Inter national Journal of Pure and Applied Mathematics Volume 113 No. 12 2017, 98 { 106 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu NSP -OPEN SETS AND NSP -CLOSED SETS IN NANO TOPOLOGICAL SPACES D. Saravanakumar1, T. Sathiyanandham2 and V. C. Shalini3 1Department of Mathematics, Kalasalingam University, Krishnankovil, India. saravana [email protected] − 2Department of Mathematics, Kamarajar Government Arts College, Surandai, India. [email protected] 3Department of Mathematics, Vaigai College of Engineering, Madurai, India. shalini [email protected] − Abstract The basic objective of this paper is to introduce and investigate the properties of nano sp-open sets in nano topological spaces. Further we obtain the notions of nano sp-interior, nano sp-closure and study some of their basic ideas. AMS (2010) Subject Classification: 54A05, 54C10, 54B05 Key Words and Phrases: nano sp-open, nano sp-closed, nano sp-interior, nano sp-closure. 1 Introduction and Preliminaries Lellis Thivagar[1] introduced the concept of nano topological space with respect to a subset X of an universe which is defined in terms of lower and upper approxima- tions of X. In this paper, we introduced the concept of nano sp-open which is analogous to nano semi-open sets and introduced the notion of NSP O(U, X) which is the set of all nano sp-open sets in a nano topological space (U, τR(X)). Moreover we defined the concept of nano sp-interior, nano sp-cloure operators and studied some of their essential properties. We recall some basic definition and notions. Let U be the universe, R be an equivalence relation on U and τR(X) = , U, LR(X),UR(X),BR(X) where X U. If τ (X) satisfies the following axioms: (i){∅,U τ (X); (ii) the union} of the elements⊆ of R ∅ ∈ R any subcollection of τR(X) is in τR(X); (iii) the intersection of the elements of any finite subcollection of τR(X) is in τR(X), then τR(X) is a topology on U called the nano topology on U with respect to X. We call (U, τR(X)) is a nano topological space (briefly NTS)[1] and the elements of τR(X) are called as nano open sets and its complements are called the nano closed sets. For A U, nano interior of A[1] is nint(A) = O : O τR(X) and O A and nano closure⊆ of A[1] is ncl(A) = C : X C τ (X∪{) and A ∈ C .A ⊆ } ∩{ − ∈ R ⊆ } subset A of a NTS (U, τR(X)) is said to be nano regular open[1] (resp. nano preopen[1], nano semi-open[1]) if A = nint(ncl(A)) (resp. A nint(ncl(A)), A ncl(nint(A))). The set of all nano regular open[1] (resp. nano preopen[1],⊆ nano semi-open[1])⊆ sets is de- noted by NRO(U, X) (resp. NPO(U, X), NSO(U, X)). A subset C of a NTS (U, τR(X)) ijpam.eu 98 2017 International Journal of Pure and Applied Mathematics Special Issue is said to be nano regular closed[1] (resp. nano preclosed[1], nano semi-closed[1]) if its complements is nano regular open (resp. nano preopen, nano semi-open). The set of all nano regular closed[1] (resp. nano preclosed[1], nano semi-closed[1])sets is denoted by NRC(U, X) (resp. NPC(U, X), NSC(U, X)). For A U, nano regular interior (resp. nano preinterior, nano semi-interior) of A[1] is nrint(A⊆) (resp. npint(A), nsint(A)) = O : O NRO(U, X) (resp. NPO(U, X), NSO(U, X)) and O A and nano regu- lar∪{ closure∈ (resp. nano preclosure, nano semi-closure) of A[1] is nrcl⊆ (A}) (resp. npcl(A), nscl(A)) = C : X C NRO(U, X) (resp. NPO(U, X), NSO(U, X)) and A C . ∩{ − ∈ ⊆ } Definition 1.1. Let (U, τR(X)) be a NTS and A U. Then A is said to be (i) nano δ-open if for each x A, there exists⊆ a nano regular open set G such that x G nint(ncl(A)) A.