open sets and open sets in Quad Topological Space

q

 open sets and qopen sets in Quad Topological Space

U.D.Tapi1,, Ranu Sharma2, (Corresponding author)and Bhagyashri A. Deole3, 1,2,3

Department of Applied Mathematics and Computational Science, SGSITS, Indore (M.P.), India,

Abstract

The main aim of this paper is to introduce new types of open sets namely qopen sets and q open sets in

quad topological spaces along with their several properties and characterization .As application to qopen sets

and q open sets we introduce qcontinuous functions ,qcontinuous functions and obtainsome of

their basic properties.

Keywords:qopen sets, qcontinuous function andqopen sets ,qcontinuous function,,

AMS Mathematics Subject Classification (2010): 54A40

Introduction

In 1965 Njastad [8 ] introduced generalization of open set in topological space called pre semi open sets (  open sets). In 1990, M. Jelic [ 3] introduced the concept of open sets in bitopological spaces. The concept of bitopological spaces was introduced by J .C. Kelly [4]. In 1986, D. Andrijevic [1] was introduced semi-pre open sets (open sets) in topological spaces. F.H. Khedr, S.M. Al-Areefi and T. Noiri [ 5] generalized the notion of semi pre open sets to bitoplogical spaces and semi pre continuity in bitopological spaces. Tri topological space is a generalization of bitopological space.The study of tri-topological space was first initiated by Martin M. Kovar [6 ]. S. Palaniammal [9 ] studied tri tri-topological space. N. F. Hameed and Moh. Yahya Abid [2] defined 123 open set in tri topological space.we [10] introduced Topen set in tri topological space. We [11 ] introduce Topen set in tri topological space.

D.V. Mukundan [7 ] introduced the concept of quad topology (4-tuple topology) and defined new types of open (closed) sets. we [ 12 ] introduce semi open set and pre open set in quad topological space. The purpose of the present paper is to introduce qand qopen sets and q and qcontinuity and their

1. Preliminaries

Definition 1.1[8]: A subset A of a topological space ( , )X  is called pre-semi open set (open sets) if

int( (int ))

A cl A .The complement of pre semi open set is called semi closed set. The class of all pre-semi open sets of X is denoted byPSO X( , ) .

Definition 1.2[ 1]: A subset A of a topological space ( , )X  is called semi-pre open set ( open sets) if

(int( ))

Acl clA .The complement of semi-pre open set is called semi-pre closed set. The class of all semi pre open sets of X is denoted by SPO X( , ) .

Definition 1.3[10]: Let ( , ,X T T T1 2, 3)be a tri topological space. A subset A of X is called Topen in

X , ifA psintpsclpsintA_{. The complement of} Topen set is called Tclosed set. The collection of all

T

 open sets of X is denoted by PSO X T T T( , ,1 2, 3).

Definition 1.4[11]: Let ( , ,X T T T1 2, 3)be a tri topological space. A subset A of X is called Topen in

X , if AspclspintspclA_{. The complement of} Topen set is called Tclosed set. The collection of all

T

 open sets of X is denoted by SPO X T T T( , ,1 2, 3).

Definition 1.5 [ 7] :Let X be a nonempty set and T T T1, 2, 3 and T3are general topologies on X.Then a subset A of space X is said to be quad-open(q-open) set if 𝐴 ⊂ 𝑇₁∪ 𝑇₂∪ 𝑇₃∪ 𝑇₄ and its complement is said to be q-closed and set X with four topologies called quad topological spaces ( , ,X T T T T1 2, 3, 4) .q-open sets satisfy all the axioms of topology.

2. q open set in quad topological space

Definition 2.1: Let ( , ,X T T T T1 2, 3, 4)be a quad topological space. A subset A of X is called qopen in

X , ifA p sq qintp s clp sq q q qintA. The complement of qopen set is called qclosed set. The collection

of all qopen sets of X is denoted byp s O X T T Tq q ( , ,1 2, 3).

Example 2.2 :LetX { , , , }a b c d ,T1{ , ,{ }}X  a , T2 { , ,{ , , }}X  a b c ,T3 { , ,{ , , }}X  b c d ,

4 { , ,{ , }} T  X  b c

q- open sets in quad topological spaces are union of all four topologies.

q

 open set of X is denoted by p s O X_{q q} ( ){ , ,{ },{ , , },{ , , },{ , }}X  a a b c b c d b c . Let A{ , , }a b c ;

{ , , }

A a b c _isqopen.

int int{ , , } int { , , }

int

q q q q q q q q q q

q q

p s p s clp s a b c p s p s cl a b c

p s X

X

 

Definition 2.3: Let ( , ,X T T T T_{1 2}, ₃, ₄)be a quad topological space. LetAX. An element xA is called

q

 interior point ofA, if there exist aqopen set Usuch that x U A. The set of all q interior

points of A is called theqinterior of A and is denoted by p sq qint( )A .

