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ISSN 2229 – 5046

On Nano Regular Generalized Star Star b-Closed Sets in Nanotopological Spaces

G. VASANTHA KANNAN*

Department of Mathematics, RVS College of Arts & Science, Sulur, Coimbatore - India.

K. INDIRANI

Nirmala College for Women, Red fields, Coimbatore - India.

(Received On: 19-09-16; Revised & Accepted On: 24-10-16)

ABSTRACT

I

n this paper we introduce the concept of nano regular generalized star star b-closed sets and investigate some of its properties in nanotopological spaces. We also define nanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-continuity, nanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open and some of its fundamental properties are given.

Keywords: Nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘-closed set, Nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed set, Nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open set, Nanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-continuity.

INTRODUCTION

In 1970, Levine [8] introduced the concept of generalized closed sets in topological spaces. Ahmed and Mohd [1] (2009) studied the class of generalized b-closed sets. Sindhu and Indirani [6] (2013) introduced the concept of regular generalized star b-closed sets in topological spaces. Banupriya and indirani [2] (2014) introduced the concept of regular generalized star star b-closed sets in topological spaces. The notion of nanotopology was introduced by Lellis Thivagar which was defined in terms of approximations and boundary regions of a subset of an universe using an equivalence relation on it and also defined nano closed, nano interior and nano closure. Smitha and Indirani [7] (2015) introduced the concept of nano regular generalized star b-closed sets in nanotopological spaces. The aim of this paper, we introduce a new class of nano sets on nanotopological spaces called nano regular generalized star star b-closed sets, nanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-continuity, nanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open and obtain their characteristics with counter examples.

2. PRELIMINARIES

Definition 2.1 [9]: Letπ‘ˆπ‘ˆbe a non-emptyfinite set of objects called the universe and 𝑅𝑅 be an equivalence relation on π‘ˆπ‘ˆ named as the indiscernibility relation, elements belonging to the same equivalence class are said to be indiscernible with one another. The pair (π‘ˆπ‘ˆ,𝑅𝑅) is said to be the approximation space. Let 𝑋𝑋 βŠ† π‘ˆπ‘ˆ.

i) The lower approximation of 𝑋𝑋 with respect to 𝑅𝑅 is the set of all objects, which can be for certain classified as 𝑋𝑋 with respect to 𝑅𝑅 and it is denoted by 𝐿𝐿𝑅𝑅(𝑋𝑋). That is𝐿𝐿𝑅𝑅(𝑋𝑋) =⋃π‘₯π‘₯βˆˆπ‘ˆπ‘ˆ{𝑅𝑅(π‘₯π‘₯):𝑅𝑅(π‘₯π‘₯)βŠ† 𝑋𝑋}, where 𝑅𝑅(π‘₯π‘₯) denotes the equivalence class determined by π‘₯π‘₯.

ii) The upper approximation of 𝑋𝑋 with respect to 𝑅𝑅 is the set of all objects, which can be possibly classified as 𝑋𝑋 with respect to 𝑅𝑅 and it is denoted by π‘ˆπ‘ˆπ‘…π‘…(𝑋𝑋). That is π‘ˆπ‘ˆπ‘…π‘…(𝑋𝑋) =⋃π‘₯π‘₯βˆˆπ‘ˆπ‘ˆ{𝑅𝑅(π‘₯π‘₯):𝑅𝑅(π‘₯π‘₯)∩ 𝑋𝑋 β‰  βˆ…} where 𝑅𝑅(π‘₯π‘₯) denotes the equivalence class determined by π‘₯π‘₯.

iii) The boundary region of 𝑋𝑋 with respect to 𝑅𝑅 is set of all objects, which can be classified neither as 𝑋𝑋 nor as not 𝑋𝑋 with respect to 𝑅𝑅 and it is denoted by𝐡𝐡𝑅𝑅(𝑋𝑋).That is 𝐡𝐡𝑅𝑅(𝑋𝑋) =π‘ˆπ‘ˆπ‘…π‘…(𝑋𝑋)βˆ’ 𝐿𝐿𝑅𝑅(𝑋𝑋).

Definition 2.2[9]: Letπ‘ˆπ‘ˆ be a non-empty, finite universe of objects and 𝑅𝑅 be an equivalence relation on π‘ˆπ‘ˆ. Let 𝑋𝑋 βŠ† π‘ˆπ‘ˆ. πœπœπ‘…π‘…(𝑋𝑋) = {π‘ˆπ‘ˆ,πœ™πœ™, 𝐿𝐿𝑅𝑅(𝑋𝑋), π‘ˆπ‘ˆπ‘…π‘…(𝑋𝑋),𝐡𝐡𝑅𝑅(𝑋𝑋)}. Thenπœπœπ‘…π‘…(𝑋𝑋) is a topology onπ‘ˆπ‘ˆ, called as the nanotopology with respect

to𝑋𝑋. Elements of nanotopology are known as the nanoopen sets in π‘ˆπ‘ˆ and (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) is called the nanotopological space. Elements of οΏ½πœπœπ‘…π‘…(𝑋𝑋)�𝑐𝑐 are called as nano closed sets.

Definition 2.3 [9]: If πœπœπ‘…π‘…(𝑋𝑋) is the nanotopology on π‘ˆπ‘ˆ with respect to𝑋𝑋, and then the set 𝐡𝐡= {π‘ˆπ‘ˆ, 𝐿𝐿𝑅𝑅(𝑋𝑋), 𝐡𝐡𝑅𝑅(𝑋𝑋)} is the basis for πœπœπ‘…π‘…(𝑋𝑋).

Corresponding Author:

G. Vasantha Kannan*

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Definition 2.4 [9]: If (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) is a nanotopological space with respect to 𝑋𝑋 where 𝑋𝑋 βŠ† π‘ˆπ‘ˆ and if 𝐴𝐴 βŠ† π‘ˆπ‘ˆ, then the nano interior of 𝐴𝐴 is defined as the union of allnano open subsets of 𝐴𝐴 and it is denoted by𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁(𝐴𝐴). That is,𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁(𝐴𝐴) is the largest nano open subset of 𝐴𝐴.Thenano closure of 𝐴𝐴 is defined as the intersection of all nano closed sets containing 𝐴𝐴 and is denoted by𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴). That is 𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴) is the smallest nano closed set containing𝐴𝐴.

Definition 2.5 [9]: Let (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) be a nanotopological space and𝐴𝐴 βŠ† π‘ˆπ‘ˆ. Then 𝐴𝐴 is said to be: i) Nano semi-open if 𝐴𝐴 βŠ† 𝑁𝑁𝑐𝑐𝑁𝑁(𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁(𝐴𝐴))

ii) Nano pre-open if 𝐴𝐴 βŠ† 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁(𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴)) iii) Nano 𝛼𝛼-open if 𝐴𝐴 βŠ† 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁(𝑁𝑁𝑐𝑐𝑁𝑁(𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁(𝐴𝐴))) iv) Nano semi pre-open if 𝐴𝐴 βŠ† 𝑁𝑁𝑐𝑐𝑁𝑁(𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁(𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴)))

𝑁𝑁𝑁𝑁𝑁𝑁(π‘ˆπ‘ˆ,𝑋𝑋),𝑁𝑁𝑁𝑁𝑁𝑁(π‘ˆπ‘ˆ,𝑋𝑋),𝑁𝑁𝛼𝛼𝑁𝑁(π‘ˆπ‘ˆ,𝑋𝑋),𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁(π‘ˆπ‘ˆ,𝑋𝑋) respectively denote the families of all nano semi-open, nano pre-open and nano semi pre-pre-open subsets of π‘ˆπ‘ˆ.

