, so đ?‘‰=(đ?‘ˆâˆŠ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)))) ∊đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??ľ) is non-empty . Since đ?‘‰âŠ‚đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)) this implies (đ?‘ˆâˆŠđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´) ∊đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??ľ) =đ?‘‰âˆŠđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´) =∅đ?‘ . It follows that Conversely, đ??´âˆŠđ??ľâŠ‚ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´) ∊ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??ľ))= đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´âˆŠđ??ľ)) i.e. đ??´ ∊đ??ľâˆˆ đ?‘ đ??śđ?›¤đ?›˝. Let đ??´âˆŠđ??ľâˆˆ đ?‘ đ??śđ?›¤đ?›˝for all đ??ľâˆˆ đ?‘ đ??śđ?›¤đ?›˝. then in particular đ??´âˆˆ đ?‘ đ??śđ?›¤đ?›˝. Assume that đ?‘ƒâˆˆđ??´âˆŠ(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ??´)∊(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)))c. Then đ?‘ƒâˆˆ đ?‘ đ??śđ?‘?đ?‘™(đ??ľ), where (đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)))c Clearly {đ?‘ƒ}âˆŞđ??ľâˆˆđ?‘ đ??śđ?›¤đ?›˝ and consequently đ??´âˆŠ{{đ?‘ƒ}âˆŞđ??ľ}∈đ?‘ đ??śđ?›¤đ?›˝. But đ??´âˆŠ{{đ?‘ƒ}âˆŞđ??ľ}={đ?‘ƒ}. Hence {đ?‘ƒ} is a neutrosophic crisp open. đ?‘ƒâˆŠ(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´))) implies đ?‘ƒâˆŠ)đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)()), contrary to assumption. Thus đ?‘ƒâˆŠđ??´ implies đ?‘ƒâˆŠ(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)) and đ??´âˆˆ đ?‘ đ??śđ?›¤đ?›ź. Thus we have found that đ?‘ đ??śđ?›¤đ?›ź is complete determined by đ?‘ đ??śđ?›¤đ?›˝ i.e. all neutrosophic crisp topologies with the same đ?‘ đ??śđ?›˝- structure also determined the same đ?‘ đ??śđ?›ź-structure, explicitly given Theorem 3.1. We shall prove that conversely all neutrosophic crisp topologies with the same đ?‘ đ??śđ?›źstructure, so that đ?‘ đ??śđ?›¤đ?›˝, is completely determined by đ?‘ đ??śđ?›¤đ?›ź Theorem 3.8 Every đ?‘ đ??śđ?›ź-structure is a đ?‘ đ??śđ?›¤.
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New Trends in Neutrosophic Theory and Applications
Proof đ?‘ đ??śđ?›¤đ?›˝ 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 đ??´ for which đ??´âˆŠđ??ľâˆˆđ?‘ đ??śđ?›¤đ?›˝ for all đ??ľâˆˆđ?‘ đ??śđ?›¤đ?›˝ constitutes a neutrosophic crisp topology, hence the theorem. We may now characterize đ?‘ đ??śđ?›¤đ?›˝, in terms of đ?‘ đ??śđ?›¤đ?›ź in the following way. Proposition 3.9 Let (đ?‘‹, đ?›¤) be a đ?‘ đ??śđ?‘‡đ?‘†. Then đ?‘ đ??śđ?›¤đ?›˝ = đ?‘ đ??śđ?›¤đ?›źđ?›˝ and hence đ?‘ đ??śđ?›ź -equivalent topologies determine the same đ?‘ đ??śđ?›˝ -structure. Proof Let đ?‘ đ??śđ?›źâ€“đ?‘?đ?‘™ andđ?›źâ€“đ?—‚đ?—‡t denote neutrosophic closure and Neutrosophic crisp interior with respect to đ?‘ đ??śđ?›¤đ?›ź. If đ?‘ƒâˆˆđ??ľâˆˆđ?‘ đ??śđ?›¤đ?›˝ and đ?‘ƒâˆˆ đ??ľâˆˆ đ?‘ đ??śđ?›¤đ?›ź, then (đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)))∊đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??ľ))≠∅đ?‘ . Since (đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´))) is a crisp neutrosophic neighbor-hood of point p, so certainly đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??ľ) meets đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)) and therefore (big neutrosophic open) meets đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´) , proving đ??´âˆŠđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??