Somewhat semicontinuous mappings via grills Essay

Slightly semipenny mappings via suffererA. 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 proviso presents the concepts of slightly G-

semipenny mapping and slightly G-semidissecretive mappings. Using

these referable attributable attributableions, some models and rare interesting specialtie s of those

mappings are discussed by resources of sufferer topological distances .

Key control and phrases: G-penny mapping, slightly

G -penny mapping, G-semipenny mapping, slightly G-

semipenny mapping, slightly G-semidissecretive mapping.

G-

semislow customary.

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

1 Introduction and Preliminaries The con-over of slightly penny comicalityctions was ?rst initiat ed by

Karl.R.Gentry et al in [4]. Although slightly penny comicality ctions

are referable attributable attributable attributable at total penny mappings it has been learned and taint eloped

considerably by some authors using topological specialties . In 1947,

Choquet [1] magni?cently customary the referable attributable attributableion of a sufferer w hich has

?

[email protected]

1

2

been milestone of tainteloping topology via sufferers.

Almost total the

foremost concepts of public topology own been seasoned to a ce rtain

extent in sufferer referable attributable attributableions by multiform intellectuals. It is broad ly known

that in frequent aspects, sufferers are further e?ective than a convinced similar

concepts relish nets and ?lters. E.Hatir and Jafari presentd the purpose of

G -penny comicalityctions in [3] and they semblanceed that the concep t of disclosed

and G-dissecretive are dogged of each other. Dhananjay Mandal and

M.N.Mukherjee[2] learned the referable attributable attributableion of G-semipenny mappings.

Our keep of this disquisition is to present and con-over innovating concepts na mely

slightly G-semipenny mapping and slightly G-semiopen

mapping. To-boot, their characterizations, interrelations an d models

are learned.

Throughout this disquisition, Xstands ce a topological distance with no

separation occurrences resultive original palpably abandoned. Ce a su bsetHof

X , the blank wwhole of Hand the inside of Hdenoted by Cl ( H) and

Int ( H) repectively. The influence customary of Xdenoted by P(X ) .

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

topological and sufferer topological distances own already choose n some

standard fashion. We rectotal those de?nitions and basic special ties as

follows:

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

) is denominated slightly

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

U ? f?

1

(V )?

= ce any dissecretive customary V ?= on ( Y ,??

) .

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

spaces X is said to be a sufferer[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 distance ( X,?) with a sufferer Gon Xdenoted by

( X, ?,G ) is denominated a sufferer topological distance.

De?nition 1.3. Permit (X,?) be a topological distance and Gbe a sufferer

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

G(

H, ?) (for

H ? P (X )) or ?

G(

H ) or singly ?( H) , denominated the operator associated

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

? G(

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

3

Then the operator ?( H) = H??( H) (ce H?X), was to-boot known

as Kuratowski’s operator[5], de?element a uncommon topology ?

G such that

? ? ? G.

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

sufferer on X. Then ce any H, K?X the controlthcoming hold:

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

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

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

De?nition 1.4. Permit (X,?,G ) be a sufferer topological distance. 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 denominated

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

1

(V ) is a G-dissecretive customary on ( X,?,G ) ce any

dissecretive customary Von ( Y ,??

) .

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

1

(V ) is a G-semidissecretive customary on ( X,?,G )

ce any dissecretive customary Von ( Y ,??

) .

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

) is denominated

slightly G-continuous[7] if there exists a G-dissecretive customary U ?= on

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

1

(V ) ?

= ce any dissecretive customary V ?= on

( Y , ??

) .

De?nition 1.7. Permit (X,?,G ) be a sufferer topological distance. A subset

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

2 Slightly G-semicontinuous

mappings

In this individuality, the concept of slightly G-semicontinuous

mapping is presentd. The referable attributable attributableion of slightly G-semicontinuous

mapping are dogged of slightly semipenny mappin g. To-boot,

we identify a slightly G-semipenny mapping.

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

) is denominated slightly

semipenny if there exists semidissecretive customary U ?= on ( X,?) such

that U ?f?

1

(V )?

= ce any dissecretive customary V ?= on ( Y ,??

) .

4

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

) is denominated

slightly G-semipenny if there exists a G-semidissecretive customary U ?=

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

1

(V )?

= ce any dissecretive customary V ?= on

( Y , ??

) .

Remark 2.1. (a)From [2], we own the controlthcoming observations:

(a)The concept of dissecretive and G-dissecretive are dogged of each other.

Hereafter the referable attributable attributableion of penny and G-penny are indepedent.

(b)The referable attributable attributableion of G-dissecretive and G-semidissecretive are dogged of each

other. Therefore, there shold be a reciprocal insurrection betw een

slightly G-penny and slightly G-semicontinuous.

Remark 2.2. The controlthcoming confliction implications are false:

(a)Total penny mapping is a slightly penny mappi ng[4].

(b)Total G-penny is slightly G-continuous[7].

(c)Total G-semipenny is semicontinuous[2].

It is apparent that total semipenny mapping is a slightly

semipenny mapping barring referable attributable attributable attributable conversely. Total G-semicontinuous

mapping is a slightly G-semipenny mapping barring the converses

are referable attributable attributable attributable penny in public as the controlthcoming models semblance.

Model 2.1 Permit 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 comicalityction f:

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

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

b . Then ce dissecretive customary {a } on ( Y ,??

) , we own {x } ? f?

1

{ a } =

{ x, z }; hereafter {x } is a G-semidissecretive customary on ( X,?,G ) . Therfore

f is slightly G-semipenny comicalityction. Barring ce dissecretive customary {a }

on ( Y ,??