Let (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) be a nanotopological space and𝐴𝐴 βŠ† π‘ˆπ‘ˆ. 𝐴𝐴 is said to be nano semi-closed, nano pre-closed, nano 𝛼𝛼-closed, nano semi pre-closed if its complement is respectively nano semi-open, nano pre-open, nano 𝛼𝛼-open, nano semi pre-open.

Definition 2.6: A subset 𝐴𝐴 of (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) is called

i) Nano generalized closed set [4] (briefly 𝑁𝑁𝑔𝑔-closed) if 𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴)βŠ† 𝐺𝐺 whenever 𝐴𝐴 βŠ† 𝐺𝐺 and 𝐺𝐺 is nano open in οΏ½π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)οΏ½.

ii) Nano generalized star closed set [11] (briefly π‘π‘π‘”π‘”βˆ—-closed) if 𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴)βŠ† 𝐺𝐺 whenever 𝐴𝐴 βŠ† 𝐺𝐺 and 𝐺𝐺 is nano 𝑔𝑔-open in οΏ½π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)οΏ½.

iii) Nano generalized 𝛼𝛼-closed set [3] (briefly 𝑁𝑁𝑔𝑔𝛼𝛼-closed) if 𝑁𝑁𝛼𝛼𝑐𝑐𝑁𝑁(𝐴𝐴)βŠ† 𝐺𝐺 whenever 𝐴𝐴 βŠ† 𝐺𝐺 and 𝐺𝐺 is nano 𝛼𝛼-open in οΏ½π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)οΏ½.

iv) Nano generalized pre-closed set [5] (briefly 𝑁𝑁𝑔𝑔𝑁𝑁-closed) if 𝑁𝑁𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴)βŠ† 𝐺𝐺 whenever 𝐴𝐴 βŠ† 𝐺𝐺 and 𝐺𝐺 is nano-open in οΏ½π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)οΏ½.

v) Nano generalized pre-regular closed set (briefly π‘π‘π‘”π‘”π‘π‘π‘Ÿπ‘Ÿ-closed) if 𝑁𝑁𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴)βŠ† 𝐺𝐺 whenever 𝐴𝐴 βŠ† 𝐺𝐺 and 𝐺𝐺 is nano π‘Ÿπ‘Ÿ-open in οΏ½π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)οΏ½.

vi) Nano regular-generalized closed set (briefly π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”-closed) if 𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴)βŠ† 𝐺𝐺 whenever 𝐴𝐴 βŠ† 𝐺𝐺 and 𝐺𝐺 is nano π‘Ÿπ‘Ÿ-open in οΏ½π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)οΏ½.

Proposition 2.7 [9]: If(π‘ˆπ‘ˆ,𝑅𝑅) is an approximation space𝑋𝑋,π‘Œπ‘Œ βŠ† π‘ˆπ‘ˆ, then 1. 𝐿𝐿𝑅𝑅(𝑋𝑋)βŠ† 𝑋𝑋 βŠ† π‘ˆπ‘ˆπ‘…π‘…(𝑋𝑋)

2. 𝐿𝐿𝑅𝑅(βˆ…) =βˆ…=π‘ˆπ‘ˆπ‘…π‘…(βˆ…) and 𝐿𝐿𝑅𝑅(π‘ˆπ‘ˆ) =π‘ˆπ‘ˆ=π‘ˆπ‘ˆπ‘…π‘…(π‘ˆπ‘ˆ) 3. π‘ˆπ‘ˆπ‘…π‘…(𝑋𝑋 βˆͺ π‘Œπ‘Œ) =π‘ˆπ‘ˆπ‘…π‘…(𝑋𝑋)βˆͺ π‘ˆπ‘ˆπ‘…π‘…(π‘Œπ‘Œ)

4. π‘ˆπ‘ˆπ‘…π‘…(𝑋𝑋 ∩ π‘Œπ‘Œ)βŠ† π‘ˆπ‘ˆπ‘…π‘…(𝑋𝑋)∩ π‘ˆπ‘ˆπ‘…π‘…(π‘Œπ‘Œ) 5. 𝐿𝐿𝑅𝑅(𝑋𝑋 βˆͺ π‘Œπ‘Œ)βŠ‡ 𝐿𝐿𝑅𝑅(𝑋𝑋)βˆͺ 𝐿𝐿𝑅𝑅(π‘Œπ‘Œ) 6. 𝐿𝐿𝑅𝑅(𝑋𝑋 ∩ π‘Œπ‘Œ) =𝐿𝐿𝑅𝑅(𝑋𝑋)∩ 𝐿𝐿𝑅𝑅(π‘Œπ‘Œ) 7. 𝐿𝐿𝑅𝑅(𝑋𝑋)βŠ† 𝐿𝐿𝑅𝑅(π‘Œπ‘Œ) and π‘ˆπ‘ˆπ‘…π‘…(𝑋𝑋)βŠ† π‘ˆπ‘ˆπ‘…π‘…(π‘Œπ‘Œ)

8. π‘ˆπ‘ˆπ‘…π‘…(𝑋𝑋𝐢𝐢) = [ 𝐿𝐿𝑅𝑅(𝑋𝑋)]𝐢𝐢 and 𝐿𝐿𝑅𝑅(𝑋𝑋𝐢𝐢) = [ π‘ˆπ‘ˆπ‘…π‘…(𝑋𝑋)]𝐢𝐢

3. Nano regular generalized star star b-closed sets (

π‘΅π‘΅π‘΅π‘΅π’ˆπ’ˆ

βˆ—βˆ—

𝒃𝒃

-closed)

Definition 3.1: A subset 𝐴𝐴 of a nanotopological space (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) is called nano regular generalized star star b-closed set (briefly π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed) if π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝐺𝐺 whenever 𝐴𝐴 βŠ† 𝐺𝐺 and 𝐺𝐺 is nano open in (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)).

Theorem 3.2: Every nano closed set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Proof: If 𝐴𝐴 is nano closed in (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) and 𝐺𝐺 is nano open in π‘ˆπ‘ˆ such that 𝐴𝐴 βŠ† 𝐺𝐺. Since 𝐴𝐴 is nano closed,𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴) = 𝐴𝐴 βŠ† 𝐺𝐺. By theorem 3.7[7] " every nano closed set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘-closed",π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴) =𝐴𝐴. Therefore π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝐺𝐺. Hence 𝐴𝐴 is nanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Converse of the above theorem need not be true as seen from the following example.

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Theorem 3.4: Every nano semi closed set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Proof: Let𝐴𝐴 benano semi closed in (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) and 𝐺𝐺 is nano open in π‘ˆπ‘ˆ such that 𝐴𝐴 βŠ† 𝐺𝐺. Since 𝐴𝐴 is nano semi closed,𝑁𝑁𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴) =𝐴𝐴 βŠ† 𝐺𝐺. By theorem 3.7[7] " every nano semi closed set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘-closed",π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝑁𝑁𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴) =𝐴𝐴. Therefore π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝐺𝐺. Hence 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Converse of the above theorem need not be true as seen from the following example.

