Somewhat semicontinuous mappings via grills Essay

Somewhat semicontinuous mappings via grillA. Swaminathan?

Department of Mathematics

Government Arts College(Autonomous)

Kumbakonam, Tamil Nadu-612 002, India.

and

R. Venugopal

Department of Mathematics,

Annamalai University, Annamalainagar,

Tamil Nadu-608 002, India.

Abstract: This article introduces the concepts of somewhat G-

semicontinuous mapping and somewhat G-semiopen mappings. Using

these notions, some examples and few interesting propertie s of those

mappings are discussed by means of grill topological spaces .

Key words and phrases: G-continuous mapping, somewhat

G -continuous mapping, G-semicontinuous mapping, somewhat G-

semicontinuous mapping, somewhat G-semiopen mapping.

G-

semidense set.

2010 Mathematics Sub ject Classi?cation: 54A10, 54A20.

1 Introduction and Preliminaries The study of somewhat continuous functions was ?rst initiat ed by

Karl.R.Gentry et al in [4]. Although somewhat continuous fun ctions

are not at all continuous mappings it has been studied and dev eloped

considerably by some authors using topological properties . In 1947,

Choquet [1] magni?cently established the notion of a grill w hich has

?

[email protected]

1

2

been milestone of developing topology via grills.

Almost all the

foremost concepts of general topology have been tried to a ce rtain

extent in grill notions by various intellectuals. It is wide ly known

that in many aspects, grills are more e?ective than a certain similar

concepts like nets and ?lters. E.Hatir and Jafari introduced the idea of

G -continuous functions in [3] and they showed that the concep t of open

and G-open are independent of each other. Dhananjay Mandal and

M.N.Mukherjee[2] studied the notion of G-semicontinuous mappings.

Our aim of this paper is to introduce and study new concepts na mely

somewhat G-semicontinuous mapping and somewhat G-semiopen

mapping. Also, their characterizations, interrelations an d examples

are studied.

Throughout this paper, Xstands for a topological space with no

separation axioms assumed unless explicitly given. For a su bsetHof

X , the closure of Hand the interior of Hdenoted by Cl ( H) and

Int ( H) repectively. The power set of Xdenoted by P(X ) .

The de?nitions and results which are used in this paper conce rning

topological and grill topological spaces have already take n some

standard shape. We recall those de?nitions and basic proper ties as

follows:

De?nition 1.1. A mappingf: ( X, ?)? (Y , ??

) is called somewhat

continuous[4] if there exists an open set U ?= on ( X,?) such that

U ? f?

1

(V )?

= for any open set V ?= on ( Y ,??

) .

De?nition 1.2. A non-empty collection Gof subsets of a topological

spaces X is said to be a grill[1] on Xif (i) /? G (ii)H? G and

H ?K ?X ? K? G and (iii) H, K?X and H?K ? G ? H? G

or K? G .

A topological space ( X,?) with a grill Gon Xdenoted by

( X, ?,G ) is called a grill topological space.

De?nition 1.3. Let (X,?) be a topological space and Gbe a grill

on X. An operator ? : P(X )? P (X ) , denoted by ?

G(

H, ?) (for

H ? P (X )) or ?

G(

H ) or simply ?( H) , called the operator associated

with the grill Gand the topology ?de?ned by[5]

? G(

H ) = {x ? X :U ?H ? G ,? U ? ? (x )}

3

Then the operator ?( H) = H??( H) (for H?X), was also known

as Kuratowski’s operator[5], de?ning a unique topology ?

G such that

? ? ? G.

Theorem 1.1. [2]Let ( X,?) be a topological space and Gbe a

grill on X. Then for any H, K?X the following hold:

(a) H, K ??(H)? ?( K) .

(b) ?( H?K ) = ?( H)? ?( K) .

(c) ?(?( H)) ? ?( H) = C l(?( H)) ? C l (H ) .

De?nition 1.4. Let (X,?,G ) be a grill topological space. A subset

H inX is said to be

(i) G-open[6] (or ? -open[3]) if H?Int ?( H) .

(ii) G-semiopen[2] if H?? (Int ( H)) .

De?nition 1.5. A mappingf: ( X, ?,G ) ? (Y , ??

) is called

(i) G-continuous[3] if f?

1

(V ) is a G-open set on ( X,?,G ) for any

open set Von ( Y ,??

) .

(ii) G-semicontinuous[2] if f?

1

(V ) is a G-semiopen set on ( X,?,G )

for any open set Von ( Y ,??

) .

De?nition 1.6. A mappingf: ( X,?,G ) ? (Y , ??

) is called

somewhat G-continuous[7] if there exists a G-open set U ?= on

( X, ?,G ) such that U ?f?

1

(V ) ?

= for any open set V ?= on

( Y , ??

) .