ľ) ≠∅đ?‘ . This means đ??ľâŠ‚ đ?‘ đ??śđ?›źđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??ľ)) .i.e. đ??ľâˆˆđ?‘ đ??śđ?›¤đ?›źđ?›˝ on the other hand let đ??´âˆˆđ?‘ đ??śđ?›¤đ?›źđ?›˝ , đ?‘ƒâˆˆ đ??´. and đ?‘ƒâˆˆ đ?‘‰âˆˆđ?‘ đ??śđ?›¤. As đ?‘‰âˆˆđ?‘ đ??śđ?›¤đ?›ź, and đ?‘ƒâˆˆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)), we have đ?‘‰âˆŠđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)≠∅N and there exist a neutrosophic trip set đ?‘Šâˆˆ đ?›¤ such that đ?‘ŠâŠ‚đ?‘‰âˆŠđ?‘ đ??śđ?›źđ?‘–đ?‘›đ?‘Ą(đ??´)⊂đ??´. In other words đ?‘‰âˆŠ (đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´))≠∅đ?‘ and đ?‘ƒâˆˆđ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)). Thus we have verified đ?‘ đ??śđ?›¤đ?›źđ?›˝ ⊂đ?‘ đ??śđ?›¤đ?›ź, and the proof is complete combining Theorem 3.1 and Proposition 3.1. and we get đ?‘ đ??śđ?›¤đ?›źđ?›ź = đ?‘ đ??śđ?›¤đ?›ź. Corollary 3.10 A neutrosophic crisp topology đ?‘ đ??śđ?›¤is a đ?‘ đ??śđ?›ź - topology iff đ?‘ đ??śđ?›¤ = đ?‘ đ??śđ?›¤đ?›ź. Evidently đ?‘ đ??śđ?›¤đ?›˝ is a neutrosophic crisp topology iffđ?‘ đ??śđ?›¤đ?›ź = đ?‘ đ??śđ?›¤đ?›˝. In this case đ?‘ đ??śđ?›¤đ?›˝đ?›˝ = đ?‘ đ??śđ?›¤đ?›źđ?›˝ = đ?‘ đ??śđ?›¤đ?›˝. Corollary 3.11 đ?‘ đ??śđ?›˝-Structure đ??ľ is a neutrosophic crisp topology, then đ??ľ= đ??ľđ?›ź= đ??ľđ?›˝. We proceed to give some results on the neutrosophic structure of neutrosophic crisp đ?‘ đ??śđ?›źâ€“ topology Proposition 3.12 The đ?‘ đ??śđ?›ź-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 a neutrosophic crisp nowhere dense set. If đ??´âˆŠ đ?‘ đ??śđ?›¤đ?›ź we have đ??´= đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´))) ∊(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´))∊đ??´c)c, where (đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´))∊đ??´c) clearly is neutrosophic crisp nowhere dense set, we easily see that đ??ľâŠ‚ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)) and consequently đ??´âŠ‚đ??ľâŠ‚đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??´)) so the proof is complete. Corollary 3.13 A neutrosophic crisp topology is a đ?‘ đ??śđ?›ź- topology iff all neutrosophic crisp nowhere dense sets are neutrosophic crisp closed. For a neutrosophic crisp đ?‘ đ??śđ?›ź-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 3.14 Neutrosophic crisp topologies which are đ?‘ đ??śđ?›ź- equivalent, determine the same class of neutrosophic crisp nowhere dense sets.
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Proposition 3.16 If a đ?‘ đ??śđ?›ź -Structure đ??ľ, is a neutrosophic crisp topology, then all neutrosophic crisp topologies đ?›¤ for which đ?›¤đ?›˝ = đ??ľ are neutrosophic crisp extremely disconnected. In particular: Either all or none of the neutrosophic crisp topologies of a đ?‘ đ??śđ?›ź – class are extremely disconnected. Proof Let đ?›¤đ?›˝ = đ??ľ, and suppose there is đ??´âˆŠđ?›¤ such that đ?‘ đ??śđ?‘?đ?‘™(đ??´)∉đ?›¤. Let đ?‘ƒâˆŠđ?‘ đ??śđ?‘?đ?‘™(đ??´)∊đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ??´))c with đ??ľ = {đ?‘ƒ} âˆŞđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ??´)), đ?› = đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ??´))c We have {đ?‘ƒ} ⊂ đ?›= (đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(A))c= đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?›)), {đ?‘ƒ}⊂ đ?‘ đ??śđ?‘?đ?‘™(đ??´) = đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ??´))⊂ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??ľ)). Hence both đ??ľ and đ?› are in đ?›¤đ?›˝. The intersection đ??ľâˆŠ đ?›= {đ?‘ƒ} is not neutrosophic crisp open, since đ?‘ƒâˆˆ đ?‘ đ??śđ?‘?đ?‘™(đ??´)∊đ?›c hence not đ?‘ đ??śđ?›˝- open. So, đ?›¤đ?›˝=đ??ľ is not a neutrosophic crisp topology. Now suppose đ??ľ is not a topology, and đ?›¤đ?›˝=đ??ľ There is a đ??ľâˆˆ đ?›¤đ?›˝ such that đ??ľâˆ‰đ?›¤đ?›ź. Assume that đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??ľ)) ∊đ?›¤. Then đ??ľâŠ‚đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??ľ))=đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??ľ)) i.e. đ??ľâˆŠ đ?›¤đ?›ź, contrary to assumption. Thus we have produced an open set whose closure is not open, which completes the proof. Corollary 3.17 A neutrosophic crisp topology đ?›¤ is a neutrosophic crisp extremally disconnected if and only if đ?›¤đ?›˝ is a neutrosophic crisp topology. Remark 3.18 The following diagram represents the relation between neutrosophic crisp nearly open sets: đ?‘ đ??śpreopen set đ?‘ đ??śopen set
đ?‘ đ??śđ?›˝open set
đ?‘ đ??śđ?›źopen set đ?‘ đ??śsemiopen set
4. Neutrosophic Crisp Continuity We, introduce and study of neutrosophic crisp continuous function and we obtain some characterizations of neutrosophic continuity. Here come the basic definitions first: Definition 4.1 Let (đ?‘‹, đ?›¤) be a đ?‘ đ??śđ?‘‡đ?‘† and đ??´ =⌊đ??´1,đ??´2,đ??´3âŒŞ be a đ?‘ đ??śđ?‘† in đ?‘‹, and đ?‘“: đ?‘‹â&#x;śđ?‘‹ then: 1) If đ?‘“đ?‘ đ??śđ?›ź-continuous â&#x;š inverse image of đ?‘ đ??śđ?›ź open set is đ?‘ đ??śđ?›ź- open set 2) If đ?‘“ đ?‘ đ??śpre-continuous â&#x;š inverse image of đ?‘ đ??śpre-open set is đ?‘ đ??śpre- open set 3) If đ?‘“ đ?‘ đ??śsemi-continuous â&#x;š inverse image of đ?‘ đ??śsemi-open set is đ?‘ đ??śsemi- open set 4) If đ?‘“ đ?‘ đ??śđ?›˝-continuous â&#x;š inverse image of đ?‘ đ??śđ?›˝-open set is đ?‘ đ??śđ?›˝- open set Definition 4.2 The following was given in [24] (a) If A  A1, A2 , A3 is a NCS in X, then the neutrosophic crisp image of A under f ,
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denoted by f ( A), is the a NCS in Y defined by
f ( A)  f ( A1 ), f ( A2 ), f ( A3 ) .
-1
(b) If đ?‘“ is a bijective map then đ?‘“ : đ?‘Œâ&#x;śđ?‘‹ 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 đ?‘ đ??śđ?‘†in X defined by f ď€1 ( B)  f ď€1 ( B1 ), f ď€1 ( B2 ), f ď€1 ( B3 ) . Definition 4.3 Let (đ?‘‹, đ?›¤1) , and (đ?‘Œ, đ?›¤2) be two đ?‘ đ??śđ?‘‡đ?‘†đ?‘ , and let đ?‘“: đ?‘‹â&#x;śđ?‘Œ be a function. Then đ?‘“ is said to be continuous if đ?‘“ the preimage of each đ?‘ đ??śđ?‘† in đ?›¤2 is a đ?‘ đ??śđ?‘† in đ?›¤1. Definition 4.4 Let (đ?‘‹, đ?›¤1) , and (đ?‘Œ, đ?›¤2) be two đ?‘ đ??śđ?‘‡đ?‘†đ?‘ and let đ?‘“: đ?‘‹â&#x;śđ?‘Œ be a function. Then đ?‘“ is said to be open iff the image of each đ?‘ đ??śđ?‘† in đ?›¤1, is a đ?‘ đ??śđ?‘† in đ?›¤2. Proposition4.5 Let X , ď ‡o and Y ,ď š o  be two đ?‘ đ??śđ?‘‡đ?‘†đ?‘ . If f : X ď‚Ž Y is continuous in the usual sense, then in this case, f is continuous in the sense of Definition 4.3 too. Proof Here we consider the đ?‘ đ??śđ?‘‡đ?‘†đ?‘ on X and Y, respectively, as follows: ď ‡1  ď ť G,ď Ś , G c : G ďƒŽ ď ‡o ď ˝ and ď ‡2  ď ť H , ď Ś , H c : H ďƒŽ ď ™o ď ˝, In this case we have, for each H , ď Ś , H c ďƒŽ ď ‡2 , H ďƒŽ ď ™o , f
ď€1
H ,ď Ś, H c  f
ď€1
( H ), f
ď€1
(ď Ś ), f
ď€1
(H c ) 
f
ď€1
H ,ď Ś, ( f
ď€1
( H ))c ďƒŽ ď ‡1
.