Example 3.5: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ =οΏ½{π‘Žπ‘Ž}, {𝑏𝑏}, {𝑐𝑐,𝑑𝑑}οΏ½ and 𝑋𝑋= {π‘Žπ‘Ž,𝑐𝑐}. Then the nanotopology πœπœπ‘…π‘…(𝑋𝑋) = οΏ½π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}, {𝑐𝑐,𝑑𝑑}οΏ½. Then the sets {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐} and {π‘Žπ‘Ž,𝑏𝑏,𝑑𝑑} are nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed but notnanosemi closed.

Theorem 3.6: Let (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) be nanotopological space and 𝐴𝐴 βŠ† π‘ˆπ‘ˆ. Then i) Every nano pre closed set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

ii) Every nano 𝛼𝛼-closed set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed. iii) Every nano regular closed set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Reverse implications of the theorem need not be true as seen from the following examples.

Example 3.7: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ =οΏ½{π‘Žπ‘Ž}, {𝑏𝑏}, {𝑐𝑐,𝑑𝑑}οΏ½ and 𝑋𝑋= {π‘Žπ‘Ž,𝑐𝑐}. Then the nanotopology πœπœπ‘…π‘…(𝑋𝑋) =οΏ½π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}, {𝑐𝑐,𝑑𝑑}οΏ½. Then the set {π‘Žπ‘Ž}is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed but not nano pre closed.

Example 3.8: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ =οΏ½{π‘Žπ‘Ž}, {𝑏𝑏}, {𝑐𝑐,𝑑𝑑}οΏ½ and 𝑋𝑋= {π‘Žπ‘Ž,𝑐𝑐}. Then the nanotopology

πœπœπ‘…π‘…(𝑋𝑋) =οΏ½π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}, {𝑐𝑐,𝑑𝑑}οΏ½. Then the set {π‘Žπ‘Ž} and {π‘Žπ‘Ž} are nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed but notnano𝛼𝛼- closed. Also the

sets {π‘Žπ‘Ž}, {𝑏𝑏} and {π‘Žπ‘Ž,𝑑𝑑} are nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed but not nano regular closed.

Theorem 3.9: Every nano𝑔𝑔-closed set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Proof: Let 𝐴𝐴 be nano𝑔𝑔-closed in (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) and 𝐺𝐺 is nano open in π‘ˆπ‘ˆ such that 𝐴𝐴 βŠ† 𝐺𝐺. Since 𝐴𝐴 is nano 𝑔𝑔-closed,𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴)βŠ† 𝐺𝐺. By theorem 3.7[7] " every nano closed set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘-closed",π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴) =𝐴𝐴.

Therefore π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝐺𝐺. Hence 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Converse of the above theorem need not be true as seen from the following example.

Example 3.10: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ =οΏ½{π‘Žπ‘Ž}, {𝑏𝑏}, {𝑐𝑐,𝑑𝑑}οΏ½ and 𝑋𝑋= {π‘Žπ‘Ž,𝑐𝑐}. Then the nanotopology πœπœπ‘…π‘…(𝑋𝑋) =οΏ½π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}, {𝑐𝑐,𝑑𝑑}οΏ½. Then the sets {𝑐𝑐} and {𝑐𝑐,𝑑𝑑} are nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed but notnano𝑔𝑔-closed.

Theorem 3.11: Every nanoπ‘”π‘”βˆ—-closed set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Proof: Let 𝐴𝐴 be nanoπ‘”π‘”βˆ—-closed in (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)). Therefore 𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴)βŠ† 𝐺𝐺 whenever 𝐴𝐴 βŠ† 𝐺𝐺 and 𝐺𝐺 is nano 𝑔𝑔-open. As every nano open set is nano 𝑔𝑔-open, we get𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴)βŠ† 𝐺𝐺 whenever 𝐴𝐴 βŠ† 𝐺𝐺 and 𝐺𝐺 is nano open. But it is true that β€œEvery nano closed set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘-closed set. Therefore π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴)βŠ† 𝐺𝐺. Hence 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Converse of the above theorem need not be true as seen from the following example.

Example 3.12: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ =οΏ½{π‘Žπ‘Ž}, {𝑏𝑏}, {𝑐𝑐,𝑑𝑑}οΏ½ and 𝑋𝑋= {π‘Žπ‘Ž,𝑐𝑐}. Then the nanotopology πœπœπ‘…π‘…(𝑋𝑋) =οΏ½π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}, {𝑐𝑐,𝑑𝑑}οΏ½. Then the sets {π‘Žπ‘Ž} and {𝑑𝑑} are nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed but notnanoπ‘”π‘”βˆ—-closed.

Theorem 3.13: Every nano𝑔𝑔𝛼𝛼-closed set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Proof: Let𝐴𝐴 benano𝑔𝑔𝛼𝛼-closed in (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) and 𝐺𝐺 is nano open in π‘ˆπ‘ˆ such that 𝐴𝐴 βŠ† 𝐺𝐺. Since 𝐴𝐴 is nano 𝑔𝑔𝛼𝛼 -closed,𝑁𝑁𝛼𝛼𝑐𝑐𝑁𝑁(𝐴𝐴)βŠ† 𝐺𝐺. By theorem 3.7[7] " every nano 𝛼𝛼-closed set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘-closed",π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝑁𝑁𝛼𝛼𝑐𝑐𝑁𝑁(𝐴𝐴) =𝐴𝐴.

Therefore π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝐺𝐺. Hence 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Converse of the above theorem need not be true as seen from the following example.

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Theorem 3.15: Every nano𝑔𝑔𝑁𝑁-closed set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Proof: Let𝐴𝐴 benano𝑔𝑔𝑁𝑁-closed in (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) and 𝐺𝐺 is nano open in π‘ˆπ‘ˆ such that 𝐴𝐴 βŠ† 𝐺𝐺. Since 𝐴𝐴 is nano 𝑔𝑔𝑁𝑁 -closed,𝑁𝑁𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴)βŠ† 𝐺𝐺. By theorem 3.7[7] " every nano pre-closed set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘-closed",π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝑁𝑁𝑁𝑁𝑐𝑐𝑁𝑁(𝐴𝐴) = 𝐴𝐴. Therefore π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝐺𝐺. Hence 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Converse of the above theorem need not be true as seen from the following example.

Example 3.16: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ =οΏ½{π‘Žπ‘Ž}, {𝑏𝑏}, {𝑐𝑐,𝑑𝑑}οΏ½ and 𝑋𝑋= {π‘Žπ‘Ž,𝑐𝑐}. Then the nanotopology πœπœπ‘…π‘…(𝑋𝑋) =οΏ½π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}, {𝑐𝑐,𝑑𝑑}οΏ½. Then the sets {π‘Žπ‘Ž} is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed but notnano𝑔𝑔𝑁𝑁-closed.

Remark 3.17: The nanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed set is an independent of the nanoπ‘”π‘”π‘π‘π‘Ÿπ‘Ÿ-closed and nanoπ‘Ÿπ‘Ÿπ‘”π‘”-closed as shown by the following example.

The set {π‘Žπ‘Ž} is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed but not nanoπ‘”π‘”π‘π‘π‘Ÿπ‘Ÿ-closed. The set {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑} is nano π‘”π‘”π‘π‘π‘Ÿπ‘Ÿ-closed but notnanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ -closed. Also the set {𝑑𝑑} is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed but notnanoπ‘Ÿπ‘Ÿπ‘”π‘”-closed. The set {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑} is nano π‘Ÿπ‘Ÿπ‘”π‘”-closed but notnanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Remark 3.18: The union of two nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed sets need not be nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed set can be seen from the following example.