Theorem 2.4: Let A Xbe a quad topological space.p s_{q q}int( )A is equal to the union of all qopen sets

contained inA.

Note 2.5: 1.p s_{q q}int( )A  A.

2. p s_{q q}int( )A isqopen sets.

Theorem 2.6: p s_{q q}int( )A is the largest qopen sets contained in A. Theorem 2.7: A is qopen if and only ifA p sq qint( )A

Theorem 2.8: p sq qint(AB) p sq qintAp sq qintB.

Definition 2.9: Let ( , ,X T T T T_{1 2}, ₃, ₄)be a quad topological space. Let AX. The intersection of all _q closed sets containing Ais called a qclosure of Aand is denoted as p s cl Aq q ( ).

Note 2.10 : Since intersection of qclosed sets is qclosed set,p s cl Aq q ( ) is a q closed set. Note 2.11:p s cl Aq q ( ) is the smallest qclosed set containingA.

Theorem 2.12: A is qclosed set if and only ifA p s cl Aq q ( ). Theorem 2.13: LetA and Bbe subsets of ( , ,X T T T T1 2, 3, 4)and xX

a) If AB, then p s cl Aq q ( )p s cl Bq q ( ).

b) xp s cl A_{q q} ( )if and only ifA U  for every qopen set U containing x.

Theorem 2.14: LetAbe a subsets of ( , ,X T T T T1 2, 3, 4), if there exist an qopen set U such that ( )

q q

A U p s cl A , then A is qopen.

Theorem 2.15: In a quad topological space ( , ,X T T T T1 2, 3, 4) , the union of any two qopen sets is always

an qopen set.

Proof: LetA and Bbe any two qopen sets inX .

Now A B p s cl p s_{q q}



_{q q}int

 

A



p s_{q q}cl



p s_{q q}int

 

B









int



q q

A B p s cl A B

    . Hence ABisqopen sets.

Theorem 2.16: Let A and Bbe subsets of X such that B A p s cl B_{q q} ( ).if Bis qopen set then Ais

also qopen set.

3. qContinuity in quad topological space

Definition 3.1: A function f from a quad topological space ( , ,X T T T T1 2, 3, 4)into another quad topological space



Y W W W W, ₁, ₂, ₃, ₄



is called _qcontinuous if f1

 

V is _qopen set in X for each quad open set V

in Y.

Example 3.2: Let, X { , , , }a b c d _,T₁{ , }X  ,T2 { , ,{ },{ , }}X  a b c ,

Open sets in quad topological spaces are union of all four topologies. Then quad open sets ofX { , ,{ },{ , },{ , , },{ , }}X  a b c a b c a d .

( )

q q

p s O X sets ofX { , ,{ },{ , },{ , , },{ , }}X  a b c a b c a d .

Let Y {1, 2,3, 4}, W1{ , }Y  ,W2 { , ,{1}.{2,3}}Y  , W3 { , ,{1, 2,3}},X  W4 { , ,{1, 4}}X  quad open sets of Y { , ,{1},{2,3},{1, 2,3},{1, 4}}Y  .

( )

q q

p s O Y sets of Y { , ,{1},{2,3},{1, 2,3},{1, 4}}Y  . Consider the function f X: Yis defined as

1

{1} { },

f  a f1{2,3} { , }, b c f1{1, 2,3} { , , } a b c f1( , )a d {1, 4},f1( ) , f1( )Y  X .

Since the inverse image of each quad open set in Yunder f is qopen set in X . Hence f is q

continuous function.

Theorem 3.3: Let f : ( ,X T1,T2,T3,T4)



Y W, 1,W2,W3,W4



be a qcontinuous open function .If A is an

q

 open set ofX , then f A( )is qopen in Y.

Proof : First ,let A be qopen set in X .There exist an quad open set UinX such that ( )

q q

A U p s cl A .since f is q open function then f U( )is quad open in Y. Since f is q continuous

function, we have f A( ) f U( ) f p s cl A( _{q q} ( )) p s cl f A_{q q} ( ( )).This show that f A( )is _q open in Y _. Let A be q open in X .There exist an q open set U such thatU A (p s cl Uq q ( )). Since f is q

continuous function, we have f U( ) f A( ) f p s cl U( q q ( )) p s cl f Uq q ( ( )).by the proof of first part,

( )

f U is qopen in X .Therefore , f A( )is q open inY.