Now we obtain some characterizations of neutrosophic continuity. Proposition 4.6 Let đ?‘“: (đ?‘‹, đ?›¤1) â&#x;ś(đ?‘Œ, đ?›¤2). Then f is neutrosophic continuous iff the preimage of each neutrosophic crisp closed set (đ?‘ đ??śđ??śđ?‘†) in đ?›¤2 is a đ?‘ đ??śđ??śđ?‘† in đ?›¤1. Proposition 4.7 The following are equivalent to each other: (a) đ?‘“: (đ?‘‹, đ?›¤1)â&#x;ś(đ?‘Œ, đ?›¤2) is neutrosophic continuous. (b) đ?‘“-1(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ??ľ) ⊆ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘“-1(đ??ľ))) for each đ?‘ đ??śđ?‘†đ??ľin đ?‘Œ. (c) (đ?‘ đ??śđ?‘?đ?‘™ đ?‘“-1(đ??ľ)) ⊆đ?‘“-1(đ?‘ đ??śđ?‘?đ?‘™(đ??ľ)). for each đ?‘ đ??śđ?‘†đ??ľ in đ?‘Œ. Corollary 4.8 Consider ( X , ď ‡1 ) and Y , ď ‡2 to be two đ?‘ đ??śđ?‘‡đ?‘†đ?‘ , and let f : X ď‚Ž Y be a function. if ď ‡1  ď ťf ď€1 ( H ) : H ďƒŽ ď ‡2 ď ˝ . Then ď ‡1 will be the coarsest NCT on X which makes the function f : X ď‚Ž Y continuous. One may call it the initial neutrosophic crisp topology with respect to f .
5. Neutrosophic Crisp Compact Space
First we present the basic concepts: Definition5.1 Let  X , ď ‡ď€Š be an NCTS. (a) If a family ď ť Gi1 , Gi2 , Gi3 : i ďƒŽ J ď ˝ of NCOSs in X satisfies the condition ďƒˆ
(b)
ď ťG
i1
, Gi2 , Gi3
ď ˝
: i ďƒŽ J  X N , then
it is called an neutrosophic open cover of X.
A finite subfamily of an open cover
ď ťG
i1 , Gi2 , Gi3
:iďƒŽ J
ď ˝ on X, which is also a
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neutrosophic open cover of X , is called a neutrosophic crisp finite open subcover. Definition5.2 A neutrosophic crisp set A A1 , A2 , A3
in a NCTS X , is called neutrosophic crisp
compact iff every neutrosophic crisp open cover of A has a finite neutrosophic crisp open subcover. Definition5.3 A family Ki1 , Ki2 , Ki3
of neutrosophic crisp compact sets in X satisfies the finite intersection property (FIP ) iff every finite subfamily K , K , K : i 1,2,..., n of the family satisfies the condition K , K , K : i 1,2,..., n . :iJ
i1
i1
i2
i3
i2
i3
N
Definition5.4 A NCTS X , is called neutrosophic crisp compact iff each neutrosophic crisp open cover of X has a finite open subcover. Corollary5.5 A NCTS X , is a neutrosophic crisp compact iff every family Gi , Gi , Gi : i J of 1
2
3
neutrosophic crisp compact sets in X having the finite intersection properties has nonempty intersection. Corollary5.6 Let X , 1 , Y , 2 be NCTSs and f : X Y be a continuous surjection. If X , 1 is a neutrosophic crisp compact, then so is Y , 2 . Definition5.7 If a family Gi1 , Gi2 , Gi3 : i J of neutrosophic crisp compact sets in X satisfies the condition A Gi1 , Gi2 , Gi3
:iJ
,
then it is called a neutrosophic crisp open cover of A.
Let’s consider a finite subfamily of a neutrosophic crisp open subcover of Gi1 , Gi2 , Gi3 : i J . Corollary5.8 Let X , 1 , Y , 2 be NCTSs and f : X Y be a continuous surjection. If A is a neutrosophic crisp compact in X , 1 , then so is f (A) in Y , 2 .
6. Conclusion
In this paper, we presented a generalization of the neutrosophic topological space. The basic definitions of the neutrosophic crisp topological space and the neutrosophic crisp compact space with some of their characterizations were deduced. Furthermore, we constructed a neutrosophic crisp continuous function, with a study of a number its properties.
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