Example 3.19: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ =οΏ½{π‘Žπ‘Ž}, {𝑏𝑏}, {𝑐𝑐,𝑑𝑑}οΏ½ and 𝑋𝑋= {π‘Žπ‘Ž,𝑐𝑐}. Then the nanotopology

πœπœπ‘…π‘…(𝑋𝑋) =οΏ½π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}, {𝑐𝑐,𝑑𝑑}οΏ½. Then the sets {π‘Žπ‘Ž} and {𝑐𝑐} are nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed sets but {π‘Žπ‘Ž}βˆͺ{𝑐𝑐} = {π‘Žπ‘Ž,𝑐𝑐} is

notnanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Remark 3.20: The intersection of two nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed sets need not be nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed set can be seen from the following example.

Example 3.21: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ =οΏ½{π‘Žπ‘Ž}, {𝑏𝑏}, {𝑐𝑐,𝑑𝑑}οΏ½ and 𝑋𝑋= {π‘Žπ‘Ž,𝑐𝑐}. Then the nanotopology

πœπœπ‘…π‘…(𝑋𝑋) =οΏ½π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}, {𝑐𝑐,𝑑𝑑}οΏ½. Then the sets {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐} and {π‘Žπ‘Ž,𝑏𝑏,𝑑𝑑} are nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed sets but {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐}∩

{π‘Žπ‘Ž,𝑏𝑏,𝑑𝑑} = {π‘Žπ‘Ž,𝑏𝑏} is notnanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Theorem 3.22: A set 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed if and only if π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βˆ’ 𝐴𝐴 contains no non-empty nano closed set.

Proof: Necessity: Let 𝐹𝐹 be a nano closed set in (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) such that 𝐹𝐹 βŠ† π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βˆ’ 𝐴𝐴. Then 𝐴𝐴 βŠ† 𝑋𝑋 βˆ’ 𝐹𝐹. Since 𝐴𝐴 is nano regular generalized star star b-closed set and 𝑋𝑋 βˆ’ 𝐹𝐹 is nano open then π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝑋𝑋 βˆ’ 𝐹𝐹.That is 𝐹𝐹 βŠ† 𝑋𝑋 βˆ’ π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴). So 𝐹𝐹 βŠ† 𝑋𝑋 βˆ’ π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)∩ π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βˆ’ 𝐴𝐴. Therefore 𝐹𝐹=βˆ….

Sufficiency: Let us assume that π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βˆ’ 𝐴𝐴 contains no non-empty nano closed set. Let 𝐴𝐴 βŠ† 𝐺𝐺 and 𝐺𝐺 is nano open. Suppose that π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴) is not contained in 𝐺𝐺,π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)∩ 𝐺𝐺𝐢𝐢 is non-empty nano closed set of π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βˆ’ 𝐴𝐴 which is a contradiction. Therefore π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝐺𝐺. Hence 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Theorem 3.23: If 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed and 𝐴𝐴 βŠ† 𝐡𝐡 βŠ† π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴) then 𝐡𝐡 is alsonanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Proof: Let BβŠ†G where 𝐺𝐺 is nano open in οΏ½π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)οΏ½. Then 𝐴𝐴 βŠ† 𝐡𝐡 implies 𝐴𝐴 βŠ† 𝐺𝐺. Since 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed, π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝐺𝐺. Also BβŠ† π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴) implies π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐡𝐡)βŠ† π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴). Thus π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐡𝐡)βŠ† 𝐺𝐺 and 𝐡𝐡 is

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Remark 3.24: The above discussions are summarized in the following diagram

4. Nano regular generalized star star b-open sets

(

π‘΅π‘΅π‘΅π‘΅π’ˆπ’ˆ

βˆ—βˆ—

𝒃𝒃 βˆ’ 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐

𝒔𝒔𝒐𝒐𝒔𝒔

)

Definition 4.1: A subset 𝐴𝐴 of a nanotopological space (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) is called nano regular generalized star star b-open set (briefly π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open) if 𝐴𝐴𝐢𝐢 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.

Theorem 4.2: Every nano open set is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open.

Proof follows from the Theorem 3.2.

Remark 4.3: For subsets 𝐴𝐴,𝐡𝐡 of a nanotopological space(π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) i) π‘ˆπ‘ˆ βˆ’ π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(𝐴𝐴) =π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(π‘ˆπ‘ˆ βˆ’ 𝐴𝐴)

ii) π‘ˆπ‘ˆ βˆ’ π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴) =π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(π‘ˆπ‘ˆ βˆ’ 𝐴𝐴)

Theorem 4.4: A subset 𝐴𝐴 βŠ† π‘ˆπ‘ˆ is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open if and only if 𝐹𝐹 βŠ† π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(𝐴𝐴) whenever F is nano closed set and 𝐹𝐹 βŠ† 𝐴𝐴.

Proof: Let 𝐴𝐴 benanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open and suppose 𝐹𝐹 βŠ† 𝐴𝐴 where 𝐹𝐹 is nano closed. Then π‘ˆπ‘ˆ βˆ’ 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed set containing in the nano open set π‘ˆπ‘ˆ βˆ’ 𝐹𝐹. Hence π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(π‘ˆπ‘ˆ βˆ’ 𝐴𝐴)βŠ† π‘ˆπ‘ˆ βˆ’ 𝐹𝐹and π‘ˆπ‘ˆ βˆ’ π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† π‘ˆπ‘ˆ βˆ’ 𝐹𝐹. Thus 𝐹𝐹 βŠ† π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(𝐴𝐴).Conversely, if 𝐹𝐹 is a nano closed set with 𝐹𝐹 βŠ† π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(𝐴𝐴) and 𝐹𝐹 βŠ† 𝐴𝐴, then π‘ˆπ‘ˆ βˆ’ π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ†

π‘ˆπ‘ˆ βˆ’ 𝐹𝐹. Thus π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(π‘ˆπ‘ˆ βˆ’ 𝐴𝐴)βŠ† π‘ˆπ‘ˆ βˆ’ 𝐹𝐹. Hence π‘ˆπ‘ˆ βˆ’ 𝐴𝐴 is nanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed set and 𝐴𝐴 benanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open.

Remark 4.5: The union of two nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open sets need not be nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open can be seen from the following example.

Example 4.6: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ =οΏ½{π‘Žπ‘Ž}, {𝑏𝑏}, {𝑐𝑐,𝑑𝑑}οΏ½ and 𝑋𝑋= {π‘Žπ‘Ž,𝑏𝑏}. Then the nanotopology

πœπœπ‘…π‘…(𝑋𝑋) = {π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž,𝑏𝑏}, {π‘Žπ‘Ž,𝑏𝑏},βˆ…}. Then the sets {𝑐𝑐} and {𝑑𝑑} are nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open sets but {𝑐𝑐}βˆͺ{𝑑𝑑} = {𝑐𝑐,𝑑𝑑} is

notnanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open.

Remark 4.7: The intersection of two nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open sets need not be nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open set can be seen from the following example.

