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International Journal of Contemporary Mathematical Sciences Vol. 10, 2015, no. 1, 1 - 11 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2015.4545 Nano PS -Open Sets and Nano PS -Continuity Mohammed M. Khalaf and Kamal N. Nimer Department of Mathematics Faculty of Science, Al-Azahar University, Asuiat, Egypt Current Address: Al-Zulfi College of Science, Majmaah University Saudi Arabia Copyright © 2014 Mohammed M. Khalaf and Kamal N. Nimer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The purpose of this paper is to introduce a new class of sets called Nano ππ -open sets in Nano-topological spaces. This class of sets lies strictly between the classes of Nano πΏ- open and Nano -preopen sets. By using this sets we defined and study the concept of Nano ππ -continuous function Keywords: Nano topological space, Nano PS -open, Nano PS -continuous function 1. Introduction Lellis Thivagar [1] introduced a nano topological space with respect to a subset X of an universe which is defined in terms of lower and upper approximations of X. The elements of a nano topological space are called the nano-open sets. He has also studied nano closure and nano interior of a set. Njastad [5], Levine [2] and Mashhour [3] respectively introduced the notions of - open, semi-open and pre-open sets. Since then these concepts have been widely investigated. It was made clear that each -open set is semi-open and pre-open but the converse of each is not true. Njastad has shown that the family of - open sets is a topology on X satisfying. The families SO (π, π) of all semiβopen sets and PO(π, π) of all preopen sets in (π, π) are not topologies. It was proved that both SO(π, π) and PO(π, π) are closed under arbitrary unions but not under finite intersection. In this paper we introduce a new class of sets called Nano PS -open sets in Nanotopological spaces and study the concept of Nano PS -continuous function. 2 Mohammed M. Khalaf and Kamal N. Nimer Throughout this paper, a space ππ (π) is a topology on U called the nano topology on with respect to X . We recall the following definitions, notions and characterizations. Definition 1.1 [1] Let π be the universe, π be an equivalence relation on π and ππ (π) = {π, β , πΏπ (π), ππ (π), π΅π (π)} where X ο U . If ππ (π) satisfies the following axioms: (i) U , ο¦ οο΄ R ( X ) (ii) The union of the elements of any subcollection of ππ (π) is in ππ (π) (iii) The intersection of the elements of any finite subcollection of ππ (π) is in ππ (π) Then ππ (π) is a topology on U called the nano topology on with respect to X . We call as the NANO TOP space The elements of ππ (π) are called as nano-open sets. Definition 1.2 [1] If (π, ππ (π)) is NANO TOP space with respect to X where X οο U and if A οο U , then the nano interior of A is defined as the union of all nano-open subsets of A and it is denoted by N int(A) . That is, N int(A) is the largest nano-open subset of A . The nano closure of A is defined as the intersection of all nano closed sets containing A and it is denoted by Ncl (A) . That is, Ncl (A) is the smallest nano closed set containing A . Definition 1.3 [1] A NANO TOP space (π, ππ (π)) is said to be extremally disconnected, if the nano closure of each nano-open set is nano-open. Definition 1.4 Let ππ (π) be a NANO TOP on U with respect to X . A subset A of U is said to be nanosemi-open [2] (by short N ππ -open) (resp. nanopreopen [3] (by short N π-open) , nano ο‘ -open [4] (by short N πΌ-open), nano- ο’ -open (by short N π½-open) or nanosemi-preopen (by short N ππ-open), nano- b -open(by short N π-open) ) if A ο cl (int ( A)) (resp. A ο int (cl ( A)), A ο int (cl (int ( A))), A ο cl (int (cl ( A))), A ο cl (int ( A)) ο int (cl ( A)) ). The family of all nano semi-open (resp. nanopreopen, nano- ο‘ -open, nano- ο’ -open, nano- b -open) sets in X is denoted by NSO(U , X ) (resp. NPO(U , X ) , Nο‘ (U , X ) , Nο’ (U , X ) , NbO(U , X ) The intersection of all NPC (U , X ) (resp. NSC (U , X ) and Nο‘C (U , X ) ) sets of U containing a subset A is called the NPclosure(U , X ) (resp. NSclosure(U , X ) and Nο‘ closure(U , X ) ) of A . The union of all NPO(U , X ) (resp. NSO(U , X ) ) sets of U contained in A is called the NPinterior (U , X ) (resp. NSinterior (U , X ) ) of A Nano ππ -open sets and nano ππ -continuity 3 Definition 1.5 Let ππ (π) be a NANO TOP on U with respect to X . A subset A of U is said to be 1- A subset A of U is said to be nanoregular-open (by short N π -open) if A ο½ N (int (cl ( A))) 2- Nano πΏ -open (by short Nο€O(U , X ) ) if for each x ο A , there exists an Nro(U , X ) G such that x ο G ο N (int (cl (G))) ο A 3- Nano π -open (by short Nο±O(U , X ) ) if for each x ο A , there exists an NO(U , X ) G such that x ο G ο N (cl (G))) ο A The intersection of all N πΏ-closed sets of U containing A is called the Nano πΏ -closure (by short ππππΏ (π΄)). The union of all N πΏ-open sets of U containing A is called the Nano πΏ -interior (by short ππππ‘πΏ (π΄)). 4- is Nano-semi dense set (by short NS dense(U , X ) ) if N (cl (int (clA))) ο½ A Definition 1.6 [1] Let ππ (π) be a NANO TOP on U with respect to X . A subset A of U is said to be nano- ο€ -semi-open (resp. nano- ο€ -preopen ) if A ο cl ( Intο€ ( A)) (resp. A ο int (clο€ ( A)) ). Definition 1.7 Let ππ (π) be a NANO TOP on U with respect to X . A subset A of U is said to be Nano ππ -open (by N ππ -open ) if for each x ο A ο NPO(U , X ) , there exists a Nano semi-closed (by short NSC (U , X ) ) set F such that x ο F ο A . The family of all N ππ -open subsets of ππ (π) is denoted by NPS O(U , X ) Proposition 1.1 A subset A of U is said to be N ππ -open if and only if A is N ππ-open and union of NSC (U , X ) sets Proof. Obvious Proposition 1.3 If A and B are N ππ -open subsets of is NANO TOP space (π, ππ (π)) and if the family of all N ππ-open sets in U forms a NANO TOP ππ (π) of U , then A β© B is N ππ -open set and hence the family of N ππ -open sets forms a NANO TOP ππ (π) of U Definition 1.8 A space (π, ππ (π)) is semi ο T1 if for each pair of distinct points x, y οU , there exists a pair of NSO(U , X ) sets containing x but not y and the other containing y but not x Theorem 1.1 A space (π, ππ (π)) singleton set {x} is NSC (U , X ) set is semi ο T1 if for any point x οU , the 4 Mohammed M. Khalaf and Kamal N. Nimer Theorem 1.2 If a space (π, ππ (π)) is semi ο