Weak Continuity Via Bioperation-Semiopen Sets

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Weak Continuity Via Bioperation-Semiopen Sets

K. Jhansi rani, G. Shanmugam and N. Rajesh

ABSTRACT. In this paper, we introduce and study the weak form of

] ,

[γ γ' -[β,β']-semicontinuous functions called weakly [γ,γ']-[β,β'] -semicontinuous functions between bioperation-topological spaces.

1. INTRODUCTION

Generalized open sets play a very important role in General Topology and they are now the research topics of many topologists worldwide. Indeed a significant theme in General Topology and Real analysis concerns the various modified forms of continuity, seperation axioms etc. by utilizing generalized open sets. Kasahara [1] defined the concept of an operation on topological spaces. Maki and Noiri [3] introduced the notion of

] , [γ γ'

τ , which is the collection of all [γ,γ']-open sets in a topological space (X,τ). In

this paper, we introduce and study the weak form of [γ,γ']

-[β,β']

-semicontinuous

functions called weakly [γ,γ']

-[β,β']

-semicontinuous functions between bioperation-topological spaces.

2 PRELIMINARIES

The closure and the interior of a subset A of (X,τ) are denoted by Cl(A) and Int(A), respectively.

Definition 2.1 [1] Let (X,τ) be a topological space. An operation γ on the topology τ is function from τ on to power set P(X) of X such that V ⊂Vγ for each V∈τ , where

γ

V denotes the value of τ at V . It is denoted by γ :τ →P(X).

Definition 2.2 [3] A topological space (X,τ) equipped with two operations namely γ

and γ' defined on τ is called a bioperation-topological space and it is denoted by

) , , ,

(X τ γ γ' .

Definition 2.3 A subset A of a topological space (X,τ) is said to be [γ,γ']-open set [3] if for each x∈A there exist open neighbourhoods U and V of x such that

A V

Uγ ∩ γ' ⊂ . The complement of a [γ,γ']-open set is called a [γ,γ']-closed set. Also

] , [γγ'

τ denotes set of all [γ,γ']-open sets in (X,τ).

2000 Mathematics Subject Classification 54D05, 54C08, 54B05.

Keywords and phrases. [γ,γ']- semiopen set, weakly [γ,γ']-semicontinuous functions. K. JHANSI RANI, G. SHANMUGAM AND N. RAJESH

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Definition 2.4. [3] For a subset A of (X,τ),

] , [γγ'

τ -Cl(A) denotes the intersection of all

] ,

[γ γ' -closed sets containing A, that is,

] , [γ γ'

τ - ( )= { : , \ }

] , [ '

F X F A F A

Cl

γ γ τ

∈ ⊂



.

Definition 2.5. Let A be any subset of X . The

] , [γ γ'

τ -Int(A) is defined as

] , [γγ'

τ

-U U A

Int( )=∪{ : is a [γ,γ']-open set and U ⊂ A}.

Definition 2.6. A subset A of a topological space (X,τ) is said to be [γ,γ']-semiopen [2] if

] , [ '

A γ γ

τ

⊂

-] , [

( _'

Cl γ γ

τ -Int(A)).

Theorem 2.7. A subset A of a bioperation-topological space (X,τ,γ,γ') is [γ,γ']-semi

open if,and only if for each x∈X there exists a [γ,γ']-semiopen set U suchthat A

U

x∈ ⊂ .

Definition 2.8. [2] A function f :(X,τ,γ,γ')→(Y,σ,β,β') is said to be [γ,γ']-[β,β']

-semicontinuous if the inverse image of every [β,β']-open set in (Y,σ,β,β') is a [γ,γ']

-semiopen set in (X,τ,γ,γ').

3 WEAK BIOPERATION-SEMICONTINUOUS FUNCTIONS

In this section, we define weakly [γ,γ']-[β,β']-semicontinuous function in bioperation-topological space and study some of their properties and theorems on them.

Definition 3.1. A function :( , , , ') ( , , , ') Y

X

f τ γ γ → σ β β is said to be weakly [γ,γ']

-] , [β β'

-semicontinuous if for each x∈X and each [β,β']

-open set V of Y containing

) (x

f , there exists a [γ,γ']

-semiopen set U containing x such that ( ) [ , '] U

f ⊂ β β

-) (V Cl .

Proposition 3.2. Every almost [γ,γ']-[β,β']-semi continuous function is weakly

] ,

[γ γ' -[β,β']-semicontinuous.

