On Nag*S Closed and Nag*S Contra Continuous Functions

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On Nag*S Closed and Nag*S Contra Continuous Functions

1M. Dharani 2V. Senthilkumaran and 3Y. Palaniappan

1M.Phil Scholar, Aringar Anna Government Arts college, Musiri-621211, Tamilnadu, India.

2Associate Professor of Mathematics, Aringar Anna Government Arts college, Musiri-621211, Tamilnadu, India.

3Associate Professor of Mathematics (Retd), Aringar Anna Government Arts college, Musiri-621211, Tamilnadu, India.

Abstract:- The aim of this paper is to give and discuss stronger form of nano continuity called nano contra continuity using Ns closed sets.

Keywords: Nano topology, Ns closed sets, Ns contra continuity.

2010 AMS subject classification: 54B05,54C05.

1.INTRODUCTION

Ganster and Reily [4] discussed Lc continuous functions. Dontchev [3] had given contra continuous functions. Lellis Thivagar [5] 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 nano open sets. Nano is a Greek word which means very small. The topology studied here is given the name nano topology as it has at most five elements Certain weak forms of

nano sets were studied by various authors. Here we study a new form of closed functions called Ns closed function and its relation to various other closed functions. Also we analyze a new form of contra continuous function namely, Ns contra

continuous function and its relation to other contra continuous functions.

2. PRELIMINARIES

Definition 2.1: [5] Let be a non-empty finite set of objects called the universe and R be an equivalence relation on named as the indiscernibility relation. Elements belonging to the same equivalence class are said to be indiscernible with one another. The

pair (, ) is said to be the approximation space. Let X .

  1. The lower approximation of X with respect to R is the set of all objects, which can be for certain classified as

    X with respect to R and it is denoted by (). That is, () =

    {(): () }, where R(x) denotes

    the equivalence class determined by .

  2. The upper approximation of X with respect to R is the set of all objects, which can be possibly classified as X

    with respect to R and it is denoted by () . That is, ()=

    {(): () }.

  3. The boundary region of X with respect to R is the set of all objects, which can be classified neither as X nor

as not-X with respect to R and it is denoted by (). That is ()=() ().

Definition 2.2:[5] let be the universe, R be an equivalence relation on and (X)= { , , (), (), ()}, where X

. (X) satisfies the following axioms. (i) and ().

  1. The union of the elements of any subcollection of (X) is in (X).

  2. The intersection of the elements of any finite subcollection of X) is in (X).

That is, (X) forms a topology on called as the nano topology on with respect to X. we call (, (X)) as the nano topological space. The elements of (X) are called as nano-open sets. A set A is said to be nano closed if its complement is nano-open.

Definition 2.3: [5] If (, (X)) is a nano topological space with respect to X where X and if A , then the nano interior

of A is defined as the union of all nano-open subsets of A and it is denoted by Nint(A). That is, Nint(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 2.4: A nano subset A of a nano topological space (, (X)) is called a

  1. Nano pre closed if Ncl Nint(A)

  2. Nano semi closed if Nint Ncl(A)

  3. Nano closed if Ncl Nint Ncl(A)

  4. Nano semi pre closed if Nint Ncl Nint(A)

  5. Nano regular closed if Ncl Nint(A) =

    For a nano subset A of (, (X)) the intersection of all nano pre closed (nano semi closed, nano closed, nano semi pre closed) sets of (, (X)) containing A is called nano pre closure of A (nano semi closure of A, nano closure of A, nano semi pre closure of A) and is denoted by Npcl(A) (Nscl(A), Ncl(A), Nspcl(A)).