∈ The set of all nano δ-open sets is denoted by NδO(U, X); (ii) nano∈ θ⊆-open set if for⊆ each x A, there exists a nano open set G such that x G ncl(A) A. The set of all nano∈ θ-open sets is denoted by NθO(U, X); ∈ ⊆ (iii) nano dense⊆ set if ncl(A) = U; (iv) nano predense set if nint(ncl(A)) = U; (v) nano semi-dense set if ncl(nint(ncl(A))) = U; (vi) nano semi-regular if A is both nano semi-open and nano semi-closed. Preposition 1.1. Let (U, τR(X)) be a NTS. Then the following statements hold: (i) If A NθO(U, X), then A NδO(U, X); (ii) A ∈NRC(U, X) if and only∈ if A NSO(U, X) NPC(U, X). ∈ ∈ ∩ Note that the converse of Proposition 1.7 need not be true. Let U = a, b, c, d with { } U/R = a , b, c , d and X = a, b . Then the topology τR(U, X) = , U, a , b, c , a, b, c {{, NδO} { (U,} X{)}} = , U, a{ , b,} c and NθO(U, X) = ,U {∅. Then{ } the{ set} {a }}NδO(U, X), but a {∅/ NθO{ (}U,{ X).}} {∅ } { } ∈ { } ∈ Definition 1.2. A NTS (U, τR(X)) is said to be (i) nano p-regular if for each x U and each nano open set G containing x, there exists a nano preopen set H such that∈ x H npcl(H) G; (ii) nano-T (resp. nano semi-T , nano∈ pre-⊆T ) if for each⊆ pair of distinct points x, y U, 1 1 1 ∈ there exists a nano open (resp. nano semi-open, nano preopen) sets A1 and A2 contains x and y respectively such that x / A2 and y / A1; (iii) nano hyperconnected if every non∈ empty nano∈ open set is nano dense; (iv) nano locally indiscrete space if every nano open set is nano closed. Proposition 1.2. Let the NTS (U, τR(X)) be nano-T1 (resp. nano pre-T1) if for any point x U, then x is nano closed (resp. nano preclosed). ∈ { } Lemma 1.1. If (U, τR(X)) is a NTS and A is a nano dense set, then A is nano predense. Proposition 1.3. Let (U, τR(X)) be a NTS. Then the below statements are equivalent: (i) (U, τR(X)) is nano hyperconnected; ijpam.eu 99 2017 International Journal of Pure and Applied Mathematics Special Issue (ii) Every non empty nano semi-open subset of (U, τR(X)) is nano dense; (iii) Every non empty nano semi-open subset of (U, τR(X)) is nano predense. Lemma 1.2. A NTS (U, τR(X)) is nano hyperconnected if and only if NRC(U, X) = ,U . {∅ } Lemma 1.3. If (U, τR(X)) is a nano locally indiscrete space, then (i) each nano semi-open subset of (U, τR(X)) is nano closed; (ii) each nano semi-closed subset of (U, τR(X)) is nano open. Definition 1.3. Let (U, τR(X)) be a NTS. If V is a subset of (U, τR(X)) and the collec- tion τ (V, X) = V G : G τ (X) is a nano topology on V with respect to X, then R { ∩ ∈ R } τR(V, X) is called a nano subspace topology. With this nano topology, V is called a nano subspace of (U, τR(X)): its nano open sets consist of all intersections of nano open sets of (U, τR(X)) with V . Proposition 1.4. Let (V, τR(V, X)) be a nano subspace of a NTS (U, τR(X)). Then, the following statements are true: (i) If A NSO(U, X) and A V , then A NSO(V, X); (ii) If A∈ NSO(V, X) and V ⊆ NSO(U, X∈), then A NSO(U, X); (iii) If F∈ NPC(U, X) and F∈ V , then F NPC(∈V, X); (iv) If F ∈ (V, X) and V NPC⊆(U, X), then∈F NPC(U, X). ∈ ∈ ∈ Lemma 1.4. If (V, τR(V, X)) is a nano closed nano subspace of a NTS (U, τR(X)) and F NPC(U, X), then F V NPC(V, X)). ∈ ∩ ∈ Lemma 1.5. If (U, τR(X)) is a NTS and A U, then A NSO(U, X) if and only if npcl(A) = ncl(nint(A)). ⊆ ∈ 2 NANO SP -OPEN SETS Definition 2.1. Let (U, τ (X)) be a NTS and A U. Then A is said to be nano s -open R ⊆ p (briefly, nsp-open) if for each x A NSO(U, X), there exists a nano preclosed set F such that x F A. The set of∈ all ns∈ -open sets is denoted by NS O(U, X). ∈ ⊆ p P Proposition 2.1. If Ai is a collection of nsp-open sets, then Ai is also nsp-open. Proof. Let A be a{ collection} of ns -open sets. Then for each∪x A NSO(U, X), { i} p ∈ ∪ i ⊆ there exists NPC(U, X) set F such that x F Ai Ai. This implies that x F A . Therefore A is ns -open. ∈ ⊆ ⊆ ∪ ∈ ⊆ ∪ i ∪ i p Remark 2.1. If A and B are two ns -open sets, then A B need not be ns -open. p ∩ p Let U = (0, 1) with U/R = (0, 1/2), [1/2, 1) and X = (0, 1). τR(U, X) = ,U . If A is the set of rational numbers{ in U and B is the} set of irrational numbers in U{∅together} with the singleton set 1/2 .