Example 4.8: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ =οΏ½{π‘Žπ‘Ž}, {𝑏𝑏}, {𝑐𝑐,𝑑𝑑}οΏ½ and 𝑋𝑋= {π‘Žπ‘Ž,𝑐𝑐}. Then the nanotopology πœπœπ‘…π‘…(𝑋𝑋) =οΏ½π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}, {𝑐𝑐,𝑑𝑑}οΏ½. Then the sets {𝑏𝑏,𝑐𝑐,𝑑𝑑} and {π‘Žπ‘Ž,𝑏𝑏,𝑑𝑑} are nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open sets but

{𝑏𝑏,𝑐𝑐,𝑑𝑑}∩{π‘Žπ‘Ž,𝑏𝑏,𝑑𝑑} = {𝑏𝑏,𝑑𝑑} is notnanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open

Theorem 4.9:π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† 𝐡𝐡 βŠ† 𝐴𝐴 and if 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open then 𝐡𝐡 is also nanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open.

(6)

Theorem 4.10: If 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘” 𝑏𝑏-closed then π‘π‘π‘Ÿπ‘Ÿπ‘”π‘” 𝑏𝑏𝑐𝑐𝑁𝑁(𝐴𝐴)βˆ’ 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘” 𝑏𝑏-open.

Proof: Let 𝐴𝐴 be nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed.Let 𝐹𝐹 be a nano closed set in (π‘ˆπ‘ˆ,πœπœπ‘…π‘…(𝑋𝑋)) such that 𝐹𝐹 βŠ† π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βˆ’ 𝐴𝐴. Then 𝐹𝐹=βˆ…. Since π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βˆ’ 𝐴𝐴 cannot have any non-empty nano closed set. Therefore 𝐹𝐹 βŠ† π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βˆ’ 𝐴𝐴. Hence π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π΄π΄βˆ’π΄π΄ is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open.

Theorem 4.11: If 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open, 𝐺𝐺 is nano open and π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(𝐴𝐴)βˆͺ 𝐴𝐴𝐢𝐢 βŠ† 𝐺𝐺, then 𝐺𝐺=π‘ˆπ‘ˆ.

Proof: Let 𝐴𝐴 be nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open and 𝐺𝐺 is nano open such that π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(𝐴𝐴)βˆͺ π΄π΄πΆπΆβŠ† 𝐺𝐺. Then πΊπΊπΆπΆβŠ† 𝐴𝐴 ∩ π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴𝐢𝐢)βŠ† π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴𝐢𝐢)βˆ’ 𝐴𝐴𝐢𝐢. Since 𝐴𝐴𝐢𝐢 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closedπ‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴𝐢𝐢)βˆ’ 𝐴𝐴𝐢𝐢 cannot contain any

non-empty nano closed set but 𝐺𝐺𝐢𝐢 is nano closed subset of π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴𝐢𝐢)βˆ’ 𝐴𝐴𝐢𝐢. Therefore 𝐺𝐺𝐢𝐢 =βˆ…. That is 𝐺𝐺 =π‘ˆπ‘ˆ.

Theorem 4.12: A subset 𝐴𝐴 βŠ† π‘ˆπ‘ˆ is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open if and only if 𝐹𝐹 βŠ† π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(𝐴𝐴) whenever𝐹𝐹 is nano closed set and 𝐹𝐹 βŠ† 𝐴𝐴.

Proof: Let 𝐴𝐴 be nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open and suppose 𝐹𝐹 βŠ† 𝐴𝐴 where 𝐹𝐹 is nano closed. Then π‘ˆπ‘ˆ βˆ’ 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed contained in the nano open setπ‘ˆπ‘ˆ βˆ’ 𝐹𝐹. Hence π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘(π‘ˆπ‘ˆ βˆ’ 𝐴𝐴)βŠ† π‘ˆπ‘ˆ βˆ’ 𝐹𝐹 and π‘ˆπ‘ˆ βˆ’ π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† π‘ˆπ‘ˆ βˆ’ 𝐹𝐹. Thus 𝐹𝐹 βŠ† π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(𝐴𝐴).Conversely, if 𝐹𝐹 is nano closed set with 𝐹𝐹 βŠ† π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(𝐴𝐴) and 𝐹𝐹 βŠ† 𝐴𝐴. Then π‘ˆπ‘ˆ βˆ’ π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ†

π‘ˆπ‘ˆ βˆ’ 𝐹𝐹. Thus π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—π‘π‘π‘π‘π‘π‘π‘π‘(π‘ˆπ‘ˆ βˆ’ 𝐴𝐴)βŠ† π‘ˆπ‘ˆ βˆ’ 𝐹𝐹. Hence π‘ˆπ‘ˆ βˆ’ 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed set and 𝐴𝐴 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-open set.

5. On

π‘΅π‘΅π‘΅π‘΅π’ˆπ’ˆ

βˆ—βˆ—

𝒃𝒃

- Continuity

Definition 5.1: Let (π‘ˆπ‘ˆ, πœπœπ‘…π‘…(𝑋𝑋)) and (𝑉𝑉, πœπœπ‘…π‘…(π‘Œπ‘Œ)) be nanotopological spaces 𝑓𝑓: (π‘ˆπ‘ˆ, πœπœπ‘…π‘…(𝑋𝑋))β†’(𝑉𝑉, πœπœπ‘…π‘…(π‘Œπ‘Œ)) is said to be i) nano continuous[10] if π‘“π‘“βˆ’1(𝐡𝐡) is nano closed in π‘ˆπ‘ˆ for every nano closed set𝐡𝐡 in 𝑉𝑉.

ii) nano𝛼𝛼- continuous if π‘“π‘“βˆ’1(𝐡𝐡) is nano𝛼𝛼- closed in π‘ˆπ‘ˆ for every nano closed set𝐡𝐡 in 𝑉𝑉. iii) nano semi continuous if π‘“π‘“βˆ’1(𝐡𝐡) is nano semi closed in π‘ˆπ‘ˆ for every nano closed set𝐡𝐡 in 𝑉𝑉. iv) nano pre continuous if π‘“π‘“βˆ’1(𝐡𝐡) is nano pre closed in π‘ˆπ‘ˆ for every nano closed set 𝐡𝐡 in 𝑉𝑉. v) nanoregular continuous if π‘“π‘“βˆ’1(𝐡𝐡) is nano regular closed in π‘ˆπ‘ˆ for every nano closed set𝐡𝐡 in 𝑉𝑉. vi) nano𝑔𝑔- continuous if π‘“π‘“βˆ’1(𝐡𝐡) is nano𝑔𝑔- closed in π‘ˆπ‘ˆ for every nano closed set𝐡𝐡 in 𝑉𝑉.

vii) nanoπ‘”π‘”βˆ—- continuous if π‘“π‘“βˆ’1(𝐡𝐡) is nano π‘”π‘”βˆ—-closed in π‘ˆπ‘ˆ for every nano closed set 𝐡𝐡 in 𝑉𝑉. viii) nano𝑔𝑔𝛼𝛼- continuous if π‘“π‘“βˆ’1(𝐡𝐡) is nano𝑔𝑔𝛼𝛼- closed in π‘ˆπ‘ˆ for every nano closed set𝐡𝐡 in 𝑉𝑉. ix) nano𝑔𝑔𝑁𝑁- continuous if π‘“π‘“βˆ’1(𝐡𝐡) is nano𝑔𝑔𝑁𝑁- closed in π‘ˆπ‘ˆ for every nano closed set𝐡𝐡 in 𝑉𝑉.