    Definition 2.5: A nano subset A of a nono topological space (, (X)) is called a

    1. Nano generalized closed (briefly Ng closed) if Ncl(A) henever A and U is nano open in .

    2. Nano generalized semi closed (briefly Ngs closed) if Nscl(A) whenever A and U is nano open in .

    3. Nano generalized regular closed (briefly Ngr closed) if Ncl(A) whenever A and U is nano regular open in .

    4. Nano generalized semi closed (briefly Ngs closed) if Ncl(A) whenever A and U is nano semi open in .

    5. Nano generalized closed (briefly Ng closed) if Ncl(A) whenever A and U is nano open in .

    6. Nano generalized semi pre closed (briefly Ngsp closed) if Nspcl(A) whenever A and U is nano open in .

    7. Nano generalized pre closed (briefly Ngp closed) if Npcl(A) whenever A and U is nano open in .

    8. Nano pre closed (briefly Np closed) if Npcl(A) whenever A and U is Ng open in .

    9. Nano generalized pre regular closed (briefly Ngpr closed) if Npcl(A) whenever A and U is nano regular open in .

    10. Nano semi generalized closed (briefly Nsg closed) if Nscl(A) whenever A and U is nano semi open in .

    11. Nano closed (briefly N closed) if Ncl(A) whenever A and U is Ng open in .

    12. Nano closed (briefly N closed) if Ncl(A) whenever A and U is Ng open in .

The complements of the above mentioned nano closed sets are respective nano open sets.

Definition: 2.6 Let (, (X)) and (, (Y)) be nano topological spaces. A function f: (, (X)) (, (Y)) s called

  1. Ns closed function if the image of every nano closed set of is Ns closed in .

  2. N closed function if the image of every nano closed set of is N closed in .

  3. N closed function if the image of every nano closed set of is N closed in .

  4. N closed function if the image of every nano closed set of is N closed in .

  5. N closed function if the image of every nano closed set of is N closed in .

  6. N closed function if the image of every nano closed set of is N closed in .

  7. N closed function if the image of every nano closed set of is N closed in .

  8. N closed function if the image of every nano closed set of is N closed in .

  9. Np closed function if the image of every nano closed set of is Np closed in .

  10. Nsg closed function if the image of every nano closed set of is Nsg closed in .

  11. N closed function if the image of every nano closed set of is N closed in .

  12. N closed function if the image of every nano closed set of is N closed in .

Definition: 2.7 Let (, (X)) and (, (Y)) be nano topological spaces. A function f: (, (X)) (, (Y)) is called

  1. Ns contra continuous function if 1() is Ns closed in for every nano open set V of .

  2. N contra continuous function if 1() is N closed in for every nano open set V of .

  3. N contra continuous function if 1() is N closed in for every nano open set V of .

  4. N contra continuous function if 1() is N closed in for every nano open set V of .

  5. N contra continuous function if 1() is N closed in for every nano open set V of .

  6. N contra continuous function if 1() is N closed in for every nano open set V of .

  7. N contra continuous function if 1() is N closed in for every nano open set V of .

  8. N contra continuous function if 1() is N closed in for every nano open set V of .

  9. Np contra continuous function if 1() is Np closed in for every nano open set V of .

  10. Nsg contra continuous function if 1() is Nsg closed in for every nano open set V of .

  11. N contra continuous function if 1() is N closed in for every nano open set V of .

  12. N contra continuous function if 1() is N closed in for every nano open set V of .

3. Ns CLOSED FUNCTION

Definition:3.1 A subset A of a nano topological space (, (X)) is said to be N semi closed (briefly Ns closed) set if Ncl(A) U, whenever AU and U is Ngs open in (, (X)).

Definition:3.2 Let (, (X)) and (, (Y)) be nano topological spaces. A function

1

f: (, (X)) (, (Y)) is called N s continuous. If

() is N s open in for every nano open set V of .

Definition:3.3 Let (, (X)) and (, (Y)) be nano topological spaces. A function

f: (, (X)) (, (Y)) is called N s closed if the image of every nano closed set of is N s closed in .

Theorem:3.4 Every nano closed map is Ns closed but not conversely.

Proof: Let f: (, (X)) (, (Y)) be a nano closed map. If A is nano closed in , then f(A) is nano closed in and hence Ns closed in .