T1 , then NPSO(U , X ) ο½ NPO(U , X ) Proof. Let A ο U and Aο NPO(U , X ) . If A ο½ ο¦ , then Aο NPO(U , X ) If A οΉ ο¦ , then for each x ο A . Since a space (U ,ο΄ R ( X )) is semi ο T1 , then by Theorem 1.1, {x} is NSC (U , X ) set and hence x ο{x} ο A . Therefore Aο NPSO(U , X ) . Hence NPO(U , X ) ο NPSO(U , X ) ). But NPSO(U , X ) ο NPO(U , X ) in general. Therefore, NPSO(U , X ) ο½ NPO(U , X ) Definition 1.9 A space (U ,ο΄ R ( X ) is Nano s-regular (by short N π π(π, π)) if for each x οU and each NO(U , X ) set G containing x , there exists a NSO(U , X ) set H such that x ο H ο NSclosure( H ) ο G Lemma 1.1 For any subset A of a space (π, ππ (π)), then Aο NPO(U , X ) if and only if NSclosure(A) ο½ N (int(cl ( A))) Theorem 1.3 For any subset A of a space (π, ππ (π)). If A ο Nο€O(U , X ) , then Aο NPSO(U , X ) Proof. Let A ο Nο€O(U , X ) . If A ο½ ο¦ , then Aο NPSO(U , X ) . If A οΉ ο¦ , then for each x ο A there exists an NO(U , X ) set B such that x ο B ο N (cl (int ( B))) ο A . Since B is NO(U , X ) and hence B ο NPO(U , X ) . By Lemma 1.1 NSclosure((B)) ο½ N (int(cl ( B))) . Then x ο B ο NSclosure(( B)) ο A implies that x ο NSclosure(( B)) ο A and NSclosure(( B)) ο NSC (U , X ) Since A ο Nο€O(U , X ) and Nο€O(U , X ) ο NPO(U , X ) in general, Aο NPO(U , X ) Therefore Aο NPSO(U , X ) Remark 1.1 1. Each N (cl ( NO(U , X )) and hence each Nro(U , X ) set is N ππ -open 2. Each Nο±O(U , X ) set is N ππ -open Lemma 1.2 A space (π, ππ (π)) is hyper connected if and only if Nro(U , X ) ο½ {U , ο¦} Theorem 2.4 A space (π, ππ (π) is hyper connected if and only if NPSO(U , X ) ο½ {U , ο¦} Nano ππ -open sets and nano ππ -continuity 5 Proof. Necessity. Suppose, if possible, that a subset Aο NPSO(U , X ) If A ο½ ο¦ , then Aο NPSO(U , X ) . If A οΉ ο¦ , then for each x ο A there exists an F ο NSC (U , X ) such that x ο F ο A . Since F ο NSC (U , X ) , then NSclosure(int (clF )) ο F . Since (U ,ο΄ R ( X ) is hyper connected space, then for every nonempty NPO(U , X ) ο U is NS dense(U , X ) , so N (cl (int (clF ))) ο½ U ο F , this implies that U ο½ F . Since F ο½ U ο A , then U ο½ A Therefore, A must be equal to U , which completes the proof. Sufficiency. Suppose that NPSO(U , X ) ο½ {U , ο¦} . Since Nro(U , X ) subset of NPSO(U , X ) in general, then Nro(U , X ) ο½ {U , ο¦} . By Lemma 1.2, we have (U ,ο΄ R ( X ) is hyper connected. Definition 1.10 A space (π, ππ (π)) is called locally indiscrete if every NO(U , X ) is NC (U , X ) Lemma 1.3 If U is locally indiscrete space, then each NSO(U , X ) of U is NC (U , X ) and hence each NSC (U , X ) of U is NO(U , X ) Theorem 2.5 If a nano topological space (π, ππ (π)) is called locally indiscrete, then NPSO(U , X ) ο½ ο΄ R ( X ) (π, ππ (π)) is locally indiscrete space and let A ο U such that A οο΄ R (U ) . Since every NO(U , X ) is NC (U , X ) , then N (int (clA)) ο½ A this implies that Aο NrO(U , X ) . Hence by Remark 1.1 (1), Aο NPSO(U , X ) . Thus, ο΄ R (U ) ο NPSO(U , X ) . On the other hand, let B ο NPSO(U , X ) , then B ο NPO(U , X ) and for each x ο B , there exists F ο NSC (U , X ) such that x ο F ο B . By Lemma 1.3, F is NO(U , X ) . It follows that B οο΄ R (U ) . Hence NPSO(U , X ) οο΄ R ( X ) . Therefore, Proof