Theorem 3.4: Let ( , ,X T T T T1 2, 3, 4)and



Y W W W W, 1, 2, 3, 4



be two quad topological space .Then f X: Y is q continuous function if and only if

1

( )

f V is qclosed in X whenever V is quad closed in Y.

Theorem 3.5: Let ( , ,X T T T T1 2, 3, 4)and



Y W W W W, 1, 2, 3, 4



be two quad topological spaces. A function

:

f X Y is qcontinuous if and only if the inverse image of every qopen set in Y is quad open in X. Theorem 3.6: Let f : ( ,X T₁,T₂,T₃,T₄)



Y W, ₁,W₂,W₃,W₄



be a _qcontinuous open function .If V is an

q

 open set of Y, then 1

( )

f V is qopen inX .

Proof : First ,let V be qopen set of Y.There exist an qset Win Y.such that V W  p s cl Vq q ( ).Since

f is quad open set ,we have f1( )V  f1( )W  f1(p s cl V_{q q} ( )) p s cl f_{q q} ( 1( ))V .since f is _q continuous, 1

( )

f W is q open set in X .By theorem 2.14 ,

1

( )

f V is qopen set in X .The proof of the

second part is shown by using the fact of first part.

Theorem 3.7: The following are equivalent for a function f : ( ,X T₁,T₂,T₃,T₄)



Y W, ₁,W₂,W₃,W₄



: a) f is qcontinuous function ;

c) For each xX and each quad open set V inWcontaining f x( ),there exist an _qopen set U of Xcontaining x such that f U( )V ;

d) p s cl f_{q q} ( 1( ))B  f1(p s cl B_{q q} ( ))for every subset BofY. e) f p s cl A( q q ( )) p s cl f Aq q ( ( ))for every subset AofX .

Theorem 3.8: If f : ( , ,X P P P P1 2, 3, 4)( ,Y W W W W1, 2, 3, 4)and g: ( ,Y W W W W1, 2, 3, 4)( , ,Z    1 2, 3, 4) be two _qcontinuous function then gof : ( , ,X P P P P1 2, 3, 4)( , ,Z    1 2, 3, 4) may not be qcontinuous

function.

Theorem 3.9: Let f : ( ,X T₁,T₂,T₃,T₄)



Y W, ₁,W₂,W₃,W₄



be bijective .Then the following conditions are equivalent:

i) f is qopen continuous function.

ii) f is qclosed continuous function and

iii) 1

f _is _qcontinuous function.

Proof:(i)(ii) Suppose B is a quad closed set in X .Then XB is an quad open set in X .Now by (i)

( )

f X B is a _qopen set in Y.Now since f is bijective so f X( B) Y f B( ).Hence f B( )is a _q closed set in Y.Therefore f is a _qclosed continuous function.

(ii)(iii) Let f is an qclosed map and Bbe q closed set of X .Since

1

f is bijective so 1 1

(f ) ( ) B

which is an qclosed set in Y. Hence

1

f is q continuous function.

(iii)(i) Let A be a quad open set in X .Since 1

f is a _qcontinuous function so 1 1

(f ) ( )A  f A( )is a

q

 open set in Y. Hence f is _qopen continuous function.

Theorem 3.10: Let X and Yare two quad topological spaces. Then

1 2 3 4 1 2 3 4

: ( , , , , ) ( , , , , )

f X P P P P  Y W W W W is q continuous function if one of the followings holds:

a) f1(p s_{q q}int ( ))B  p s_{q q}int (f1( ))B , for every quad open set BinY. b) p s cl f_{q q} ( 1( ))B  f1(p s cl B_{q q} ( )), for every quad open set BinY .

Proof: LetBbe any quad open set in Yand if condition (i) is satisfied then

1 1

( _{q q}int ( )) _{q q}int ( ( ))

f p s B p s f B .

We get f1( )B  p s_{q q}int (f1( ))B .Therefore 1 ( )

f B is a quad pre semi open set in X .Hence f is q

continuous function. Similarly we can prove (ii).

Theorem 3.11: A function f : ( , ,X P P P P1 2, 3, 4)( ,Y W W W W1, 2, 3, 4) is called qopen continuous

function if and only if f p s( _{q q}int ( ))A  p s_{q q}int ( ( ))f A , for every q open set Ain X . Proof: Suppose that f is a qopen continuous function.

Since p sq qint( )A  Asof p s( q qint( ))A  f A( ).

By hypothesis f p s( q qint( ))A is an q open set and p sq qint( ( ))f A is largest qopen set contained in ( )

Conversely, suppose A is an q open set in X .So f p s( q qint( ))A  p sq qint( ( ))f A .

Now since A p sq qint( )A so f A( ) p sq qint( ( ))f A .Therefore f A( )is a qopen set in Yand f is q

open continuous function.