Example:3.5 Let = {, , }, /R={{, }, {}} , X = {}, (X) = {, , {}}

Define f: (, (X)) (, (X)) by f(a)= a, f(b)= a, f(c)= b.

f is Ns closed but not nano closed function as f ({, }) = {} is not nano closed.

Theorem:3.6

  1. Every Ns closed function is Ns closed

  2. Every Ns closed function is N closed

  3. Every Ns closed function is N closed

  4. Every Ns closed function is N closed

  5. Every Ns closed function is N closed

  6. Every Ns closed function is N closed

  7. Every Ns closed function is N closed

  8. Every Ns closed function is N closed

  9. Every Ns closed function is Np closed

  10. Every Ns closed function is Nsg closed

  11. Every Ns closed function is N closed

  12. Every Ns closed function is N closed

Proof: Obvious from [2]

The converse of the above statements need not be true can be seen from the following examples.

Example:3.7 Let = {, , , }, /R={{}, {}, {, }} , X = {, },

(X) = { , , {}, {, , }, {, }}

Define f: (, (X)) (, (X)) by

f(a)= b, f(b)= a, f(c)= c, f(d)= d

f is Ns closed function but not Ns closed function as f{, }={, } is not Ns closed.

Example: 3.8 Refer Example 3.7

f is N closed function but not Ns closed function as f{, }={, } is not Ns closed.

Example: 3.9 Refer Example 3.7

f is N closed function but not Ns closed function as f{, }={, } is not Ns closed.

Example: 3.10 Refer Example 3.7

f is N closed function but not Ns closed function as f{, }={, } is not Ns closed.

Example: 3.11 Refer Example 3.7

f is N closed function but not Ns closed function as f{, }={, } is not Ns closed.

Example: 3.12 Refer Example 3.7

f is N closed function but not Ns closed function as f{, }={, } is not Ns closed.

Example: 3.13 Refer Example 3.7

f is N closed function but not Ns closed function as f{, }={, } is not Ns closed.

Example: 3.14 Refer Example 3.7

f is N closed function but not Ns closed function as f{, }={, } is not Ns closed.

Example: 3.15 Refer Example 3.7

f is N closed function but not Ns closed function as f{, }={, } is not Ns closed.

Example: 3.16 Refer Example 3.7

f is N closed function but not Ns closed function as f{, }={, } is not Ns closed.

Example: 3.17 Refer Example 3.7

f is N closed function but not Ns closed function as f{, }={, } is not Ns closed.

Example: 3.18 Refer Example 3.7

f is N closed function but not Ns closed function as f{, }={, } is not Ns closed.

Theorem:3.19 If f: (, (X)) (, (Y)) is nano closed and g: (, (Y)) (, (Z)) is N s closed, then g o f: (, (X))

( , (Z)) is N s closed.

Proof: obvious.

Theorem:3.20 A map f: (, (X)) (, (Y)) is N s closed if and only if for each nano subset S of and each nano open set U of such that 1() U, there is Ns open subset V of such that SV and 1(V) U.

Proof: Let f be Ns closed. Let S V and U be a nano open set of such that 1() U.

is closed in . f ( ) is Ns closed in .

V= f ( ) is Ns open in .

1(V)= -1(f ( )) – ( )=U Conversely, let F be nano closed in .

1(f ()) and is nano open in .

By assumption, there exists a Ns open subset V of such that f () and

1(V) .

This implies F (1(V))

Hence (()) = () (1(V))

So, f(F)=Vc , which is Ns closed.

Definition:3.21 Let and be a nano topological space. A map f: (, (X)) (, (Y)) is called N s open map if the image of every nano open set of is Ns open in .

Theorem:3.22 For any bijection map f: (, (X)) (, (Y)), the following are equivalent:

1. 1: (, (Y)) (, (X)) is N s continuous map.

  1. f is Ns open map.

  2. f is Ns closed map. Proof: (1)(2)

    Let U be nano open in . (1)1(U) is Ns open in . That is f(U) is Ns open in .

