Closed Sets in Nano Topological Spaces D

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 denoted by NRO(U, X) (resp. NPO(U, X),NSO(U, X)). A subset C of a NTS (U, τR(X))

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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;

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(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 collection 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 . Then A NSO(U, X). Since (U, τ (X)) is a nano T -space { } ∈ R 1

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and by Theorem 1.10, every singleton set is nano closed and hence is nano preclosed, then A NSP O(U, X) and B NSO(U, X) NPC(U, X), then B NSP O(U, X). But A B =∈ 1/2 / NS O(U, X),∈ because 1/2∩ / NSO(U, X). ∈ ∩ { } ∈ P { } ∈

Remark 2.2. The concept of nano open and nano sp-open sets are independent. Let U = a, b, c, d with U/R = a , b, c , d and X = a, b . Then τ (X) = , U, a , b, c , { } {{ } { } { }} { } R {∅ { } { } a, b, c and NSP O(U, X) = , U, a, d , b, c , b, c, d . Then a, b, c τR(X), {a, b, c}}/ NS O(U, X) and a, d{∅ NS{ O}(U,{ X}), {a, d }}/ τ (X). { } ∈ { } ∈ P { } ∈ P { } ∈ R

Proposition 2.2. A subset A of a NTS (U, τR(X)) is nsp-open if and only if A is nano semi-open set and A is a union of nano preclosed sets.

Note that the converse of Proposition 2.2 need not be true. From the Remark 2.2, NSO(U, X) = , U, a , a, d , b, c , a, b, c , b, c, d and NPC(U, X) = , U, b , c , d , a, d , b, d{∅, c,{ d }, {a, b, d} ,{ a,} c,{ d , b,} c,{ d .}} Then a, b, c NSO{∅(U, X{)} and{ } {a,} b,{ c is} not{ the} { union} { of nano} preclosed{ } { sets.}} Also a, b, c{ / NS}O ∈(U, X). { } { } ∈ P Proposition 2.3. If A NS O(U, X), then A NSO(U, X). ∈ P ∈ Note that the converse of Proposition 2.3 need not be true. From the Remark 2.2, NSO(U, X) = , U, a , a, d , b, c , a, b, c , b, c, d and NSP O(U, X) = , U, a, d , b, c , b, c, d {∅. The{ set} {a is} nano{ } semi-open{ } { set but}} not nano s -open set.{∅ { } { } { }} { } p

Proposition 2.4. If A, B NSP O(U, X) and NSO(U, X) forms a nano topology, then A B NS O(U, X) and ∈NS O(U, X) forms a nano topology. ∩ ∈ P P

Proposition 2.5. If a NTS (U, τR(X)) is nano pre-T1, then NSP O(U, X) = NSO(U, X). Proof. Let A be a subset of (U, τR(X)) and A NSO(U, X). If A = , then A NS O(U, X). If A = , then for each x A. Since∈ U is nano pre-T and∅ by Theorem∈ P ̸ ∅ ∈ 1 1.1, x is nano preclosed set and hence x x A. Therefore, A NSP O(U, X). Therefore{ } NSO(U, X) NS O(U, X). By Proposition∈ { } ⊆ 2.3, NS O(U, X)∈ NSO(U, X). ⊆ P P ⊆ Hence NSP O(U, X) = NSO(U, X).

Note that the converse of Proposition 2.5 need not be true. From the Remark 2.2, NθO(U, X) = ,U and NSP O(U, X) = , U, a, d , b, c , b, c, d . Then a, d NS O(U, X) but{∅ a,} d / NθO(U, X). {∅ { } { } { }} { } ∈ P { } ∈ Proposition 2.6. If A NθO(U, X), then A NS O(U, X). ∈ ∈ P Note that the converse of Proposition 2.6 need not be true. From the Remark 2.2, NθO(U, X) = ,U and NSP O(U, X) = , U, a, d , b, c , b, c, d . Then a, d is a nano semi-open{∅ set} but not nano θ-open. {∅ { } { } { }} { }

Remark 2.3. The concept of nano δ-open and nano sp-open sets are independent.