Theorem 3.12: A function f : ( , ,X P P P P1 2, 3, 4)( ,Y W W W W1, 2, 3, 4) is called qclosed continuous

function if and only if p s cl f Aq q ( ( ))  f p s cl A( q q ( )), for every q closed set Ain X .

Proof: Suppose that f is a qclosed continuous function. Since A p s cl Aq q ( ) so f A( ) f pscl A( ( )).By

hypothesis, f p s cl A( q q ( ))is a qclosed set and p s cl f Aq q ( ( )) is smallest qclosed set containing f A( )

so p s cl f Aq q ( ( )) f p s cl A( q q ( )).

Conversely, supposeAis an _q closed set in X .Sop s cl f A_{q q} ( ( )) f p s cl A( _{q q} ( )).

Since A p s cl A_{q q} ( )so p s cl f A_{q q} ( ( )) f A( ).Therefore f A( )is a _qclosed set in Yand f is _qclosed continuous function.

Theorem 3.13: Let ( , ,X T T T T1 2, ,3 4)and



Y W W W W, 1, 2, 3, 4



be two quad topological space. Then, f X: Y is

q

 continuous function if and only if f p s cl A( q q ( )) p s cl f Aq q ( ( )) A X.

Proof: Suppose f X: Y is qcontinuous function. Since p s cl f Aq q [ ( )]is q closed in Y. Then by

theorem (3.4) 1

[ _{q q} ( ( ))]

f p s cl f A is _q closed in X ,

1 1

( ( ( ))) ( ( ))

q q q q q q

p s cl f p s clf A  f p s clf A − − − −(1).

Now : 1 1

( ) _{q q} ( ( ) ), ( ( )) ( _{q q} ( ))

f A  p s cl f A A f f A  f p s clf A .

Then 1 1

( ) ( ( ( ))) ( ( ))

q q q q q q q q

p s cl A  p s cl f p s clf A  f p s clf A by (1). Then f p s cl A( _{q q} ( )) p s cl f A_{q q} ( ( )).

Conversely, Let f p s cl A( q q ( )) p s cl f Aq q ( ( )) A X .

Let F be q closed set in Y, so that p s cl Fq q ( )F. Now

1

( )

f F  X _{, by hypothesis,}

1 1

( _{q q} ( ( ))) _{q q} ( ( ( )) _{q q} ( ) f p s cl f F  p s cl f f F  p s cl F F.

Therefore 1 1

( ( ) ( )

q q

p s cl f F  f F . But 1 1

( ) _{q q} ( ( )

f F  p s cl f F .

Hence 1 1

( ( ) ( )

q q

p s cl f F  f F and so 1 ( )

f F is q closed in X .

Hence by theorem (3.4) 𝑓 is qcontinuous function.

Theorem 3.14: Let ( , ,X T T T T1 2, ,3 4)and



Y W W W W, 1, 2, 3, 4



be two quad topological spaces. Then, f X: Y is

q

 continuous function if and only if p s cl f_{q q} ( 1( ))B  f1(p s cl B_{q q} ( )) B Y.

Proof: Suppose f X: Y is _qcontinuous. Since p s cl B_{q q} ( )is _q closed in Y, then by theorem (3.4) 1

( q q ( ))

f p s cl B is q closed in X and therefore ,

1 1

( ( ( ))) ( ( ))...(2)

q q q q q q

p s cl f p s cl B  f p s cl B Now,B p s cl B_{q q} ( ),then 1 1

( ) ( _{q q} ( ))

f B  f p s cl B ,then

1 1 1

( ( )) ( ( ( ))) ( )

q q q q q q q q

Conversely: Let the condition hold and let F be any quad closed set in Yso that p s cl Fq q ( )F. By

hypothesis, 1 1 1

( ( )) ( ( )) ( )

q q q q

p s cl f F  f p s cl F  f F .But 1 1

( ) _{q q} ( ( ))

f F  p s cl f F always.Hence

1 1

( ( )) ( )

q q

p s cl f F  f F and so 1 ( )

f F is q closed in X . It follows from theorem (3.4) that f is q

continuous function.

Theorem 3.15 Let ( , ,X T T T T1 2, ,3 4)and



Y W W W W, 1, 2, 3, 4



be two quad topological spaces. Then, f X: Y is

q

 continuous open function if and only if f1(p s_{q q}int( ))B  p s_{q q}int(f1( ))B  B Y.