    (2)(3)

    Let F be a nano closed set of . Then is open in . By assumption, f () is Ns open in .

    f () = f () is Ns closed open in . f(F) is Ns closed in . (3)(1)

    Let F be nano closed in . f(F) is Ns closed in . f(F)= (1)1(F) is Ns closed in .

    Hence 1 is Ns continuous map.

  3. Ns CONTRA CONTINOUS FUNCTIONS

1

Definition:4.1 A function f: (, (X)) (, (Y)) is called N s contra continuous function if

for every nano open set V of .

Example:4.2 Refer Example 3.5

Define f: (, (X)) (, (X)) by f(a)= c, f(b)= a, f(c)= b f is Ns contra continuous function.

(V) is N s closed in

1

Definition:4.3 A function f: (, (X)) (, (Y)) is called almost N s contra continuous if

for every nano regular open set V of .

Theorem:4.4 Every Ns contra continuous function is almost Ns contra continuous.

Proof: The proof is obvious as every nano regular open set is nano open.

The converse of the above theorem need not be true can be seen from the following example.

Example:4.5 Refer Example 3.5

  1. is N s closed in

    Let f be the identity function. f is almost Ns contra continuous but not Ns contra continuous an 1({}) = {} is not Ns closed.

    Theorem:4.6

    1. Every Ns contra continuous function is Ns contra continuous.

    2. Every Ns cotra continuous function is N contra continuous.

    3. Every Ns contra continuous function is N contra continuous.

    4. Every Ns contra continuous function is N contra continuous.

    5. Every Ns contra continuous function is N contra continuous.

    6. Every Ns contra continuous function is N contra continuous.

    7. Every Ns contra continuous function is N contra continuous.

    8. Every Ns contra continuous function is N contra continuous.

    9. Every Ns contra continuous function is Np contra continuous.

    10. Every Ns contra continuous function is Nsg contra continuous.

    11. Every Ns contra continuous function is N contra continuous.

    12. Every Ns contra continuous function is N contra continuous.

Proof: Obviuous from[2]

The converse of the above statements need not be true can be seen from the following examples.

Example: 4.7 Refer Example 3.7

Define f: (, (X)) (, (X)) by f(a)= a, f(b)= b, f(c)= d, f(d)= c.

f is Ns contra continuous but not Ns contra continuous as 1({}) = {} is not Ns closed in .

Example: 4.8 Refer Example 3.7

Define f: (, (X)) (, (X)) by f(a)= d, f(b)= b, f(c)= a, f(d)= c.

f is N contra continuous but not Ns contra continuous as 1({, }) = {, } is not Ns closed in .

Example: 4.9 Refer Example 3.7

Define f: (, (X)) (, (X)) by f(a)= a, f(b)= a, f(c)= a, f(d)= c.

f is N contra continuous but not Ns contra continuous as 1({}) = {, , } is not Ns closed in .

Example: 4.10 Refer previous Example

f is N contra continuous but not Ns contra continuous as 1({}) = {, , } is not Ns closed in .

Example: 4.11 Refer Example 3.7

Define f: (, (X)) (, (X)) by f(a)= a, f(b)= b, f(c)= d, f(d)= d.

f is N contra continuous but not Ns contra continuous as 1({}) = {} is not Ns closed in .

Example: 4.12 Refer Example 4.9

f is Np contra continuous function but not Ns contra continuous function.

Example: 4.13 Refer Example 4.8

f is Np contra continuous function but not Ns contra continuous function.

Example: 4.14 Refer Example 4.8

f is Npr contra continuous function but not Ns contra continuous function.

Example: 4.15 Refer Example 4.8

f is Np contra continuous function but not Ns contra continuous function.

Example: 4.16 Refer Example 4.11

f is Nsg contra continuous function but not Ns contra continuous function.

Example: 4.17 Refer Example 4.9

f is N contra continuous function but not Ns contra continuous function.

Example: 4.18 Refer Example 4.9

f is N contra continuous function but not Ns contra continuous function.