Let NPSO(U , X ) ο½ ο΄ R ( X ) The following diagram show relations among NPSO(U , X ) , NPO(U , X ) , NrO(U , X ) , Nο€O(U , X ) , ππ (π)and Nο‘ (U , X ) NrO(U , X ) ο NPSO(U , X ) ο Nο€O(U , X ) ο NPO(U , X ) ο Nο‘ (U , X ) ο ππ (π) 6 Mohammed M. Khalaf and Kamal N. Nimer Remark 1.2 In above Diagram, we have the following notes 1. ππ (π) is not comparable with NPSO(U , X ) 2. Nο‘ (U , X ) is not comparable with NPSO(U , X ) Definition 1.11 A subset B of a space (π, ππ (π)) is called N ππ -closed if U B is N ππ -open set. The family of all N ππ -closed subsets of a NANO TOP space (π, ππ (π)is denoted by NPS C (U , X ) Proposition 1.2 A subset B of a space (π, ππ (π)) and only B is an intersection of NSO(U , X ) is called NPC (U , X ) if Proof. Obvious Corollary 2.1 For any A of a space (π, ππ (π)) . The following conditions are equivalent: 1. 2. 3. 4. 2. A A A A is is is is N closure( NO(U , X )) N ππ -open and NC (U , X ) Nο‘ (U , X ) and NC (U , X ) NPO(U , X ) and NC (U , X ) N π·πΊ -Continuous Functions Definition 2.1 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and Y respectively. is called N π·πΊ -continuous at a point x οU if for each x οU and each open set B of a space (π, π β π (π)) Containing f (x) , there exists a NPS O(U , X ) A of a space (π, ππ (π)) containing x such that f ( A) ο B . If f is N π·πΊ -continuous at every point of U , then it is called N π·πΊ -continuous Definition 2.2 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and Y respectively. is called N -continuous if the inverse image of each NO(U , X ) subset of a space (π, π β π (π)) is NO(U , X ) in (π, ππ (π)) Nano ππ -open sets and nano ππ -continuity 7 Definition 2.3 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and Y respectively. is called NP -continuous ( resp. Nο‘ -continuous, N -perfectly continuous, N -complete continuous, N super-continuous) if the inverse image of each NO(U , X ) subset of A of a space (π, π β π (π)) is NPO(U , X ) ( resp. Nο‘ (U , X ) , N closureO(U , X ) , NRO(U , X ) , Nο€O(U , X ) ) in (π, ππ (π)) Definition 2.4 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and Y respectively. is called N -Almost pre continuous ( resp. N -Almost continuous,) if the inverse image of each NRO(U , X ) A of a space (π, π β π (π)) is NPO(U , X ) ( resp. NO(U , X ) )in (π, ππ (π)) Definition 2.5 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and Y respectively. is called Nο€ - continuous ( resp. Nο± -Almost continuous ) if for each x οU and each NO(U , X ) A of a space (π, π β π (π)) containing f (x) , there exists an NO(U , X ) B of a space (π, ππ (π)) containing x such that f ( N (int(clB ))) ο N (int(clA)) ( resp. f ( N (cl ( B)) ο N (cl ( A)) ) Definition 2.6 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and Y respectively. is called N - contra continuous ( resp. N -contra semi continuous ) if the inverse image of every NO(U , X ) subset of (π, π β π (π)) is NC (U , X ) ( resp. NSC (U , X ) ) of a space (π, ππ (π)) Lemma 2.1 Every Ncomplete -continuous and hence every N super-continuous function is N π·πΊ -continuous and every N π·πΊ -continuous function NP -continuous. Proof. Follows directly from their definitions. The