Proof: Let f X: Y be a quad continuous. Since p s_{q q}int( )B is qopen inY , then by theorem (3.3)

1

( _{q q}int( ))

f p s B is qopen in X and therefore ,

1 1

int ( ( int( ))) ( int( ))...(3)

q q q q q q

p s f p s B  f p s B

Now, p s_{q q}int( )B B,then 1 1

( _{q q}int( )) ( )

f p s B  f B ,then 1 1

int( ( int( ))) int( ( ))

q q q q q q

p s f p s B  p s f B by (3)

Conversely: Let the condition hold and let G be any q open set in Yso thatp sq qint( )G G. By

hypothesis, 1 1

( _{q q}int( )) _{q q}int ( )

f p s G  p s f G .Since 1 1

( _{q q}int( )) ( ) f p s G  f G then

1 1

( ) q qint( ( ))

f G  p s f G But 1 1

int( ( )) ( )

q q

p s f G  f G always and so 1 1

int( ( )) ( )

q q

p s f G  f G .Therefore 1

( )

f G is _q open inX . Consequently by theorem (3.3) f is _q continuous function.

4. qopen sets in quad topological space

Definition 4.1: Let ( , ,X T T T T1 2, 3, 4)be a quad topological space. A subset A of X is called qopen in

X , if As p cls pq q q qints p clAq q . The complement of qopen set is called qclosed set. The collection

of all _qopen sets of X is denoted by S P O X T T_{q q} ( , ,_{1 2}).

Example 4.2: Let X { , , , }a b c d ,T1{ , ,{ },{ , , }}X  a a b d , T2 { , ,{ , }}X  a d , T3{ , ,{ }}X  b

4 { , ,{ , }}

T  X  a b

Quad open sets in quad topological spaces are union of all four topologies.

q

 Open set of X is denoted by s p O X_{q q} ( ){ , ,{ },{ },{ , },{ , },{ , , }}X  a b a b a d a b d . LetA{ }a ;

int { } int

q q q q q q q q q q

q q

s p cls p s p cl a s p cls p X

s p clX

X  



{ }

A a isqopen.

Definition 4.3: Let ( , ,X T T T T1 2, 3, 4)be a quad topological space. LetA X. An element xA is called

q

 interior point ofA, if there exist aqopen set V such thatx V A. The set of all q interior points

Theorem 4.4: Let A Xbe a quad topological space.s pq qint( )A is equal to the union of all qopen sets

contained inA.

Note 4.5: 1.s pq qint( )A  A.

2. s p_{q q}int( )A isqopen sets.

Theorem 4.6: s p_{q q}int( )A is the largest qopen sets contained in A. Theorem 4.7: A is qopen if and only ifAs pq qint( )A

Theorem 4.8: s p_{q q}int(AB)s p_{q q}intAs p_{q q}intB.

Definition 4.9: Let ( , ,X T T T T1 2, 3, 4)be a quad topological space. Let AX. The intersection of all q

closed sets containing Ais called a_q closure of Aand is denoted as s p cl A_{q q} ( ).

Note 4.10 : Since intersection of _qclosed sets is _qclosed set,s p cl A_{q q} ( ) is a _q closed set.

Note 4.11:s p cl A_{q q} ( ) is the smallest qclosed set containingA. Theorem 4.12:A is qclosed set if and only ifAs p cl Aq q ( ).

Theorem 4.13: LetA and Bbe subsets of ( , ,X T T T T1 2, 3, 4)and xX a) If AB, then s p cl Aq q ( )s p cl Bq q ( ).

b) xs p cl A_{q q} ( )if and only ifA U  for every _qopen set U containing x.

Theorem 4.14: LetAbe a subsets of( , ,X T T T T1 2, 3, 4), if there exist an qopen set U such that ( )

q q

A U s p cl A , then A is qopen.

Theorem 4.15: In a quad topological space ( , ,X T T T T_{1 2}, ₃, ₄) , the union of any two _qopen sets is always an qopen set.

Theorem 4.16: Let A and Bbe subsets of X such that B A s p cl B_{q q} ( ).if Bis qopen set then Ais

also qopen set.

5. TContinuity in quad topological space

Definition 5.1: A function f from a quad topological space ( , ,X T T T T1 2, 3, 4)into another quad topological space



Y W W W W, 1, 2, 3, 4



is called qcontinuous if

 

1

f V is qopen set in X for each quad open set V

in Y.

Example 5.2: Let, X { , , , }a b c d _,T1{ , }X  ,T2 { , ,{ }}X  d ,T3 { , ,{ , }},X  c d T4 { , ,{ }}X  c Open sets in quad topological spaces are union of four topologies.

Then quad open sets ofX { , ,{ },{ },{ , }}X  d c c d .

( )

q q

s p O X sets ofX { , ,{ },{ },{ , }}X  d c c d .