Theorem:4.19 Let arbitrary union of Ns open sets be Ns open. Then the following statements are equivalent for a map f: (, (X)) (, (Y))

  1. f is Ns contra continuous.

  2. For every nano closed set F of , 1(F) is Ns open in .

  1. For each x and each nano closed set F of containing f(x), there exists Ns open set U containing x such that f(U)

    .

  2. For each x and each nano open set V of not containing f(x), there exists Ns closed set K not containing x such that 1(V) .

Proof: (1)(2)

Let F be nano closed in . Then is nano open in . By (1) 1( F) = 1(F) is Ns closed in . So, 1(F)

is Ns open. Hence (2) holds. (2)(1)

Let G be nano open in . Then is nano closed in . By (2) 1( G) = 1(G) is Ns open in . So, 1(G) is closed in . Hence (1) holds.

(2)(3)

Let F be nano closed in containing f(x). Hence x 1(F). By (2), 1(F) is Ns open in . Let U=1(F). This implies U is Ns open in containing x and f(U)=F . So (3) holds.

(3)(2)

Let F be a nano closed set of containing f(x). So, x 1(F). By (3), there exists Ns open set of containing x such that f () . So, 1(F) = {: 1(F)}. This is a union of Ns open sets, hence it is Ns open. Hence (2) holds. (3)(4)

Let V be nano open in not containing f(x). Then is closed in containing f(x). By (3), there exists Ns open set U in

containing x such that f(U) V. This implies U 1( V) = 1(). Hence1() -U. Let K= -U, K is

Ns closed in not containing x such that 1() . Hence (4) holds. (4)(3)

Let F be nano closed in containing f(x). Then is nano open in not containing f(x). From (4), there exists Ns closed set K in not containing x such that 1( F) .

This implies 1() . Hence 1(). That is, f ( ) . Let U= . Then U is Ns open in

containing x such that f(U) . So (3) holds.

Theorem:4.20 The following are equivalent for a map f: (, (X)) (, (Y))

1. f is almost Ns contra continuous.

  1. 1((())) is Ns closed in for every nano open set G of .

  2. 1((())) is Ns open in for every nano closed set F of .

Proof: (1)(2)

Let G be nano open in . Then Nint (Ncl(G)) is nano regular open in . Hence 1((())) is Ns closed in .

(2)(1) obvious (1)(3)

Let F be nano closed in . Then (()) is nano regular closed in . Hence 1((())) is Ns open in . (3)(1) obvious

Definition:4.21 A map f: (, (X)) (, (Y)) is said to be nano R-map if

regular open set V of .

1() is nano regular open in for each nano

Definition:4.22 A map f: (, (X)) (, (Y)) is said to be nano perfectly continuous if

nano open set V of .

1() is nano clopen in for each

Theorem:4.23 For two mappings f: (, (X)) (, (Y)) and g: (, (Y)) (, (Z)), then for g o f: , the following properties hold:

  1. If f is almost Ns contra continuous and g is a nano R map then g o f is almost Ns contra continuous.

  2. If f is almost Ns contra continuous and g is nano perfectly continuous, then g o f is almost Ns contra continuous and almost Ns continuous.

Proof: (1) obvious

(2) Let V be nano regular open in . 1() is nano clopen in and hence nano regular open and nano regular closed.

1(1()) = ( )1() is Ns open and Ns closed in as f is almost Ns contra continuous. So, g o f is almost contra Ns continuous and almost Ns continuous.

Definition:4.24 A nano topological space (, (X)) is called Ns space if every Ns open set is nano open.

Theorem:4.25 Let f: (, (X)) (, (Y)) be a N s contra continuous map and g: (, (Y)) (, (Z)) be N s continuous. If is a Ns space, then g o f: is an almost Ns contra continuous map.

Proof: Let V be nano regular open in . 1() is Ns open in . As is Ns space, Ns space, 1() is nano open in

. 1(1()) = ( )1() is Ns closed in . Hence g o f is almost Ns contra continuous map.