) , f?

1

{ a } = {x, z }which is referable attributable attributable attributable a G-semipenny on

( X, ?,G ) .

Now we own the controlthcoming diagram from our comparision:

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

) is slightly G-

semipenny and g: ( Y , ??

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

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

5

Proof. PermitKbe a non-empty dissecretive customary in Z. Past gis penny,

g ?

1

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

1

(K ) = f?

1

(g ?

1

(K )) ?

= .

Past g?

1

(K ) is dissecretive in Yand fis slightly G-semicontinuous,

then there exists a G-semidissecretive customary H?

= in X such that

H ?f?

1

(g ?

1

(K )) = ( g?f)?

1

(K ) . Hereafter g?f is slightly G-

semicontinuous.

De?nition 2.3. Permit (X,?,G ) be a sufferer topological distance. A subset

H ?X is denominated G-semislow in Xif semi- ?( H) = X.

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

) is slightly G-

semipenny and Ais a G-semislow subcustomary of Xand G

H is

the vital sufferer topology ce H, then f?

H : (

X, ?,G

H )

? (Y , ??

) is

slightly G-semicontinuous.

The controlthcoming model is sufficient to vindicate the exclusion i s

slightly G-semicontinuous.

Model 2.2 Permit 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 };

Permit Y={a, b }and ??

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

a subcustomary of ( X,?,G ) and the vital sufferer topology ce 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 ce total the customarys {w },{ x, w },{ z, w }, H on ( X,?,G

H ) , we own

? G? (

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

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

customary on ( X,?,G

H ) . To-boot there is no non-empty

G-semidissecretive customary smaller

than f?

1

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

H )

? (Y , ??

) is referable attributable attributable

slightly G-semipenny comicalityctions.

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

) be a mapping, then the

forthcoming are equivalent:

(1) fis slightly G-semicontinuous.

(2) If Vis a secretive customary of ( Y ,??

) such that f?

1

(V )?

= X , then there

exists a G-semisecretive customary U ?= X of ( X,?,G ) such that f?

1

(V )? U .

(3) If Uis a G-semislow customary on ( X,?,G ) , then f(U ) is a slow customary

on ( Y ,??

) .

6

Proof. (1)?(2) :Permit Vbe a secretive customary on Ysuch that f?

1

(V )?

= X.

Then Vc

is an dissecretive customary in Yand f?

1

(V c

) = ( f?

1

(V )) c

?

= . Past f

is slightly G-semicontinuous, there exists a G-semidissecretive customary U ?=

on Xsuch that U ?f?

1

(V c

) . Permit U= Vc

. Then U ?XisG-

semisecretive such that f?

1

(V ) = X?f?

1

(V c

) ? X ? U c

= U.

(2) ?(3): Permit Ube a G-semislow customary on Xand conjecture f(U )

is referable attributable attributable attributable slow on Y. Then there exists a secretive customary Von Ysuch

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

1

(V )?

= X, there exists a

G -semisecretive customary W ?=X such that U ?f?

1

(f (U )) ? f?

1

(V )? W .

This contradicts to the boldness that Uis a G-semislow customary on

X . Hereafter f(U ) is a slow customary on Y.

(3) ?(1): Permit V ?= be a dissecretive customary on Yand f?

1

(V ) ?

= .

Conjecture there exists no G-semidissecretive U ?= on Xsuch that

U ? f?

1

(V ). Then ( f?

1

(V )) c

is a customary on Xsuch that there

is no G-semisecretive customary WonXwith ( f?

1

(V )) c

? W ? X.

In occurrence, if there exists a G-semidissecretive customary Wc

such that Wc

?

f ?

1

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

1

(V )) c

is a G-semidense

customary on X. Then f(( f?

1

(V )) c

) is a slow customary on Y. Barring

f (( f?

1

(V )) c

) = f(( f?

1

(V )) c

)) ?

= Vc

? X. This contradicts to the

occurrence that f(( f?

1

(V )) c

) is fuzzy slow on Y. Hereafter there exists a

G -semidissecretive customary U ?= on Xsuch that U ?f?

1

(V ) . Consequently,

f is slightly G-semicontinuous.

Theorem 2.4. Permit (X

1,

?

1,

G ) , ( X

2,

?

2,

G ) , ( Y

1,

? ?

1 ,

G ) and

( Y

2,

? ?

2 ,

G ) be sufferer topological distances. Permit ( X

1,

?

1,

G ) be result

kindred to ( X

2,

?

2,

G ) and permit ( Y

1,

? ?

1 ,

G ) be result kindred 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 slightly G-semicontinuous, then the result f

1?

f

2 :

( X

1,

?

1,

G )? (X

2,

?

2,

G ) ? (Y

1,

? ?

1 ,

G )? (Y

2,

? ?

2 ,

G ) is to-boot slightly

G -semipenny mappings.

Proof. PermitG=

i,j (

M

i?

N

j) be an dissecretive customary on

Y

1 ?

Y

2 where

M i?

=

Y1 and

N

j?

=

Y2 are dissecretive customarys 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))

.Past f

1 is slightly

G -semicontinuous, there exists a G-semidissecretive customary U

i?

=

X 1 such that

7

U i?

f?

1

1 (

M

i)

?

=

X 1.

And, past f

2 is slightly

G-semicontinuous,

there exists a G-semidissecretive customary 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. Hereafter

i,j (

M

i?

N

j)

?

=

X 1 ?

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