Let Y {1, 2,3, 4}, W1{ , }Y  ,W2 { , ,{1}}Y  , W3 { , ,{2}},Y  W4 { , ,{1, 2}}Y  quad open sets of Y { , ,{1},{2},{1, 2}}Y  .

( )

q q

1

{1} { },

f  c f1{2} { }, d f1{1, 2} { , } c d ,f1( ) , f1( )Y X .

Since the inverse image of each quad open set in Yunder f is qopen set in X . Hence f is q

continuous function.

Theorem 5.3: Let f : ( ,X T₁,T₂,T₃,T₄)



Y W, ₁,W₂,W₃,W₄



be a qcontinuous open function .If A is an

q

 open set ofX , then f A( )is qopen in Y.

Proof : First ,let A be qopen set in X .There exist an quad open set UinX such that ( )

q q

A U s p cl A .since f is q open function then f U( )is quad open in Y. Since f is q continuous

function, we have f A( ) f U( ) f s p cl A( _{q q} ( ))s p cl f A_{q q} ( ( )).This show that f A( )is q open in Y .

Let A be _q open in X .There exist an _q open set U such thatU A (p s cl U_{q q} ( )). Since f is _q continuous function, we have f U( ) f A( ) f s p cl U( _{q q} ( ))s p cl f U_{q q} ( ( )).by the proof of first part,

( )

f U is qopen in X .Therefore , f A( )is q open inY.

Theorem 5.4: Let ( , ,X T T T T1 2, 3, 4)and



Y W W W W, 1, 2, 3, 4



be two quad topological space .Then f X: Y is q continuous function if and only if

1

( )

f V is qclosed in X whenever V is quad closed in Y.

Theorem 5.5: Let ( , ,X T T T T1 2, 3, 4)and



Y W W W W, 1, 2, 3, 4



be two quad topological spaces. A function

:

f X Y is qcontinuous if and only if the inverse image of every qopen set in Yis quad open in

. X

Theorem 5.6: Let f : ( ,X T1,T2,T3,T4)



Y W, 1,W2,W3,W4



be a qcontinuous open function .If V is an

q

 open set of Y, then 1

( )

f V is qopen inX .

Proof : First ,let V be qopen set of Y.There exist an qset Win Y.such that V W s p cl Vq q ( ).Since

f is quad open set ,we have f1( )V  f1( )W  f1(s p cl V_{q q} ( ))s p cl f_{q q} ( 1( ))V .since f is q

continuous, 1

( )

f W is q open set in X .By theorem 4.14,

1

( )

f V is qopen set in X .The proof of the

second part is shown by using the fact of first part.

Theorem 5.7: The following are equivalent for a function f : ( ,X T₁,T₂,T₃,T₄)



Y W, ₁,W₂,W₃,W₄



: f) f is qcontinuous function ;

g) The inverse image of each _q closed set of Yis _q closed in X ;

h) For each xX and each quad open set V inWcontaining f x( ),there exist an qopen set U of

Xcontaining x such that f U( )V ;

Theorem 5. 8: If f : ( , ,X P P P P_{1 2}, ₃, ₄)( ,Y W W W W₁, ₂, ₃, ₄)and g: ( ,Y W W W W₁, ₂, ₃, ₄)( ,Z    ₁, ₂, ₃, ₄)

be two _qcontinuous function then gof : ( , ,X P P P P_{1 2}, ₃, ₄)( , ,Z    _{1 2}, ₃, ₄) may not be _qcontinuous function .

Theorem 5.9: Let f : ( , ,X P P P P_{1 2}, ₃, ₄)( ,Y W W W W₁, ₂, ₃, ₄) be bijective .Then the following conditions are equivalent:

i) f is q open continuous function.

ii) f is qclosed continuous function and

iii) 1

f _is _qcontinuous function.

Proof:(i)(ii) Suppose B is a quad closed set in X .Then XB is an quad open set in X .Now by (i)

( )

f X B is aqopen set in Y.Now since f is bijective so f X( B) Y f B( ).Hence f B( )is a q

closed set in Y.Therefore f is a qclosed continuous function.

(ii)(iii) Let f is an qclosed map and Bbe q closed set of X .Since

1

f is bijective so 1 1

(f ) ( ) B

which is an _qclosed set in Y. Hence 1

f is _qcontinuous function. (iii)(i) Let A be a quad open set in X .Since 1

f is a qcontinuous function so

1 1

(f ) ( )A  f A( )is a

q

 open set in Y. Hence f is _qopen continuous function.