Definition:4.26 A map f: (, (X)) (, (Y)) is called strongly N s open (strongly N s closed) if (V) is N s open (Ns closed) in for every Ns open (Ns closed) set V of .

Theorem:4.27 If f: (, (X)) (, (Y)) is surjective strongly N s nano open (strongly N s closed) map and g: (, (Y))

(, (Z)) is a map such that g o f: is an almost N s nano contra continuous, then g is almost N s contra

continuous.

Proof: Let V be any regular closed (nano regular open) set in . As g o f is almost Ns contra continuous, ( )1() =

1(1()) is Ns open (Ns closed) in . Since f is surjective and strongly Ns open (nano strongly Ns closed), f (1(1()))= 1() is Ns open (Ns closed) in . Hence g is almost Ns contra continuous.

Definition:4.28 A map f: (, (X)) (, (Y)) is called weakly N s continuous if for each x and each nano open set V of containing f(x), there exists Ns open set U of such that f(U) ().

Theorem:4.29 If a map f: (, (X)) (, (Y)) is N s contra continuous, then f is weekly N s continuous map.

Proof: Let x and V be an open set in containing f(x). Ncl(V) is closed in containing f(x). Since f is Ns contra continuous, 1(Ncl(V)) is Ns open in . Let U=1(Ncl(V)). f(U)=f(1(Ncl(V))) Ncl(V). So, f is weakly Ns

continuous.

Definition:4.30 A space is called locally Ns indiscrete if every Ns open set is nano closed in .

Theorem:4.31 If a map f: (, (X)) (, (Y)) is almost N s contra continuous and is locally N s indiscrete, then f

is almost nano continuous.

Proof: Let V be nano regular open in . As f is almost Ns contra continuous. 1() is Ns closed in . Since is locally Ns indiscrete space, 1() is nano open in . Hence f is almost nano continuous.

5. DIFFERENT Ns FUNCTIONS

Definition:5.1 A function f: (, (X)) (, (Y)) is said to be N s continuous if and only if the inverse image of every nano open set in is Ns open in .

Definition:5.2 A function f: (, (X)) (, (Y)) is said to be N s nano contra continuous if and only if the inverse image of every closed set in is Ns open in .

1

1

Theorem:5.3 Let f: (, (X)) (, (Y)) be N s contra continuous. Thus

B .

() N s (Nint (

())), for every

Proof: f is Ns contra continuous. Ncl(B) is nano closed in . 1(Ncl(B)) is Ns open in . So, Ns (Nint(1(Ncl(B)))= 1(Ncl(B))

1(B) 1(Ncl(B)) = Ns (Nint(1(Ncl(B)))

Theorem:5.4 For a function f: (, (X)) (, (Y)), the following conditions are equivalent.

  1. f is Ns contra continuous.

  2. For each x and each nano closed set V in with f(x) , there exists an Ns open set U in such that x and f(U) , if arbitrary union of Ns open sets Ns open.

  3. The inverse image of each nano open set in is Ns closed in .

Proof: (1)(2)

Let f be Ns contra continuous.

Let x and V be a nano closed in containing f(x). So, x 1(), which is Ns open in .

Let 1() = . Hence x . f(U)=f1() .

(2)(1)

Let V be nano closed in . Let x 1(). So, f(x) V. Hence, there exists Ns open set containing x such that f () .

That is, x 1(). Therefore 1() is Ns open in X. (3)(1) is obvious

Definition:5.5 A subset S of is called locally nano Ns closed if S=U ,

where U is Ns open and F is Ns closed.

Definition: 5.6 f: (, (X)) (, (Y)) is said to be Lc- N s continuous if and only if the pre image of every nano open set of is locally Ns closed.

Theorem:5.7 Every Ns continuous function is Lc- Ns continuous.

1 1

1

Proof: Let f: (, (X)) (, (Y)) be N s continuous. Let V be nano open in .

which is locally Ns closed in . Hence f is Lc- Ns continuous.

() =

() N s (Ncl (

())),

Definition:5.8 Let (, (X)) be a nano topological space. A subset A is said to be Ns clopen if and only if it is Ns closed and Ns open.