Proof. Let U be any [β,β']-regular open subset in Y. Then, since f is almost [γ,γ']

-] ,

[β β' -semicontinuous, f−1(U) is a [γ,γ']-semiopen set in X . From the fact

] , [ '

U ⊂ β β -Int([β,β']-Cl(U)), it follows f−1(U)=[γ,γ']-sInt(f−1(U))⊂[γ,γ']

-] , ([ (f 1 '

sInt − β β -Int([β,β']-Cl(U))))⊂[γ,γ']-sInt(f−1([β,β']Cl(U))). Thus, f is

weakly [γ,γ']-[β,β']-semicontinuous.

The following example shows that the converse of Proposition 3.2 is not true, in general.

K. JHANSI RANI, G. SHANMUGAM AND N. RAJESH

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Example 3.3 Let X ={a,b,c}, τ ={∅,{b},{c},{a,b},{b,c},X} and take the

} }, , { }, , { }, { }, { , {

= ∅ a b a b b c X

σ . Let γ,γ' :τ →P(X) and β,β' :σ →P(X) be operations defined as follows:

  

≠ ∪

  

≠

∪ { } { },

}, { = =

}, { }

{

}, { = =

c A if b A

c A if A A

b A if c A

b A if A

Aγ γ'

  

≠ ∪

  

≠

∪ { } { },

}, { = =

}, { }

{

}, { = =

b A if a A

b A if A A

and a A if b A

a A if A

Aβ β'

Then the identity function f :(X,τ,γ,γ')→(Y,σ,β,β') is weak [γ,γ']-[β,β']-semi

continuous but not almost [γ,γ']-[β,β']-semicontinuous.

Corollary 3.4. Every [γ,γ']-[β,β']-semi continuous function is weakly [γ,γ']-[β,β'] -semicontinuous.

Theorem 3.5. For a function f :(X,τ,γ,γ')→(Y,σ,β,β'), the following properties are equivalent:

(1). f is weakly [γ,γ']-[β,β']-semicontinuous;

(2). [γ,γ']-sCl(f−1([β,β']-Int([β,β']-Cl(B))))⊂ f−1([β,β']-Cl(B)) for every subset B of Y;

(3). [γ,γ']-sCl(f−1([β,β']-Int(F)))⊂ f−1(F) for every [β,β']-regular closed set F of Y;

(4). [γ,γ']-sCl(f−1(V))⊂ f−1([β,β']-Cl(V)) for every [β,β']-open set V of Y;

(5). f−1(V)⊂[γ,γ']-sInt(f−1([β,β']-Cl(V))) for every [β,β']-open set V of Y.

Proof. (1) ⇒ (2): Let B be any subset of Y. Suppose that x∈X \ f−1([β,β']-Cl(B)).

Then f(x)∈Y\[β,β']-Cl(B) and there exists a [β,β']-open set V of Y containing

) (x

f such that V∩B=∅. Hence, V∩[β,β']-Int([β,β']-Cl(B))=∅ and then

] ,

[β β' -Cl(V)∩[β,β']-Int([β,β']-Cl(B))=∅. By assumption, there exists a [γ,γ']

-semiopen set U containing x such that f(U)⊂[β,β']-Cl(V). Hence, we have

] , ([

1 '

f

U∩ − β β -Int([β,β']-Cl(B)))=∅ and x∈X \[γ,γ']-sCl(f−1([β,β']

-] , ([ '

Int β β -Cl(B)))). Thus, we obtain [γ,γ']-sCl(f−1([β,β']-Int([β,β']

-] , ([ ))))

(B f 1 '

Cl ⊂ − β β -Cl(B)).

(2) ⇒ (3): Let F be a [β,β']-regular closed subset of Y. Then F =[β,β']

-] , ([ '

Cl β β -Int(F)) and we have [γ,γ']-sCl(f−1([β,β']-Int(F)))=[γ,γ']

-] , ([ (f 1 '

sCl − β β -Int([β,β']-Cl([β,β']-Int(F))))⊂ f−1([β,β']-Cl([β,β']

-) ( = )))

(F f 1 F

Int − .

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(3) ⇒ (4): Let V be a [β,β']-open subset of Y. Then [β,β']-Cl(V) is [β,β']

-regular closed. Then we obtain [γ,γ']-sCl(f−1(V))⊂[γ,γ']-sCl(f−1([β,β']

-] , ([ '

Int β β -Cl(V)))) ⊂ f−1([β,β']-Cl(V)).