Theorem 5.10: Let X and Yare two quad topological spaces. Then





1 2 3 4 1 2 3 4

: ( , , , , ) Y W, , , ,

f X T T T T  W W W is q continuous function if one of the followings holds:

c) f1(s p_{q q}int ( ))B s p_{q q}int (f1( ))B , for every quad open set BinY. d) s p cl f_{q q} ( 1( ))B  f1(s p cl B_{q q} ( )), for every quad open set BinY .

Proof: LetBbe any quad open set in Yand if condition (i) is satisfied then

1 1

( _{q q}int ( )) _{q q}int ( ( ))

f s p B s p f B .

We get f1( )B  s p_{q q}int (f1( ))B .Therefore 1 ( )

f B is a q open set in X .Hence f is qcontinuous

function. Similarly we can prove (ii).

Theorem 5.11: A function f : ( ,X T₁,T₂,T₃,T₄)



Y W, ₁,W₂,W₃,W₄



is called qopen continuous function

if and only if f s p( q qint ( ))A  s pq qint ( ( ))f A , for every quad open set Ain X . Proof: Suppose that f is a_qopen continuous function.

sinces p_{q q}int( )A Aso f s p( _{q q}int( ))A  f A( ).

By hypothesis f s p( _{q q}int( ))A is an q open set and s pq qint( ( ))f A is largest qopen set contained in

( )

f A so f s p( _{q q}int( ))A s p_{q q}int( ( ))f A .

Conversely, suppose A is an quad open set in X .So f s p( _{q q}int( ))A s p_{q q}int( ( ))f A .

Now since As pq qint( )A so f A( )s pq qint( ( ))f A .Therefore f A( )is aqopen set in Yand f is q

Theorem 5.12:A function f : ( ,X T₁,T₂,T₃,T₄)



Y W, ₁,W₂,W₃,W₄



is called qclosed continuous

function if and only if s p cl f A_{q q} ( ( ))  f s p cl A( _{q q} ( )), for every quad closed set Ain X .

Proof: Suppose that f is a qclosed continuous function. sinceAs p cl Aq q ( ) so f A( ) f s p cl A( q q ( ))

.By hypothesis, f s p cl A( _{q q} ( ))is a qclosed set and s p cl f Aq q ( ( )) is smallest qclosed set containing

( )

f A so s p cl f A_{q q} ( ( )) f s p cl A( _{q q} ( )).

Conversely, supposeAis an quad closed set in X .Sos p cl f A_{q q} ( ( )) f s p cl A( _{q q} ( )).

Since As p cl Aq q ( )so s p cl f Aq q ( ( )) f A( ).Therefore f A( )is a qclosed set in Yand f is qclosed

continuous function.

Theorem 5.13: Let ( , ,X T T T T1 2, 3, 4)and



Y W W W W, 1, 2, 3, 4



be two quad topological space .Then,

:

f X Yisqcontinuous function if and only if

1

( )

f V is qclosed in X whenever V is quad closed in

Y

Theorem 5.14: Let ( , ,X T T T T1 2, 3, 4)and



Y W W W W, 1, 2, 3, 4



be two quad topological space . Then, f X: Y is

q

 continuous function if and only if f s p cl A( q q ( ))s p cl f Aq q ( ( )) A X.

Proof: Suppose f X: Yis _qcontinuous function. Since s p cl f A_{q q} [ ( )]is _q closed in Y. Then by theorem (5.4) 1

[ _{q q} ( ( ))]

f s p cl f A is q closed in X ,

1 1

( ( ( ))) ( ( ))

q q q q q q

s p cl f s p clf A  f s p clf A − − − −(1).

Now: 1 1

( ) _{q q} ( ( ) ), ( ( )) ( _{q q} ( ))

f A s p cl f A A f f A  f s p clf A .

Then 1 1

( ) ( ( ( ))) ( ( ))

q q q q q q q q

s p cl A s p cl f s p clf A  f s p clf A by (1). Then f s p cl A( q q ( ))s p cl f Aq q ( ( )).

Conversely, Let f s p cl A( q q ( ))s p cl f Aq q ( ( )) A X .

Let F beq closed set in Y ,so that s p cl Fq q ( )F.Now

1

( )

f F  X , by hypothesis,

1 1

( _{q q} ( ( ))) _{q q} ( ( ( )) _{q q} ( ) f s p cl f F s p cl f f F s p cl F F.

Therefore 1 1

( ( ) ( )

q q

s p cl f F  f F . But 1 1

( ) _{q q} ( ( )

f F s p cl f F always.

Hence 1 1

( ( ) ( )

q q

s p cl f F  f F and so 1 ( )

f F is q closed in X .

Hence by theorem (5.14) 𝑓 is qcontinuous function.