Definition: 5.9 A function f: (, (X)) (, (Y)) is said to be strongly N s continuous if and only if the inverse of every subset of is Ns clopen.

Theorem:5.10 Every strongly nano Ns continuous function is Ns contra continuous.

1

Proof: Let f: (, (X)) (, (Y)) be strongly N s continuous. Let V be nano closed in . Then

clopen and hence 1() is Ns open. So f is Ns contra continuous.

The converse of the above theorem need not be true can be seen from the following example.

Example:5.11 Let = {, , }, /R={{, }, {}} , X = {},

(X) = { , , {}}

Define f: (, (X)) (, (X)) by

f(a)= c, f(b)= a, f(c)= b

f is Ns contra continuous but not Ns strongly continuous as 1{, }={, } is not Ns clopen.

Theorem:5.12 Every perfectly Ns continuous function is Ns contra continuous.

() is N s is N s

1

Proof: Let f: (, (X)) (, (Y)) be perfectly N s continuous. Let V be nano open in .

() is N s clopen which is

Ns closed. Hence f is Ns contra continuous. The converse of the above theorem need not be true can be seen from the

following example.

Example:5.13 Refer Example:5.11

f is Ns contra continuous but not Ns perfectly continuous as 1{}={} is not Ns clopen.

Definition: 5.14 A nano topological space (, (X)) is called strongly S- Ns closed if and only if every Ns closed cover of has a finite sub cover.

Theorem:5.15 Ns nano contra continuous images of strongly S- Ns closed spaces are nano compact.

Proof: Let f: (, (X)) (, (Y)) be N s contra continuous and onto. Let be strongly S- N s closed. Let ( ) be an nano open cover of . Then (1() ) is a Ns closed cover of . Since f is Ns contra continuous, then for some finite

J I, we have =

1( ). As f is onto =

. That is, is nano compact.

Definition: 5.16 A map f: (, (X)) (, (Y)) is called N s irresolute if and only if the inverse image of every N s closed set in is Ns closed in .

Remark:5.17 f is Ns irresolute the inverse image of every Ns open set under f is Ns open.

Theorem:5.18 Let f: (, (X)) (, (Y)), g: (, (Y)) (, (Z)) be functions. Then

  1. If f is Ns continuous and g is nano continuous, then g o f is Ns continuous.

  2. If f and g are Ns irresolute, g o f is Ns irresolute.

  3. If f is Ns irresolute and g is Ns continuous, then g o f is Ns continuous.

Proof: obvious

Definition: 5.19 f: (, (X)) (, (Y)) is said to be almost N s continuous if and only if the inverse image of every nano regular open set is Ns open in .

Theorem:5.20 f: (, (X)) (, (Y)) be a function. f is N s continuous f is almost N s continuous.

Proof: Let f be Ns continuous. Let U be nano regular open in . As nano regular open sets are nano open, U is nano open in

. So, 1() is Ns open in . Hence f is almost Ns continuous.

Definition: 5.21 A nano topological space (, (X)) is said to be Ns Hausdorff if and only if x and y are distinct points of

, there exists disjoint Ns open sets U and V containing x and y respectively.

Theorem:5.22 Let (, (X)) be a topological space and let (, (Y)) be a nano Hausdorff space. If f: (, (X)) (, (Y)) is injective and Ns continuous, then is a Ns Hausdorff space.

Proof: Let x and y be distinct points of . Since f is injective f(x) and f(y) are distinct points of . As is nano Hausdorff, there

exists disjoint nano open sets U and V containing f(x) and f(y) respectively.

Since f is Ns continuous and U and V are disjoint, 1() and, 1() are disjoint Ns open sets. So is Ns

Hausdorff.

Following the same lines, we can prove the following.

Theorem:5.23 Let (, (X)) be a nano topological space and let be Ns Hausdorff. If f: (, (X)) (, (Y)) is injective and Ns irresolute, then is Ns Hausdorff.

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