(4) ⇒ (5): Let V be a [β,β']-open subset of Y. Then Y\[β,β']-Cl(V) is

] ,

[β β' -open and we have [γ,γ']-sCl(f−1(Y\[β,β']-Cl(V)))⊂ f−1([β,β']

-] , [ \ (Y '

Cl β β -Cl(V))) and hence X \[γ,γ']-sInt(f−1([β,β']-Cl(V)))⊂ X \ f−1([β,β']

-] , ([ '

Int β β -Cl(V))) ⊂ X \ f−1(V).There fore, we obtain f−1(V)⊂[γ,γ']

-] , ([ (f 1 '

sInt − β β -Cl(V))).

(5) ⇒ (1): Let x∈X and V be a [β,β']-open set containing f(x). We have

] , [ ) (

1 '

V f

x∈ − ⊂ γ γ -sInt(f−1([β,β']-Cl(V))). Put U =[γ,γ']-sInt(f−1([β,β']

-))) (V

Cl . Then U is a [γ,γ']-semiopen set containing x and f(U)⊂[β,β']-Cl(V).

This shows that f is weakly [γ,γ']-[β,β']-semicontinuous.

1) f is weakly [γ,γ']

-[β,β']

-semicontinuous.

2) [γ,γ']

- ( 1([ , '] f

sCl − β β - ( ))) 1( ) K f K

Int ⊂ − for every [β,β']

-regular closed subset K of Y.

3) [γ,γ']-sCl(f−1([β,β']-Int([β,β']-Cl(V))))⊂ f−1([β,β']-Cl(V)) for every

] ,

[β β' -semi-preopen subset V of Y.

4) [γ,γ']-sInt(f−1([β,β']-Int([β,β']-Cl(U))))⊂ f−1([β,β']-Cl(U)) for every

] ,

[β β' -semiopen subset U of Y.

Proof. (1)⇒(2): Let K be any [β,β']-regular clsoed set of Y. Then since [β,β']

-) (K

Int is a [β,β']-open set in Y, from Theorem 3.5 (4) and K =[β,β']-Cl([β,β']

-)) (K

Int , we have [γ,γ']-sCl(f−1([β,β']-Int(K)))⊂ f−1([β,β']-Cl([β,β']

-) ( = )))

(K f 1 K

Int − . Thus [γ,γ']-sCl(f−1([β,β']-Int(K)))⊂ f−1(K).

(2)⇒(3): Let V be any [β,β']-semiopen set. Then [β,β']-Cl(V)=[β,β']

-] , ([ '

Cl β β -Int([β,β']-Cl(V))), so we know that [β,β']-Cl(V) is [β,β']-regular

closed. By (2), [γ,γ']-sCl(f−1([β,β']-Int([β,β']-Cl(V))))⊂ f−1([β,β']-Cl(V)).

(3)⇒(4): Since every [β,β']-semiopen set is [β,β']-semi-preopen, it is obvious.

(4)⇒(1): Let V be any [β,β']-open subset of Y. Then V is also a [β,β']

-semiopen set and so from (4), [γ,γ']-sCl(f−1(V))⊂[γ,γ']-sCl(f−1([β,β']-Int([β,β']

-] , ([ ))))

(V f 1 '

Cl ⊂ − β β -Cl(V)). Hence by Theorem 3.5 (4), f is weakly [γ,γ']-[β,β'] -semicontinuous.

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1) f is weakly [γ,γ']-[β,β']-semicontinuous;

2) [γ,γ']-sCl(f−1([β,β']-Int([β,β']-Cl(G))))⊂ f−1([β,β']-Cl(G)) for every

] ,

[β β' -preopen set G of Y;

3) [γ,γ']-sCl(f−1(V))⊂ f−1([β,β']-Cl(V)) for every [β,β']-preopen set V of Y;

4) f−1(V)⊂[γ,γ']-sInt(f−1([β,β']-Cl(V))) for every [β,β']-preopen set V of Y;

5) [γ,γ']

- ( 1([ , '] f

sCl − β β - ( ))) 1( ) G f G

Int ⊂ − for every [β,β']

-preclosed set G of Y.