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From the Remark 2.2, NδO(U, X) = , U, a , b, c and NS O(U, X) = , U, a, d , {∅ { } { }} P {∅ { } b, c , b, c, d . Then a, d NSP O(U, X) but a, d / NδO(U, X) and a NδO(U, X) {but }a{ / NS}} O(U, X{). Therefore} ∈ NS O(U, X){ and}NδO∈ (U, X) are independent.{ } ∈ { } ∈ P P

Theorem 2.1. Let (U, τR(X)) be a NTS and A U. (i) If A NθO(U, X) NSO(U, X), then A NS O(U, X); (ii) If A NRC(U, X),⊆ then A NS∈ O(U, X). ∩ ∈ P ∈ ∈ P Proof (i). Let A NθO(U, X) NSO(U, X). If A = , then A NSP O(U, X). If A = , then for each x ∈A, there exists∩ a nano open set A ∅such that∈x A ncl(A ) ̸ A∅. ∈ 0 ∈ 0 ⊆ 0 ⊆ Since A0 is nano open, A0 NSO(U, X). By Lemma 1.5, ncl(nint(A0)) = npcl(A0). Then x A npcl(A )) A∈implies that x npcl(A ) A and npcl(A ) NPC(U, X). ∈ 0 ⊆ 0 ⊆ ∈ 0 ⊆ 0 ∈ Therefore A NSP O(U, X). (ii). Since A ∈is nano regular closed, A is both nano semi-open and nano preclosed. Then for each x A NSO(U, X), a nano preclosed set A such that x A A. by the Deﬁnition 2.1,∈ A∈ NS O(U, X). ∈ ⊆ ∈ P Note that the converse of Theorem 2.1 need not be true. From the Remark 2.2, NθO(U, X) = ,U , NRC(U, X) = , U, a, d , b, c, d . Then the set a, d NSP O(U, X) but a,{∅ d /}NθO(U, X) and {∅b, c { NS} O{ (U, X}}) but b, c / NRC{ (U,} X ∈ ). { } ∈ { } ∈ P { } ∈

Theorem 2.2. A NTS (U, τR(X)) is nano hyperconnected if and only if NSP O(U, X) = ,U . {∅ } Proof. Let us assume that (U, τR(X)) is nano hyperconnected and let A NSP O(U, X). If A = , then for each x A, there exists F NPC(U, X) such that x ∈ F A. Since F NPC̸ ∅ (U, X), ncl(nint∈(F )) F . This implies∈ that npcl(ncl(nint(F )))∈ npcl⊆ (F )) = ∈ ⊆ ⊆ F . Since (U, τR(X)) is nano hyperconnected space and by Theorem 1.2(iii), we have that npcl(ncl(nint(F ))) = U F and hence U = A. Therefore NSP O(U, X) = ,U . Conversely, suppose that NS⊆O(U, X) = ,U . Since NRC(U, X) NS O(U,{∅ X) in} P {∅ } ⊆ P general, NRC(U, X) = ,U . By Lemma 1.2, we have (U, τR(X)) is a nano hyperconnected space. {∅ }

Proposition 2.7. If a NTS (U, τ (X)) is nano preregular, then τ (X) NS O(U, X). R R ⊆ P Proof. Let A be a subset of (U, τR(X)) and A τR(X). If A = , then A NSP O(U, X). If A = , since (U, τ (X)) is nano preregular, for∈ each x A, there∅ exists V∈ NPO(U, X) ̸ ∅ R ∈ ∈ such that x V npclV A. Thus, we have that x npclU A. Since A τR(X) and hence A∈ NSO⊆ (U, X).⊆ Therefore, A NS O(U, X∈). Hence⊆τ (X) NS O∈(U, X). ∈ ∈ P R ⊆ P

Theorem 2.3. If a NTS (U, τR(X)) is nano locally indiscrete, then NSP O(U, X) = τR(X). Proof. Suppose that (U, τ (X)) is nano locally indiscrete space and let V U such R ⊆ that V τR(X). Since every nano open set is nano closed, ncl(nintV )) = V this implies that∈ V NRC(U, X). Hence by Theorem 2.1(ii), V NS O(U, X). Thus, ∈ ∈ P τR(X) NSP O(U, X). On the other hand, let V NSP O(U, X), then V NSO(U, X). By Lemma⊆ 1.3, V is nano closed. Since every nano∈ closed set is nano open,∈ V is open.

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Thus V τ (X) and hence NS O(U, X) τ (X). Therefore, NS O(U, X) = τ (X). ∈ R P ⊆ R P R

Proposition 2.8. Let (U, τR(X)) be a NTS. (i) If τR(X)(resp.NSO(U, X)) is indiscrete, then NSP O(U, X) is also indiscrete; (ii) If NSP O(U, X) is discrete, then NSO(U, X) is discrete; (iii) τR(X) is discrete if and only if NSP O(U, X) is discrete.

Proposition 2.9. Let (U, τR(X)) be a NTS and x U. (i) x NSP O(U, X) if and only if x NRC(U, X); (ii) If x NS O(U, X),∈ then x{ }NPC ∈ (U, X). { } ∈ { } ∈ P { } ∈ Corollary 2.1. For any subset A U, The following conditions are equivalent: ⊆ (i) A is nano clopen; (ii) A is nsp-open and nano closed; (iii) A is nano semi-open and nano closed.