following diagram show relations among N -complete continuous, N super-continuous, N π·πΊ -continuous, N -continuous, Nο‘ -continuous and NP -continuous 8 Mohammed M. Khalaf and Kamal N. Nimer N -complete continuous β N super-continuous, β β N -continuous β -continuous Nο‘ N π·πΊ -continuous β NP -continuous Theorem 2.1 For a function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and Y respectively, the following statements are equivalent: 1. π is N π·πΊ -continuous 2. π β1 (π΄) is N π·πΊ -open set in (π, ππ (π)) of each NO(U , X ) A in (π, π β π (π)) 3. π β1 (πΉ) is N π·πΊ -closed set in (π, ππ (π)) of each NC (U , X ) F in (π, π β π (π)) 4. For each NO(U , X ) A in (π, ππ (π)), π( N π·πΊ (πππ(π΄))) β π(ππ(π(π΄)) 5. For each NC (U , X ) F in (π, π β π (π)), ππ·πΊ (πππ(π β1 (πΉ))) β π β1 (πππ(πΉ) 6. For each NO(U , X ) B in (π, π β π (π)), π β1 (ππππ‘(π΅)) β πππ (ππππ‘(π β1 π΅)) Proof. Straight forward. Theorem 2.2 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and Y respectively, is N π·πΊ -continuous if and only if π is NP -continuous and for each x οU and each NO(U , X ) B of V containing f (x) , there exists a NSC (U , X ) F in U containing x such that f ( F ) ο B Proof. Necessity. Let x οU and let V be any NO(U , X ) B containing f (x) . Since π is N π·πΊ -continuous, there exists a N ππ -open set A of U containing x such that f ( A) ο B . Since A is N ππ -open set. Then for each x οU , there exists a NSC (U , X ) F in U such that x ο F ο U Therefore, we have f ( F ) ο B . And also since π is N π·πΊ -continuous,. Then π is NP -continuous Sufficiency. let V be any NO(U , X ) V . We have to show that π β1 (π΅) is N ππ -open set in U . Since π is NP -continuous , thenπ β1 (π΅) is NPO(U , X ) in U . Let π₯ β π β1 (π΅). Then f ( x) ο B By hypothesis, there exists a NSC (U , X ) Nano ππ -open sets and nano ππ -continuity 9 F of U containing x such that f ( F ) ο B . Which implies that x ο F ο f ο1 (B) Therefore, π β1 (π΅) is N ππ -open set in U . Hence by Theorem 2.1, π is N π·πΊ -continuous Theorem 2.3 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and Y respectively, is N π·πΊ -continuous and NO(U , X ) function and B is NPO(U , X ) in V , π β1 (π΅) is NPO(U , X ) in U Proof. Let B be a NPO (U , X ) in V , then there exists an N -open set G in V ο1 ο1 ο1 , such that B ο G ο N (cl ( B)) . Then f ( B) ο f (G) ο f ( N ((cl ( B))) . Since f is N -continuous function, f ο1 (G) is NO(U , X ) in U and since f is an NO(U , X ) function, then f ο1 (cl ( B)) ο N (cl ( f ο1 (G)) ). Hence we obtain that f ο1 ( B) ο f ο1 (G) ο N ο cl ( f ο1 (( B))) ). Therefore, π β1 (π΅) is NPO(U , X ) in U Theorem 2.4 If π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and Y respectively, is N -continuous and NO(U , X ) function and F is a NSC (U , X ) of V , then f ο1 ( F ) is a NSC (U , X ) of U Proof. Let be a NSC (U , X ) of V , then there exists a NC (U , X ) ο1 ο1 E in V ο1 such that N (int(E) ο F ο E . Then f ( N (int(E) ο f ( F ) ο f ( E) ). Since f is N -continuous function, then f ο1 ( F ) is a NC (U , X ) in U and since f is an NO(U , X ) ο1 ο1 function, then N (int( f ( E)) ο f ( N (int(F )) ). ο1 ο1 ο1 Hence we obtain that N ο int( f ( E )) ο f ( F ) ο f ( E) ). Therefore, f ο1 ( F ) is NSC (U , X ) of U Theorem 2.4 A function π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and Y respectively, is a N -continuous and NO(U , X ) function and B is a N ππ -open set of V , then π β1 (π΅) is N ππ -open set in U Proof. Let B is a N ππ -open set of V , then B is NPO(U , X ) in V and B ο½ οFi is where Fi is NSC (U , X ) of V for each i . Then π β1 (π΅) = π β1 (βͺ πΉπ ) =βͺ π β1 (πΉπ ), where Fi is NSC (U , X ) of V for each i . Since f is a N -continuous and NO(U , X ) function Then by Theorem 2.3 π β1 (π΅) is 10 Mohammed M. Khalaf and Kamal N. Nimer NPO(U , X ) in U and by Theorem 2.4 π β1 (βͺ πΉπ ) NSC (U , X ) of U for each i . Hence by Proposition 1.1, π β1 (π΅) is N ππ -open set in U Corollary 2.1 If π: (π, ππ (π)) β (π, π β π (π)) . with respect to X and Y respectively, is N -continuous and NO(U , X ) function and F is a N ππ -closed set of V , then π β1 (πΉ) is N ππ -closed set in U Theorem 2.5 The following statements are equivalent for a function π: (π, ππ (π)) β (π, π β π (π)). With respect to X and Y respectively, 1. f 2. f 3. f 4. f is is is is N -perfectly continuous N π·πΊ -continuous and N - contra continuous Nο‘ -continuous and N - contra continuous NP -continuous and N - contra continuous Proof. This is an immediate consequence of Corollary 2.1 Theorem 2.6 Let π: (π, ππ (π)) β (π, π β π (π)). with respect to X and Y respectively, be a function. Let πβπ β be any basis for (π, π β π (π)) . Then f is N π·πΊ -continuous if and only if for each πβ β π β1 (πβ β ) is π π N π·πΊ -open subsets of U Proof. Necessity. Suppose that f is N π·πΊ -continuous. Then since each πβπ β N π·πΊ -open subsets of V and f is N π·πΊ -continuous. Therefore, by Theorem 2.1, π β1 (πβπ β ) is N π·πΊ -open subsets of U Sufficiency. Let B is a N ππ -open set of V . Then B = {βͺ (πβ β ) : π β πΌ} π π where every πβ β is a member of πβπ β and I is a suitable index set. It follows π π that f ο1 ( B) ο½ π β1 {βͺ (πβ π πβ ) : π β πΌ} =βͺ π β1 {(πβ β ) : π β πΌ} }. Since π π ο1 π β1 {(πβ β )} is N π·πΊ -open subsets of U for each β πΌ . Hence f ( B) is π π Nano ππ -open sets and nano ππ -continuity 11 the union of a family of N π·πΊ -open subsets of U and hence is N π·πΊ -open subsets of U Therefore, by Theorem 2.1, f is N π·πΊ -continuous References [1] M. Lellis Thivagar and Carmel Richard, Note on nano topological spaces, Communicated. [2] N. Levine, Semiβopen sets and semicontinuity in topological spaces, Amer. Math. Monthly 70 (1963), 36 β 41. http://dx.doi.org/10.2307/2312781 [3] A. S. Mashhour, M. E. Abd El-Monsef and S.N. El-Deeb, On pre-topological spaces, Bull.Math. de la Soc. R. S. de Roumanie 28 (76) (1984), 39 β 45. [4] Miguel Caldas, A note on some applications of -open sets, IJMMS, 2 (2003), 125-130 O. Njastad, on some classes of nearly open sets, Pacific J. Math. 15 (1965), 961 β 970. [5] Z. Pawlak, Rough sets, International journal of computer and Information Sciences, 11 (1982), 341 - 356. http://dx.doi.org/10.1007/bf01001956 [6] I. L. Reilly and M. K. Vamanamurthy, On-sets in topological spaces, Tamkang J. Math., 16 (1985), 7 - 11. Received: May 15, 2014; Published: January 3, 2015