Proof. (1) ⇒ (2): Let G be any [β,β']-preopen subset of Y. Then [β,β']

-] , [ = )

(G '

Cl β β -Cl([β,β']-Int([β,β']-Cl(G))), so [β,β']-Cl(G) is [β,β']-regular

closed. From Theorem 3.6 (2), we have [γ,γ']-sCl(f−1([β,β']-Int([β,β']

-] , ([ ))))

(G f 1 '

Cl ⊂ − β β -Cl(G)).

(2) ⇒ (3): The proof is obvious.

(3) ⇒ (4): Let V be any [β,β']-preopen subset of Y. By (3), [β,β']

-preopenness of Y \[β,β']-Cl(V) we have f−1(V)⊂ f−1([β,β']-Int([β,β']

-] , ([ \ = )))

(V X f 1 '

Cl − β β -Cl(Y \[β,β']-Cl(V)))⊂ X \[γ,γ']-sCl(f−1(Y\[β,β']

-] , [ = )))

(V '

Cl γ γ -sInt(f−1([β,β']-Cl(V))).

(4) ⇒ (1): Let V be any [β,β']-open set of Y. Then V is [β,β']-preopen set in

Y and by (4) f−1(V)⊂[γ,γ']-sInt(f−1([β,β']-Cl(V))). By Theorem 3.5 (5), f is

weakly [γ,γ']-[β,β']-semicontinuous. (4) ⇔ (5): The proof is obvious.

Theorem 3.8. The set of all points x of X at which a function )

, , , ( ) , , , (

: ' '

Y X

f τ γ γ → σ β β is not weakly [γ,γ']

-[β,β']

-semicontinuous is identical

with the union of all of the [γ,γ']

-[β,β']

-semifrontiers of the inverse images of the

] , [β β'

-closure of [β,β']

-open sets of Y containing f(x).

Proof. Let x be a point of X at which f(x) is not weakly [γ,γ']-[β,β']

-semicontinuous. Then, there exists a [β,β']-open set V of Y containing f(x) such that

] , ([ \

(X f 1 '

U∩ − β β -Cl(V)))≠∅ for every [γ,γ']-semiopen set U of X containing x.

Then x∈[γ,γ']-sCl(X \ f−1([β,β']-Cl(V))). Simultaneously, since x∈f−1([β,β']

-))) (V

Cl , we have x∈[γ,γ']-sCl(f−1([β,β']-Cl(V))) and hence

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] , [ '

x∈ γ γ -sFr(f−1([β,β']-Cl(V))). Conversely, if f is weakly [γ,γ']-[β,β']

-semicontinuous at x, then for each [β,β']-open set V of Y containing f(x), there

exists a [γ,γ']-semiopen set U containing x such that f(U)⊂[β,β']-Cl(V) and hence

] , ([ 1 ' f U

x∈ ⊂ − β β -Cl(V)). Therefore, we obtain that x∈[γ,γ']-sInt(f−1([β,β']

-))) (V

Cl . This contradicts that x∈[γ,γ']-sFr(f−1([β,β']-Cl(V)).

Now we characterize the set of points at which a function f :(X,τ,γ,γ')→(Y,σ,β,β')

is not weakly [γ,γ']-[β,β']-semicontinuous.

For a function f :(X,τ,γ,γ')→(Y,σ,β,β'), we define D_wsc[γ,γ']−[β,β'](f) as follows:

) ( ] , [ ] , [ f

D_wscγγ' −ββ' = {x∈X : f is not weakly [γ,γ']-[β,β']-semicontinuous at x}.

Theorem 3.9 For a function f :(X,τ,γ,γ')→(Y,σ,β,β'), the following properties hold: )))}. ( ] , ([ ( ] , [ \ ) ( { = ))} ( ] , ([ \ )) ( ( ] , {[ = )} ( \ ))) ( ] , ([ ( ] , {[ = ))} ( ] , ([ \ )))) ( ] , ([ ] , ([ ( ] , {[ = ) ( 1 1 ] , [ 1 1 ] , [ 1 1 ) , ( ] , [ 1 1 ) ( ] , [ ] , [ V Cl f sInt V f V Cl f V f sCl F f F Int f sCl B Cl f B Cl Int f sCl f D ' ' ' V ' ' ' V ' ' Y RC ' F ' ' ' ' Y B ' ' wsc − − ∪ − − ∪ − − ∪ − − − − ∪ − − ∈ − − ∈ − − − ∈ − − ∈ − β β γ γ β β γ γ β β γ γ β β β β β β γ γ β β σ β β σ σ β β β β γ γ P

Proof. We shall show only the equality since the proof of the other are similar to the

first. Let x Dwsc[ ,'] [ , '](f)