Corollary 2.2. Let (U, τR(X)) be a NTS. Then the following conditions are equivalent: (i) A is nano regular closed; (ii) A is nsp-open and nano preclosed; (iii) A is nano open and nano preclosed; (iv) A is nano semi-open and nano preclosed.

Proposition 2.10. Let (V, τR(V,X)) be a nano subspace of a NTS (U, τR(X)). If A NS O(U, τ (X)) and A is a subset of (V, τ (V,X)), then A NS O(V, τ (V,X)). ∈ P R R ∈ P R Proof. Let A NSP O(U, τR(X)). Then A NSO(U, τR(X)) and for each x A, there exists F ∈NPC(U, τ (X)) such that x ∈F A. Since A NSO(U, τ (X)),∈A ∈ R ∈ ⊆ ∈ R ⊆ (V, τR(V,X)) and by Theorem 1.3(i), A NSO(V, τR(V,X)). Since F NPC(U, τR(U, X)) and F V and by Theorem 1.3(iii), F ∈NPC(V, τ (V,X)) and A NS∈ O(V, τ (V,X)). ⊆ ∈ R ∈ P R

Proposition 2.11. Let (V, τR(V,X)) be a nano subspace of a NTS (U, τR(X)). If A NSP O(V, τR(V,X)) and V NRC(U, τR(X)), then A NSP O(U, τR(X)). Proof.∈ Let A NS O(V, τ (V,X∈ )), then A NSO(V, τ ∈(V,X)) and for each x A, ∈ P R ∈ R ∈ there exists F NPC(V, τR(V,X)) such that x F A. Since V NRC(U, τR(X)), V NSO(U, τ ∈(X)), A NSO(V, τ (V,X)) and∈ by Theorem⊆ 1.3(ii), A∈ NSO(U, τ (X)). ∈ R ∈ R ∈ R Since V NRC(U, τR(X)), V NPC(U, τR(X)), F NPC(V, τR(V,X)) and by Theo- rem 1.3(iv),∈ F NPC(U, τ (X∈)). Hence A NS O(∈U, τ (X)). ∈ R ∈ P R

Corollary 2.3. Let (V, τR(V,X)) be a nano regular closed nano subspace of a NTS (U, τ (X)) and A V . Then A NS O(V, τR(V,X)) if and only if A NS O(U, τ (X)). R ⊆ ∈ P ∈ P R Proposition 2.12. If A NS O(U, X), then nsint(A) NS O(U, X) and npcl(A) ∈ P ∈ P ∈ NSP O(U, X).

Deﬁnition 2.2. A subset B of a NTS (U, τ (X)) is called ns -closed if U B is ns -open R p − p set. The set of all nsp-closed sets is denoted by NSP C(U, τR(X)) or NSP C(U, X).

Proposition 2.13. A nano semi-closed subset B of a NTS (U, τR(X)) is nsp-closed if and only if B is an intersection of nano preopen sets.

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Deﬁnition 2.3. A point x U is said to be a nsp-interior point of A if there exists a ns -open set V containing x∈such that V A. The set of all ns -interior points of A is p ⊆ p said to be nsp-interior of A and is denoted by nspint(A).

Theorem 2.4. Let A be any subset of a NTS (U, τR(X)). If a point x nspint(A) then there exists F NPC(U, X) containing x such that F A. ∈ Proof. Suppose∈ that x ns int(A). Then V NS ⊆O(U, X) containing x such that ∈ p ∈ P V A. Since V NSP O(U, X), there exists F NPC(U, X) containing x such that F ⊆ U A. Hence∈ x F A. ∈ ⊆ ⊆ ∈ ⊆

Theorem 2.5. Let A be any subset of a nano topological space (U, τR(X)). Then the following statements are true. (i) The nsp-interior of A is the union of all nsp-open sets which are contained in A; (ii) nspint(A) is nsp-open set in U contained in A; (iii) ns int(A) is the largest ns -open set contained in A; (iv) A NS O(U, X) if and only if p p ∈ P A = nspint(A).

Deﬁnition 2.4. Let A be any subset of a NTS (U, τR(X)). Then A point x U is in the ns -closure of A if and only if A H = for every H NS O(U, X) containing∈ x and is p ∩ ̸ ∅ ∈ P denoted by nspcl(A).