De?nition 1.7. Let (X,?,G ) be a grill topological space. A subset

H ?X is called G-dense[3] in Xif ?( H) = X.

2 Somewhat G-semicontinuous

mappings

In this section, the concept of somewhat G-semicontinuous

mapping is introduced. The notion of somewhat G-semicontinuous

mapping are independent of somewhat semicontinuous mappin g. Also,

we characterize a somewhat G-semicontinuous mapping.

De?nition 2.1. A mappingf: ( X, ?)? (Y , ??

) is called somewhat

semicontinuous if there exists semiopen set U ?= on ( X,?) such

that U ?f?

1

(V )?

= for any open set V ?= on ( Y ,??

) .

4

De?nition 2.2. A mappingf: ( X,?,G ) ? (Y , ??

) is called

somewhat G-semicontinuous if there exists a G-semiopen set U ?=

on ( X,?,G ) such that U ?f?

1

(V )?

= for any open set V ?= on

( Y , ??

) .

Remark 2.1. (a)From [2], we have the following observations:

(a)The concept of open and G-open are independent of each other.

Hence the notion of continuous and G-continuous are indepedent.

(b)The notion of G-open and G-semiopen are independent of each

other. Therefore, there shold be a mutual independence betw een

somewhat G-continuous and somewhat G-semicontinuous.

Remark 2.2. The following reverse implications are false:

(a)Every continuous mapping is a somewhat continuous mappi ng[4].

(b)Every G-continuous is somewhat G-continuous[7].

(c)Every G-semicontinuous is semicontinuous[2].

It is clear that every semicontinuous mapping is a somewhat

semicontinuous mapping but not conversely. Every G-semicontinuous

mapping is a somewhat G-semicontinuous mapping but the converses

are not true in general as the following examples show.

Example 2.1 Let X={x, y, z, w },? ={ , {x },{ y, w },{ x, y, w }, X }

and G= {{x},{ x, y },{ x, z },{ x, w },{ x, y, z },{ x, y, w },{ x, z, w }, X };

Y ={a, b }and ??

= { , {a }, Y }. We de?ne a function f:

( X, ?,G ) ? (Y , ??

) as follows: f(x ) = f(z ) = aand f(y ) = f(w ) =

b . Then for open set {a } on ( Y ,??

) , we have {x } ? f?

1

{ a } =

{ x, z }; hence {x } is a G-semiopen set on ( X,?,G ) . Therfore

f is somewhat G-semicontinuous function. But for open set {a }

on ( Y ,??

) , f?

1

{ a } = {x, z }which is not a G-semicontinuous on

( X, ?,G ) .

Now we have the following diagram from our comparision:

Theorem 2.1. Iff: ( X,?,G ) ? (Y , ??

) is somewhat G-

semicontinuous and g: ( Y , ??

) ? (Z, J) is continuous, then g?f :

( X, ?,G ) ? (Z, J) is somewhat G-semicontinuous.

5

Proof. LetKbe a non-empty open set in Z. Since gis continuous,

g ?

1

(K ) is open in Y. Now ( g?f)?

1

(K ) = f?

1

(g ?

1

(K )) ?

= .

Since g?

1

(K ) is open in Yand fis somewhat G-semicontinuous,

then there exists a G-semiopen set H?

= in X such that

H ?f?

1

(g ?

1

(K )) = ( g?f)?

1

(K ) . Hence g?f is somewhat G-

semicontinuous.

De?nition 2.3. Let (X,?,G ) be a grill topological space. A subset

H ?X is called G-semidense in Xif semi- ?( H) = X.

Theorem 2.2. Iff: ( X,?,G ) ? (Y , ??

) is somewhat G-

semicontinuous and Ais a G-semidense subset of Xand G

H is

the induced grill topology for H, then f?

H : (

X, ?,G

H )

? (Y , ??

) is

somewhat G-semicontinuous.

The following example is enough to justify the restriction i s

somewhat G-semicontinuous.

Example 2.2 Let X={x, y, z, w },? ={ , {w },{ x, z },{ x, z, w }, X }and

G = {{w},{ x, w },{ y, w },{ z, w },{ x, y, w },{ x, z, w },{ y, z, w }, X };

Let Y={a, b }and ??

= { , {y }, Y }; Let A= {x, z, w }be

a subset of ( X,?,G ) and the induced grill topology for G

H is

G H =

{{w},{ x, w },{ z, w }, H }. Then f: ( X, ?,G

H )

? (Y , ??

)

is de?ned as follows: f(x ) = f(z ) = yand f(y ) = f(w ) = x.

Now for all the sets {w },{ x, w },{ z, w }, H on ( X,?,G

H ) , we have

? G? (

H ) = {y, w }. Then for H={w }, ?( H) = {y, w }; for

H ={x, w }, ?( H) = {x, y, w }; H ={z, w }, ?( H) = {y, z, w };

H ={x, z, w }, ?( H) = {x, z, w }. Therefore there is no G-semidense

set on ( X,?,G

H ) . Also there is no non-empty

G-semiopen set smaller

than f?