β β γ γ −

∈ . By Theorem 3.5, for every subset B of Y such that ]

, [ '

x∈ γ γ -sCl(f−1([β,β']-Int([β,β']-Cl(B)))) and x∉f−1([β,β']-Cl(B)). Therefore,

we have x∈[γ,γ']-sCl(f−1([β,β']-Int([β,β']-Cl(B))))\ f−1([β,β']-Cl(B))⊂

] , {[ ) ( ' Y B γ γ ∪ ∈P

- ( 1([ , '] f

sCl − β β - ([ , ']

Int β β - ( ))))\ 1([ , '] f

B

Cl − β β -Cl(B))}. Conversely,

let {[ , ]

) (

' Y B

x∈ ∪ γ γ ∈P

- ( 1([ , '] f

sCl − β β - ([ , ']

Int β β - ( ))))\ 1([ , '] f

B

Cl − β β -Cl(B))}. Then

) (Y P

B∈ such that x∈{[γ,γ']-sCl(f−1([β,β']-Int([β,β']-Cl(B))))\ f−1([β,β']

-))} (B

Cl . By Theorem 3.5, we obtain x D[wsc,'] [ , '](f)

β β γ γ −

∈ .

Similarly, by Theorems 3.6 and 3.7, we obtain the following theorems

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Theorem 3.10 For a function f :(X,τ,γ,γ')→(Y,σ,β,β'), the following properties hold: ))}. ( ] , ([ \ )))) ( ] , ([ ] , ([ ( ] , {[ = ))} ( ] , ([ \ )))) ( ] , ([ ] , ([ ( ] , {[ = )} ( \ ))) ( ] , ([ ( ] , {[ = ) ( 1 1 ) , ( ] , [ 1 1 ) , ( ] , [ 1 1 ) , ( ] , [ ] , [ ] , [ U Cl f U Cl Int f sInt V Cl f V Cl Int f sCl K f K Int f sCl f D ' ' ' ' Y SO ' U ' ' ' ' Y SPO ' V ' ' Y RC ' K ' ' wsc − − − − ∪ − − − − ∪ − − ∪ − − − ∈ − − − ∈ − − − ∈ − β β β β β β γ γ β β β β β β γ γ β β γ γ σ β β σ β β σ β β β β γ γ

Theorem 3.11 For a function f :(X,τ,γ,γ')→(Y,σ,β,β'), the following properties hold: )}. ( \ ))) ( ] , ([ ( ] , {[ = )))} ( ] , ([ ( ] , [ \ ) ( { = ))} ( ] , ([ \ )) ( ( ] , {[ = ))} ( ] , ([ \ )))) ( ] , ([ ] , ([ ( ] , {[ = ) ( 1 1 ) , ( ] , [ 1 1 ) , ( ] , [ 1 1 ) , ( ] , [ 1 1 ) , ( ] , [ ] , [ ] , [ G f G Int f sCl V Cl f sInt V f V Cl f V f sCl G Cl f G Cl Int f sCl f D ' ' Y PC ' G ' ' Y PO ' V ' ' Y PO ' V ' ' ' ' Y PO ' G ' ' wsc − − − ∈ − − − ∈ − − − ∈ − − − ∈ − − − ∪ − − ∪ − − ∪ − − − − ∪ β β γ γ β β γ γ β β γ γ β β β β β β γ γ σ β β σ β β σ β β σ β β β β γ γ

R

EFERENCES

[1] S.Kasahara, Operation-compact spaces, Math. Japonica 24 (1979), 97 -105

[2] C. Carpintero, N. Rajesh and E. Rosas, On [γ,γ']-semiopen, Boletin de Matematicas Colombia, 17(2), 125-136, (2010).

[3] H. Maki and T. Noiri, Bioperations and some separation axioms, Scientiae Math. Japonicae, 53(1)(2001), 165-180.

82/1,ARULANANDAAMMAL NAGAR, THANJAVUR-613007, TAMILNADU, INDIA.

E-mail address:[email protected]

DEPARTMENT OF MATHEMATICS, JEPPIAAR ENGINEERING COLLEGE, CHENNAI 600119, TAMILNADU, INDIA.

E-mail address: [email protected]

DEPARTMENT OF MATHEMATICS, RAJAH SERFOJI GOVT. COLLEGE, THANJAVUR-613005, TAMILNADU, INDIA.

E-mail address: [email protected]