Definition 5.2: Let (π‘ˆπ‘ˆ, πœπœπ‘…π‘…(𝑋𝑋)) and (𝑉𝑉, πœπœπ‘…π‘…(π‘Œπ‘Œ)) be nanotopological spaces. Then a mapping 𝑓𝑓: (π‘ˆπ‘ˆ, πœπœπ‘…π‘…(𝑋𝑋))β†’ (𝑉𝑉, πœπœπ‘…π‘…(π‘Œπ‘Œ)) is said to be nano regular generalized star star𝑏𝑏-closed continuous (π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘- continuous) if the inverse

images of every nano closed set in 𝑉𝑉 is nano π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘-closed in π‘ˆπ‘ˆ.

Theorem 5.3: Let (π‘ˆπ‘ˆ, πœπœπ‘…π‘…(𝑋𝑋)) and (𝑉𝑉, πœπœπ‘…π‘…(π‘Œπ‘Œ)) be nanotopological spaces and a mapping 𝑓𝑓: (π‘ˆπ‘ˆ, πœπœπ‘…π‘…(𝑋𝑋))β†’(𝑉𝑉, πœπœπ‘…π‘…(π‘Œπ‘Œ)). Then every nano continuous, nano 𝛼𝛼- continuous, nano semi continuous, nano pre continuous, nano regular continuous, nano𝑔𝑔- continuous, nano π‘”π‘”βˆ—- continuous, nano𝑔𝑔𝛼𝛼- continuous, nano𝑔𝑔𝑁𝑁- continuous functions areπ‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘- continuous.

Remark 5.4: The converse of the above theorem need not be true whichcan be seen from the following example.

Example 5.5: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ = {{π‘Žπ‘Ž}, {𝑏𝑏}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}} and𝑋𝑋= {π‘Žπ‘Ž,𝑐𝑐}.Then the nano topology is defined asπœπœπ‘…π‘…(𝑋𝑋) = {π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}, {𝑐𝑐,𝑑𝑑}}. Let 𝑉𝑉= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with 𝑉𝑉 𝑅𝑅⁄ β€² = {{π‘Žπ‘Ž}, {𝑏𝑏,𝑐𝑐}, {𝑑𝑑}} andπ‘Œπ‘Œ= {π‘Žπ‘Ž,𝑏𝑏}.Then the nanotopology is defined asπœπœπ‘…π‘…(π‘Œπ‘Œ) = {𝑉𝑉,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐}, {𝑏𝑏,𝑐𝑐}}. Define 𝑓𝑓:π‘ˆπ‘ˆ β†’ 𝑉𝑉 as 𝑓𝑓(π‘Žπ‘Ž) =𝑑𝑑,𝑓𝑓(𝑏𝑏) =𝑐𝑐,𝑓𝑓(𝑐𝑐) = 𝑏𝑏,𝑓𝑓(𝑑𝑑) =π‘Žπ‘Ž. Then 𝑓𝑓is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘- continuous, but 𝑓𝑓is not nano continuous and nano semi continuous since π‘“π‘“βˆ’1{𝑏𝑏,𝑐𝑐,𝑑𝑑} = {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐} and π‘“π‘“βˆ’1{𝑏𝑏} = {𝑐𝑐} which not nano closed are and nano semi closed in π‘ˆπ‘ˆ where as {𝑏𝑏,𝑐𝑐,𝑑𝑑}, {𝑏𝑏} are nano closed in𝑉𝑉. Thus aπ‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘- continuous function is not nano continuous and nano semi continuous.

Example 5.6: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ = {{π‘Žπ‘Ž}, {𝑏𝑏}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}} and𝑋𝑋= {π‘Žπ‘Ž,𝑐𝑐}.Then the nano topology is defined asπœπœπ‘…π‘…(𝑋𝑋) = {π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}, {𝑐𝑐,𝑑𝑑}}. Let 𝑉𝑉= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with 𝑉𝑉 𝑅𝑅⁄ β€² = {{π‘Žπ‘Ž}, {𝑏𝑏,𝑐𝑐}, {𝑑𝑑}} andπ‘Œπ‘Œ= {π‘Žπ‘Ž,𝑏𝑏}.Then the nanotopology is defined asπœπœπ‘…π‘…(π‘Œπ‘Œ) = {𝑉𝑉,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐}, {𝑏𝑏,𝑐𝑐}}. Define 𝑓𝑓:π‘ˆπ‘ˆ β†’ 𝑉𝑉 as 𝑓𝑓(π‘Žπ‘Ž) =𝑐𝑐,𝑓𝑓(𝑏𝑏) =π‘Žπ‘Ž,𝑓𝑓(𝑐𝑐) = 𝑑𝑑,𝑓𝑓(𝑑𝑑) =𝑏𝑏. Then 𝑓𝑓is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘- continuous but 𝑓𝑓is not nano 𝛼𝛼- continuous since π‘“π‘“βˆ’1{π‘Žπ‘Ž,𝑏𝑏} = {𝑏𝑏,𝑑𝑑}which is not nano𝛼𝛼- closed in π‘ˆπ‘ˆ where as {π‘Žπ‘Ž,𝑏𝑏} are nano closed in 𝑉𝑉. Thus a π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘- continuous is not nano𝛼𝛼- continuous.

(7)

pre-Example 5.8: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ = {{π‘Žπ‘Ž}, {𝑏𝑏}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}} and𝑋𝑋= {π‘Žπ‘Ž,𝑐𝑐}.Then the nano topology is defined asπœπœπ‘…π‘…(𝑋𝑋) = {π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}, {𝑐𝑐,𝑑𝑑}}. Let 𝑉𝑉= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with 𝑉𝑉 𝑅𝑅⁄ β€² = {{π‘Žπ‘Ž}, {𝑏𝑏,𝑐𝑐}, {𝑑𝑑}} andπ‘Œπ‘Œ= {π‘Žπ‘Ž,𝑏𝑏}.Then the nanotopology is defined asπœπœπ‘…π‘…(π‘Œπ‘Œ) = {𝑉𝑉,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐}, {𝑏𝑏,𝑐𝑐}}. Define 𝑓𝑓:π‘ˆπ‘ˆ β†’ 𝑉𝑉 as 𝑓𝑓(π‘Žπ‘Ž) =𝑑𝑑,𝑓𝑓(𝑏𝑏) =𝑐𝑐,𝑓𝑓(𝑐𝑐) = π‘Žπ‘Ž,𝑓𝑓(𝑑𝑑) =𝑏𝑏. Then π‘“π‘“π‘π‘π‘π‘π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘- continuous is but 𝑓𝑓is not nano𝑔𝑔- continuous and nano π‘”π‘”βˆ—-continuous since π‘“π‘“βˆ’1{π‘Žπ‘Ž,𝑏𝑏} = {𝑐𝑐,𝑑𝑑}and π‘“π‘“βˆ’1{𝑏𝑏} = {𝑑𝑑} which are not nano𝑔𝑔- closed and nanoπ‘”π‘”βˆ—-closed in π‘ˆπ‘ˆ where as {π‘Žπ‘Ž,𝑏𝑏} and {𝑏𝑏}nano

closed in 𝑉𝑉. Thus a π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’ 𝑐𝑐𝑐𝑐𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑐𝑐𝑐𝑐𝑐𝑐𝑁𝑁 function is not nano 𝑔𝑔- continuous and nano π‘”π‘”βˆ—-continuous.