Theorem 5.15: Let ( , ,X T T T T1 2, 3, 4)and



Y W W W W, 1, 2, 3, 4



be two quad topological spaces. Then, f X: Y is

q

 continuous function if and only if s p cl f_{q q} ( 1( ))B  f1(s p cl B_{q q} ( )) B Y.

Proof: Suppose f X: Yis qcontinuous. Since s p cl Bq q ( )is q closed in Y , then by theorem (5.4)

1

( _{q q} ( ))

f s p cl B is q closed in X and therefore ,

1 1

( ( ( ))) ( ( ))...(2)

q q q q q q

s p cl f s p cl B  f s p cl B Now, Bs p cl Bq q ( )then

1 1

( ) ( _{q q} ( ))

f B  f s p cl B ,

then 1 1 1

( ( )) ( ( ( ))) ( )

q q q q q q q q

Conversely: Let the condition hold and let F be any quad closed set in Yso thatspcl F( )F. By

hypothesis, 1 1 1

( ( )) ( ( )) ( )

q q q q

s p cl f F  f s p cl F  f F .But 1 1

( ) _{q q} ( ( ))

f F s p cl f F always. Hence

1 1

( ( )) ( )

q q

s p cl f F  f F and so 1 ( )

f F is _q closed inX . It follows from theorem (5.6) that f is _q continuous function.

Theorem 5.16: Let ( , ,X T T T T1 2, 3, 4)and



Y W W W W, 1, 2, 3, 4



be two quad topological spaces. Then, f X: Y is

q

 open continuous function if and only if f1(s p_{q q}int( ))B s p_{q q}int(f1( ))B  B Y.

Proof: Let f X: Ybe a quad continuous. Since s p_{q q}int( )B is qopen in Y, then by theorem (5.3)

1

( _{q q}int( ))

f s p B is qopen in X and therefore ,

1 1

int ( ( int( ))) ( int( ))...(3)

q q q q q q

s p f s p B  f s p B Now,s pq qint( )B B, then

1 1

( _{q q}int( )) ( )

f s p B  f B ,then 1 1

int( ( int( ))) int( ( ))

q q q q q q

s p f s p B s p f B by (3)

Conversely: Let the condition hold and let G be any q open set in Y so thats pq qint( )G G.

Byhypothesis, 1 1

( _{q q}int( )) _{q q}int ( )

f s p G s p f G .Since 1 1

( _{q q}int( )) ( ) f s p G  f G then

1 1

( ) _{q q}int( ( ))

f G s p f G But 1 1

int( ( )) ( )

q q

s p f G  f G always and so 1 1

int( ( )) ( )

q q

s p f G  f G .Therefore 1

( )

f G is _q open inX . Consequently by theorem (5.2) f is _q continuous function.

CONCLUSION: In this paper we studied new forms of _qopen sets and_qopen set in quad topological space. We also studied _q continuous function and _qcontinuous function in quad topological space. It is established that composition of any two qopen sets is again aqopen set in quad topological space

and composition of any two qopen sets is again aqopen set in quad topological space. In q

continuity, inverse image of every quad open is q open set and in qcontinuity, inverse image of every

quad open is q open set.

REFERENCES

[1] Andrijevic D., Semi-pre open sets , Matematicki Vesnik,1986,38,93,24-32.

[2]Hameed N.F. & Mohammed Yahya Abid , Certain types of separation axioms in tri topological spaces, Iraqi journal of science, 2011,Vol 52,(2),212-217.

[3] Jelic M., A decomposition of pairwise continuity. J. Inst. Math. Comput. Sci. Math. Ser, 1990 , 3,25-29.

[4] Kelly J. C , Bitopological spaces,Proc.LondonMath.Soc.,1963,3, 17-89 .

[5] Khedr F.H. , Al-Areefi S.M. &NoiriT. ,Pre continuity and semi pre continuity in bitopological spaces Indian J. Pure and Appl. Math.,1992 , 23, 625-633.

[7] Mukundan D.V. , Introduction to Quad topological spaces, Int. Journal of Scientific & Engg. Research , Vol.4,Issue 7,2483-2485,July-2013.

[8] Njastad O , On some classes of nearly open sets ,Pacific J. Math.,1965, 15 ,961-970.

[9] Palaniammal S. , Study of Tri topological spaces, Ph.D Thesis, 2011.

[10] Tapi U.D. and R. Sharma,Topen sets in Tri Topological space, International Journal of Innovative

Research in Science and Engineering ,Vol. 3, No. 3, 26 .

[11] Tapi U.D., R. Sharma and B. Deole,Topen sets in Tri Topological space, accepted for publication in

international journal of scientific Research.

[12] Tapi U.D., R. Sharma and B. Deole , semi open and pre open sets in quad topological space,under communication.