1

{ b} = {x, z }. Hence f: ( X, ?,G

H )

? (Y , ??

) is not

somewhat G-semicontinuous functions.

Theorem 2.3. Iff: ( X, ?,G ) ? (Y , ??

) be a mapping, then the

following are equivalent:

(1) fis somewhat G-semicontinuous.

(2) If Vis a closed set of ( Y ,??

) such that f?

1

(V )?

= X , then there

exists a G-semiclosed set U ?= X of ( X,?,G ) such that f?

1

(V )? U .

(3) If Uis a G-semidense set on ( X,?,G ) , then f(U ) is a dense set

on ( Y ,??

) .

6

Proof. (1)?(2) :Let Vbe a closed set on Ysuch that f?

1

(V )?

= X.

Then Vc

is an open set in Yand f?

1

(V c

) = ( f?

1

(V )) c

?

= . Since f

is somewhat G-semicontinuous, there exists a G-semiopen set U ?=

on Xsuch that U ?f?

1

(V c

) . Let U= Vc

. Then U ?XisG-

semiclosed such that f?

1

(V ) = X?f?

1

(V c

) ? X ? U c

= U.

(2) ?(3): Let Ube a G-semidense set on Xand suppose f(U )

is not dense on Y. Then there exists a closed set Von Ysuch

that f(U )? V ? X. Since V ?Xand f?

1

(V )?

= X, there exists a

G -semiclosed set W ?=X such that U ?f?

1

(f (U )) ? f?

1

(V )? W .

This contradicts to the assumption that Uis a G-semidense set on

X . Hence f(U ) is a dense set on Y.

(3) ?(1): Let V ?= be a open set on Yand f?

1

(V ) ?

= .

Suppose there exists no G-semiopen U ?= on Xsuch that

U ? f?

1

(V ). Then ( f?

1

(V )) c

is a set on Xsuch that there

is no G-semiclosed set WonXwith ( f?

1

(V )) c

? W ? X.

In fact, if there exists a G-semiopen set Wc

such that Wc

?

f ?

1

(V ) , then it is a contradiction. So ( f?

1

(V )) c

is a G-semidense

set on X. Then f(( f?

1

(V )) c

) is a dense set on Y. But

f (( f?

1

(V )) c

) = f(( f?

1

(V )) c

)) ?

= Vc

? X. This contradicts to the

fact that f(( f?

1

(V )) c

) is fuzzy dense on Y. Hence there exists a

G -semiopen set U ?= on Xsuch that U ?f?

1

(V ) . Consequently,

f is somewhat G-semicontinuous.

Theorem 2.4. Let (X

1,

?

1,

G ) , ( X

2,

?

2,

G ) , ( Y

1,

? ?

1 ,

G ) and

( Y

2,

? ?

2 ,

G ) be grill topological spaces. Let ( X

1,

?

1,

G ) be product

related to ( X

2,

?

2,

G ) and let ( Y

1,

? ?

1 ,

G ) be product related to

( Y

2,

? ?

2 ,

G ) . If f

1 : (

X

1,

?

1,

G ) ? (Y

1,

? ?

1 ,

G ) and f

2 : (

X

2,

?

2,

G ) ?

( Y

2,

? ?

2 ,

G ) are somewhat G-semicontinuous, then the product f

1?

f

2 :

( X

1,

?

1,

G )? (X

2,

?

2,

G ) ? (Y

1,

? ?

1 ,

G )? (Y

2,

? ?

2 ,

G ) is also somewhat

G -semicontinuous mappings.

Proof. LetG=

i,j (

M

i?

N

j) be an open set on

Y

1 ?

Y

2 where

M i?

=

Y1 and

N

j?

=

Y2 are open sets on

Y

1 and

Y

2 respectively.

Then ( f

1 ?

f

2)?

1

(G ) =

i,j (

f ?

1

1 (

M

i)

? f?

1

2 (

N

j))

.Since f

1 is somewhat

G -semicontinuous, there exists a G-semiopen set U

i?

=

X 1 such that

7

U i?

f?

1

1 (

M

i)

?

=

X 1.

And, since f

2 is somewhat

G-semicontinuous,

there exists a G-semiopen set V

j ?

=

X 2 such that

V

j ?

f?

1

2 (

N

j)

?

=

X 2.

Now U

i?

V

j ?

f?

1

1 (

M

i)

? f?

1

2 (

N

j) = (

f

1 ?

f

2)?

1

(M

i?

N

j) and

U i?

V

j ?

=

X 1?

X

2. Hence

i,j (

M

i?

N

j)

?

=

X 1 ?