Example 5.9: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ = {{π‘Žπ‘Ž}, {𝑏𝑏}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}} and𝑋𝑋= {π‘Žπ‘Ž,𝑐𝑐}.Then the nano topology is defined asπœπœπ‘…π‘…(𝑋𝑋) = {π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}, {𝑐𝑐,𝑑𝑑}}. Let 𝑉𝑉= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with 𝑉𝑉 𝑅𝑅⁄ β€² = {{π‘Žπ‘Ž}, {𝑏𝑏,𝑐𝑐}, {𝑑𝑑}} andπ‘Œπ‘Œ= {π‘Žπ‘Ž,𝑏𝑏}.Then the nanotopology is defined asπœπœπ‘…π‘…(π‘Œπ‘Œ) = {𝑉𝑉,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐}, {𝑏𝑏,𝑐𝑐}}. Define 𝑓𝑓:π‘ˆπ‘ˆ β†’ 𝑉𝑉 as 𝑓𝑓(π‘Žπ‘Ž) =𝑑𝑑,𝑓𝑓(𝑏𝑏) =𝑐𝑐,𝑓𝑓(𝑐𝑐) = π‘Žπ‘Ž,𝑓𝑓(𝑑𝑑) =𝑏𝑏. Then 𝑓𝑓is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’ 𝑐𝑐𝑐𝑐𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑐𝑐𝑐𝑐𝑐𝑐𝑁𝑁 but 𝑓𝑓is not nano𝑔𝑔𝛼𝛼- continuous and nano 𝑔𝑔𝑁𝑁-continuous since π‘“π‘“βˆ’1{π‘Žπ‘Ž,𝑏𝑏} = {𝑐𝑐,𝑑𝑑} is not nano𝑔𝑔𝛼𝛼- closed and nano𝑔𝑔𝑁𝑁-closed in π‘ˆπ‘ˆ where as {π‘Žπ‘Ž,𝑏𝑏} is nano closed in 𝑉𝑉. Thus a nano π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’ continuous function is not nano𝑔𝑔𝛼𝛼- continuous and nano 𝑔𝑔𝑁𝑁-continuous.

Theorem 5.10: If (π‘ˆπ‘ˆ, πœπœπ‘…π‘…(𝑋𝑋)) and (𝑉𝑉,πœπœπ‘…π‘…(π‘Œπ‘Œ)) be nanotopological spaces with respect to𝑋𝑋 βŠ† π‘ˆπ‘ˆ and π‘Œπ‘Œ βŠ† 𝑉𝑉 respectively, then any function𝑓𝑓:π‘ˆπ‘ˆ β†’ 𝑉𝑉, the following conditions are equivalent:

i) 𝑓𝑓 is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’continuous.

ii) The inverse of each nano open set in π‘Œπ‘Œ is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’open in𝑋𝑋. iii) For each subset 𝐴𝐴 of𝑋𝑋, 𝑓𝑓(π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴))βŠ† 𝑁𝑁𝑐𝑐𝑁𝑁�𝑓𝑓(𝐴𝐴)οΏ½.

Proof:

(π’Šπ’Š)β‡’(π’Šπ’Šπ’Šπ’Š): Let 𝐡𝐡 be an nano open subset of π‘Œπ‘Œ. Thenπ‘Œπ‘Œ βˆ’ 𝐡𝐡 is nano closed in π‘Œπ‘Œ. Since 𝑓𝑓 is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’continuous,π‘“π‘“βˆ’1(π‘Œπ‘Œ βˆ’ 𝐡𝐡) is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’ closed in𝑋𝑋. That is,𝑋𝑋 βˆ’ π‘“π‘“βˆ’1(𝐡𝐡) is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’closed in𝑋𝑋. Hence

π‘“π‘“βˆ’1(𝐡𝐡) is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’ open in𝑋𝑋.

(π’Šπ’Šπ’Šπ’Š) β‡’(π’Šπ’Šπ’Šπ’Šπ’Šπ’Š):Let 𝐴𝐴 be a subset of 𝑋𝑋. Since 𝐴𝐴 βŠ‚ π‘“π‘“βˆ’1�𝑓𝑓(𝐴𝐴)οΏ½, 𝐴𝐴 βŠ‚ π‘“π‘“βˆ’1�𝑁𝑁𝑐𝑐𝑁𝑁�𝑓𝑓(𝐴𝐴)οΏ½οΏ½. Now 𝑁𝑁𝑐𝑐𝑁𝑁�𝑓𝑓(𝐴𝐴)οΏ½ is a nano closed set in π‘Œπ‘Œ. Then by (𝑁𝑁𝑁𝑁),π‘“π‘“βˆ’1�𝑁𝑁𝑐𝑐𝑁𝑁�𝑓𝑓(𝐴𝐴)οΏ½οΏ½ is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’ closed in𝑋𝑋 containing 𝐴𝐴. But π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴) is smallest π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’ closed set in𝑋𝑋 containing𝐴𝐴. Therefore π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)βŠ† π‘“π‘“βˆ’1�𝑁𝑁𝑐𝑐𝑁𝑁�𝑓𝑓(𝐴𝐴)οΏ½οΏ½. Hence π‘“π‘“οΏ½π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)οΏ½ βŠ†

𝑁𝑁𝑐𝑐𝑁𝑁�𝑓𝑓(𝐴𝐴)οΏ½.

(π’Šπ’Šπ’Šπ’Šπ’Šπ’Š) β‡’(π’Šπ’Šπ’Šπ’Š): Let 𝐡𝐡 be a nano closed subset ofπ‘Œπ‘Œ. Then π‘“π‘“βˆ’1(𝐡𝐡) is a subset of 𝑋𝑋. By (𝑁𝑁𝑁𝑁𝑁𝑁)𝑓𝑓 οΏ½π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘π‘π‘π‘π‘οΏ½π‘“π‘“βˆ’1(𝐡𝐡)οΏ½οΏ½ βŠ† 𝑁𝑁𝑐𝑐𝑁𝑁 οΏ½π‘“π‘“οΏ½π‘“π‘“βˆ’1(𝐡𝐡)οΏ½οΏ½ βŠ† 𝑁𝑁𝑐𝑐𝑁𝑁(𝐡𝐡) =𝐡𝐡. This impliesπ‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘π‘π‘π‘π‘οΏ½π‘“π‘“βˆ’1(𝐡𝐡)οΏ½ βŠ† π‘“π‘“βˆ’1(𝐡𝐡). But π‘“π‘“βˆ’1(𝐡𝐡)βŠ† π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘π‘π‘π‘π‘οΏ½π‘“π‘“βˆ’1(𝐡𝐡)οΏ½. Hence π‘“π‘“βˆ’1(𝐡𝐡) =π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘π‘π‘π‘π‘οΏ½π‘“π‘“βˆ’1(𝐡𝐡)οΏ½ and π‘“π‘“βˆ’1(𝐡𝐡) is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’ closed in𝑋𝑋. This implies that 𝑓𝑓 is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’continuous.

Remark 5.11: If 𝑓𝑓: (π‘ˆπ‘ˆ, πœπœπ‘…π‘…(𝑋𝑋))β†’(𝑉𝑉, πœπœπ‘…π‘…(π‘Œπ‘Œ)) is nanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘- continuous then 𝑓𝑓(π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)) is not necessarily equal to𝑁𝑁𝑐𝑐𝑁𝑁�𝑓𝑓(𝐴𝐴)οΏ½.