Theorem 2.6. Let A be any subset of a NTS (U, τR(X)). If A F = for every F NPC(U, X) containing x, then x ns cl(A). ∩ ̸ ∅ ∈ ∈ p Proof. Let V NSP O(U, X) containing x. Since V NSP O(U, X), there exists F NPC(U, X)∈ containing x such that F V . By hypothesis,∈ we have that A F = . Therefore∈ A V = . Hence x ns cl(A). ⊆ ∩ ̸ ∅ ∩ ̸ ∅ ∈ p

Theorem 2.7. Let A be any subset of a nano topological space (U, τR(X)).Then the following statements are true. (i) The nsp-closure of A is the intersection of all nsp-closed sets containing A. (ii) nspcl(A) is nsp-closed set in (U, τR(X)) containing A. (iii) nspcl(A) is the smallest nsp-closed set containing A. (iv) A NSP C(U, X) if and only if A = nspcl(A). Proof.∈Obvious.

Theorem 2.8. For any subsets A and B of a nano topological space U. The following statements are true. (i) nspint(nspintA) = nspint(A) and nspcl(nspclA) = nspcl(A). (ii) ns int(A) = U nspcl(U A) and ns cl(A) = U ns int(U A). p − − p − p − (iii) If A B, then nspint(A) nspint(B) and nspcl(A) nspcl(B). (iv) ns int⊆(A) ns int(B) ns⊆ int(A B). ⊆ p ∪ p ⊂ p ∪ (v) nspint(A B) nspint(A) nspint(B). (vi) ns cl(A)∩ ns ⊂cl(B) ns cl∩(A B). p ∪ p ⊂ p ∪ (vii) nspcl(A B) nspcl(A) nspcl(B). Proof. Straightforward.∩ ⊂ ∩

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The inclusions in (iv), (v), (vi) and (vii) cannot be replaced by equality in general, as it is shown in the following two examples.

Example 2.1. Let U = a, b, c, d , U/R = a , b, c , d and X = b, d and τ (U, X) = , U, d , b, c {, b, c, d }; so NS O{{(U,} X{) =} { ,}} U, a, d , b, c ,{ a,} b, c . R {∅ { } { } { }} P {∅ { } { } { }} Then nspint a, b nspint a, c = = but nspint( a, b a, c ) = nspint a, b, c = a, b, c . Therefore,{ }∪ ns int{a, b} ns∅∪∅int a,∅ c = ns int({ a, b}∪{a, c}). Also ns {cl( a, d} { } p { }∪ p { } ̸ p { }∪{ } p { }∩ b, d ) = nspcl d = d but nspcl a, d nspcl b, d = a, d U = a, d . Therefore, {ns cl}( a, d {b,} d ) ={ ns} cl a, d { ns}cl ∩ b, d .{ } { } ∩ { } p { } ∩ { } ̸ p { } ∩ p { }

Example 2.2. From Remark 2.1, we have that A, B NSP O(U, X). Therefore nspintA ns intB = A B = 1/2 . But ns int(A B) = ns ∈int 1/2 = . Therefore, ns int(A ∩ p ∩ { } p ∩ p { } ∅ p ∩ B) = nspintA nspintB. Also, by Theorem 2.8(iii), nspcl(F E) = U nspint(U (F E))̸ = U (ns ∩int((U F ) (U E)) where F = U A and E∪= U B.− So ns cl(F −E) =∪ − p − ∩ − − − p ∪ U nspint((U F ) (U E)) = U nspint(A B). Since nspint(A B) = nspint 1/2 = . So−ns cl(F −E) =∩ U − = U.− But ns clF∩ ns clE = ns cl(U∩ A) nspcl{ (U } B∅), p ∪ − ∅ p ∪ p p − ∪ − by Theorem 2.8 (iii), nspcl(U A) nspcl(U B) = (U nspintA) (U nspintB) = U (ns intA ns intB). Since− ns ∪intA ns −intB = A −B = 1/2 ∪, ns clF− ns clE = − p ∩ p p ∩ p ∩ { } p ∪ p U (nspintA nspintB) = U (A B) = U 1/2 . Therefore nspclF nspclE = ns−cl(F E). ∩ − ∩ − { } ∪ ̸ p ∪ References

[1] M. Lellis Thivagar and C. Richard, On nano forms of weakly open sets, Int. J. Math. Stat. Inven. 1(1) (2013), 31 - 37.

[2] N. Levine, Semi-open sets and semicontinuity in topological spaces, Amer. Math. Monthly 70 (1963), 36 - 41.

[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 - 45.

[4] O. Njastad, on some classes of nearly open sets, Paciﬁc J. Math. 15 (1965), 961 - 970.

[5] Z. Pawlok, Rough sets: Theoretical Aspects of Reasoning about Data, Kluwer Aca- demic Publishers, Boston, 1991.

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