Example 5.12: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ = {{π‘Žπ‘Ž}, {𝑏𝑏}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}} and𝑋𝑋= {π‘Žπ‘Ž,𝑐𝑐}.Then the nano topology is defined asπœπœπ‘…π‘…(𝑋𝑋) = {π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}, {𝑐𝑐,𝑑𝑑}}. Let 𝑉𝑉= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with 𝑉𝑉 𝑅𝑅⁄ β€² = {{π‘Žπ‘Ž}, {𝑏𝑏,𝑐𝑐}, {𝑑𝑑}} andπ‘Œπ‘Œ= {π‘Žπ‘Ž,𝑏𝑏}.Then the nanotopology is defined asπœπœπ‘…π‘…(π‘Œπ‘Œ) = {𝑉𝑉,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐}, {𝑏𝑏,𝑐𝑐}}. Define 𝑓𝑓:π‘ˆπ‘ˆ β†’ 𝑉𝑉 as 𝑓𝑓(π‘Žπ‘Ž) =𝑑𝑑,𝑓𝑓(𝑏𝑏) =𝑐𝑐,𝑓𝑓(𝑐𝑐) = 𝑏𝑏,𝑓𝑓(𝑑𝑑) =π‘Žπ‘Ž. Then 𝑓𝑓is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’continuous.

i) Let𝐴𝐴= {π‘Žπ‘Ž}βŠ† 𝑉𝑉. Then 𝑓𝑓(π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴)) =𝑓𝑓({π‘Žπ‘Ž}) =𝑑𝑑. But𝑁𝑁𝑐𝑐𝑁𝑁�𝑓𝑓(𝐴𝐴)οΏ½=𝑁𝑁𝑐𝑐𝑁𝑁({𝑑𝑑}) = {𝑏𝑏,𝑐𝑐,𝑑𝑑}. Thus 𝑓𝑓(π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘π‘π‘π‘π‘(𝐴𝐴))β‰  𝑁𝑁𝑐𝑐𝑁𝑁�𝑓𝑓(𝐴𝐴)οΏ½, even though 𝑓𝑓is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’continuous.

Remark 5.13: If 𝑓𝑓: (π‘ˆπ‘ˆ, πœπœπ‘…π‘…(𝑋𝑋))β†’(𝑉𝑉, πœπœπ‘…π‘…(π‘Œπ‘Œ)) is nanoπ‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘- continuous then is not π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘π‘π‘π‘π‘οΏ½π‘“π‘“βˆ’1(𝐡𝐡)οΏ½ is not necessarily equal toπ‘“π‘“βˆ’1(𝑁𝑁𝑐𝑐𝑁𝑁(𝐡𝐡).

Example 5.14: Let π‘ˆπ‘ˆ= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with π‘ˆπ‘ˆ 𝑅𝑅⁄ = {{π‘Žπ‘Ž}, {𝑏𝑏}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}} and𝑋𝑋= {π‘Žπ‘Ž,𝑐𝑐}.Then the nano topology is defined asπœπœπ‘…π‘…(𝑋𝑋) = {π‘ˆπ‘ˆ,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑐𝑐,𝑑𝑑}, {𝑐𝑐,𝑑𝑑}}. Let 𝑉𝑉= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑} with 𝑉𝑉 𝑅𝑅⁄ β€² = {{π‘Žπ‘Ž}, {𝑏𝑏,𝑐𝑐}, {𝑑𝑑}} andπ‘Œπ‘Œ= {π‘Žπ‘Ž,𝑏𝑏}.Then the nanotopology is defined asπœπœπ‘…π‘…(π‘Œπ‘Œ) = {𝑉𝑉,πœ™πœ™, {π‘Žπ‘Ž}, {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐}, {𝑏𝑏,𝑐𝑐}}. Define 𝑓𝑓:π‘ˆπ‘ˆ β†’ 𝑉𝑉 as 𝑓𝑓(π‘Žπ‘Ž) =𝑑𝑑,𝑓𝑓(𝑏𝑏) =𝑐𝑐,𝑓𝑓(𝑐𝑐) = π‘Žπ‘Ž,𝑓𝑓(𝑑𝑑) =𝑏𝑏. Then 𝑓𝑓is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’continuous.

(8)

Theorem 5.15: If 𝑓𝑓 ∢ 𝑋𝑋 β†’ π‘Œπ‘Œ is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’ continuous and 𝑔𝑔 ∢ π‘Œπ‘Œ β†’ 𝑍𝑍 is nano continuous then their composition 𝑓𝑓 ∘ 𝑔𝑔:𝑋𝑋 β†’ 𝑍𝑍 is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’ continuous.

Proof: Let 𝑓𝑓 ∢ 𝑋𝑋 β†’ π‘Œπ‘Œ is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’ continuous and 𝑔𝑔 ∢ π‘Œπ‘Œ β†’ 𝑍𝑍 is nano continuous. Let π‘ˆπ‘ˆ be a nano closed set in 𝑍𝑍.Therefore π‘”π‘”βˆ’1(π‘ˆπ‘ˆ) is nano closed in Y and π‘“π‘“βˆ’1(π‘”π‘”βˆ’1(π‘ˆπ‘ˆ)) is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’ closed in 𝑋𝑋. Therefore 𝑓𝑓 ∘ 𝑔𝑔 is π‘π‘π‘Ÿπ‘Ÿπ‘”π‘”βˆ—βˆ—π‘π‘ βˆ’ continuous.

Remark 5.16: The above discussions are summarized in the following diagram

6. REFERENCES

1. Ahmed Al-Omari and Mohd. Salim Md. Noorani, On generalized b-closed sets, Bulletin of Malaysian Mathematical Sciences Society (2) 32(1), 2009, 19-30.

2. Banupriya P, Indirani K, On regular generalized star star b-closed sets in topological spaces, M.Phil Thesis, 2014, Nirmala College for Women, Coimbatore.

3. Bhuvaneshwari K, ThangaNachiyar R, On generalized 𝛼𝛼-closed sets (communicated).

4. Bhuvaneshwari K, MythiliGnanapriya K, Nano generalized closed sets, International Journal of Scientific and Research Publications, 14(5), 2014, 1-3.

5. Bhuvaneshwari K, Mythili Gnanapriya K, On nano generalized pre closed sets and nano pre generalized closed sets in nanotopological spaces, International Journal of Innovative Research in Science Engineering and Technology, 3(10), 2014, 16825-16829.

6. Sindhu K and Indirani K, On regular generalized star b-closed sets, IJMA- 4(10), 2013, 85-92.

7. Smitha M.G. and Indirani K, On nano regular generalized star b-closed sets in nanotoplogical spaces, International Journal of Physics and Mathematical Sciences, 5(3), 2015, 22-26.

8. Levine N, Generalised closed sets in topological spaces, Rend Circ, Mat. Palermo (2) 19, 1970, 89-96.

9. LellisThivagar.M and Carmel Richard, On nano forms of weakly open sets, International Journal of Mathematical and Statistics Invention, 1(1), 2012, 31-37.

10. LellisThivagar.M and Carmel Richard, β€œOn nano continuity”, Mathematical theory and modelling, 7, 2013, 32-37.

11. Rajendran V, Sathishmohan P, and Indiarni K, On nano generalized star closed sets in nanotopological spaces, communicated.

Source of support: Nil, Conflict of interest: None Declared

References

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