Somewhat semicontinuous mappings via grills Essay

Somewhat semicontinuous mappings via grillA. Swaminathan?

Division of Arithmetic

Authorities Arts Faculty(Autonomous)

Kumbakonam, Tamil Nadu-612 002, India.

and

R. Venugopal

Division of Arithmetic,

Annamalai College, Annamalainagar,

Tamil Nadu-608 002, India.

Summary: This text introduces the ideas of considerably G-

semicontinuous mapping and considerably G-semiopen mappings. Utilizing

these notions, some examples and few attention-grabbing propertie s of these

mappings are mentioned by way of grill topological areas .

Key phrases and phrases: G-continuous mapping, considerably

G -continuous mapping, G-semicontinuous mapping, considerably G-

semicontinuous mapping, considerably G-semiopen mapping.

G-

semidense set.

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

1 Introduction and Preliminaries The examine of considerably steady capabilities was ?rst initiat ed by

Karl.R.Gentry et al in [4]. Though considerably steady enjoyable ctions

are by no means steady mappings it has been studied and dev eloped

significantly by some authors utilizing topological properties . In 1947,

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

?

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1

2

been milestone of creating topology via grills.

Nearly all of the

foremost ideas of basic topology have been tried to a ce rtain

extent in grill notions by varied intellectuals. It’s extensive ly identified

that in lots of facets, grills are extra e?ective than a sure related

ideas like nets and ?lters. E.Hatir and Jafari launched the concept of

G -continuous capabilities in [3] they usually confirmed that the concep t of open

and G-open are unbiased of one another. Dhananjay Mandal and

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

Our goal of this paper is to introduce and examine new ideas na mely

considerably G-semicontinuous mapping and considerably G-semiopen

mapping. Additionally, their characterizations, interrelations an d examples

are studied.

All through this paper, Xstands for a topological house with no

separation axioms assumed except explicitly given. For a su bsetHof

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

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

The de?nitions and outcomes that are used on this paper conce rning

topological and grill topological areas have already take n some

customary form. We recall these de?nitions and fundamental correct ties as

follows:

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

) known as considerably

steady[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 assortment Gof subsets of a topological

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

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

or Ok? G .

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

( X, ?,G ) known as a grill topological house.

De?nition 1.three. Let (X,?) be a topological house and Gbe a grill

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

G(

H, ?) (for

H ? P (X )) or ?

G(

H ) or just ?( H) , known as the operator related

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

? G(

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

three

Then the operator ?( H) = H??( H) (for H?X), was additionally identified

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

G such that

? ? ? G.

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

grill on X. Then for any H, Ok?X the next maintain:

(a) H, Ok ??(H)? ?( Ok) .

(b) ?( H?Ok ) = ?( H)? ?( Ok) .

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

De?nition 1.four. Let (X,?,G ) be a grill topological house. A subset

H inX is claimed 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 , ??

) known as

(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 , ??

) known as

considerably 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 house. A subset

H ?X known as G-dense[3] in Xif ?( H) = X.

2 Somewhat G-semicontinuous

mappings

On this part, the idea of considerably G-semicontinuous

mapping is launched. The notion of considerably G-semicontinuous

mapping are unbiased of considerably semicontinuous mappin g. Additionally,

we characterize a considerably G-semicontinuous mapping.

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

) known as considerably

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

that U ?f?

1

(V )?

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

) .

four

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

) known as

considerably 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 , ??

) .

Comment 2.1. (a)From [2], we have now the next observations:

(a)The idea of open and G-open are unbiased of one another.

Therefore the notion of steady and G-continuous are indepedent.

(b)The notion of G-open and G-semiopen are unbiased of every

different. Subsequently, there shold be a mutual independence betw een

considerably G-continuous and considerably G-semicontinuous.

Comment 2.2. The next reverse implications are false:

(a)Each steady mapping is a considerably steady mappi ng[4].

(b)Each G-continuous is considerably G-continuous[7].

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

It’s clear that each semicontinuous mapping is a considerably

semicontinuous mapping however not conversely. Each G-semicontinuous

mapping is a considerably G-semicontinuous mapping however the converses

are usually not true typically as the next examples present.

Instance 2.1 Let X=x, y, z, w ,? = , x ,, x, y, w , X

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

Y =a, b and ??

= . We de?ne a perform 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 now x ? f?

1

=

; therefore x is a G-semiopen set on ( X,?,G ) . Therfore

f is considerably G-semicontinuous perform. However for open set a

on ( Y ,??

) , f?

1

= x, z which isn’t a G-semicontinuous on

( X, ?,G ) .

Now we have now the next diagram from our comparision:

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

) is considerably G-

semicontinuous and g: ( Y , ??

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

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

5

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

g ?

1

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

1

(Ok ) = f?

1

(g ?

1

(Ok )) ?

= .

Since g?

1

(Ok ) is open in Yand fis considerably G-semicontinuous,

then there exists a G-semiopen set H?

= in X such that

H ?f?

1

(g ?

1

(Ok )) = ( g?f)?

1

(Ok ) . Therefore g?f is considerably G-

semicontinuous.

De?nition 2.three. Let (X,?,G ) be a grill topological house. A subset

H ?X known as G-semidense in Xif semi- ?( H) = X.

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

) is considerably 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

considerably G-semicontinuous.

The next instance is sufficient to justify the restriction i s

considerably G-semicontinuous.

Instance 2.2 Let X=x, y, z, w ,? =and

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

Let Y=a, b and ??

= ; Let A= be

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

H is

G 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 of the units ,,, H on ( X,?,G

H ) , we have now

? G? (

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

H =x, w , ?( H) = ; H =z, w , ?( H) = ;

H =, ?( H) = . Subsequently there isn’t a G-semidense

set on ( X,?,G

H ) . Additionally there isn’t a non-empty

G-semiopen set smaller

than f?

1

b = x, z . Therefore f: ( X, ?,G

H )

? (Y , ??

) shouldn’t be

considerably G-semicontinuous capabilities.

Theorem 2.three. Iff: ( X, ?,G ) ? (Y , ??

) be a mapping, then the

following are equal:

(1) fis considerably 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 .

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

shouldn’t be 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 belief that Uis a G-semidense set on

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

(three) ?(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

isn’t any G-semiclosed set WonXwith ( f?

1

(V )) c

? W ? X.

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

such that Wc

?

f ?

1

(V ) , then it’s 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. However

f (( f?

1

(V )) c

) = f(( f?

1

(V )) c

)) ?

= Vc

? X. This contradicts to the

proven fact that f(( f?

1

(V )) c

) is fuzzy dense on Y. Therefore there exists a

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

1

(V ) . Consequently,

f is considerably G-semicontinuous.

Theorem 2.four. Let (X

1,

?

1,

G ) , ( X

2,

?

2,

G ) , ( Y

1,

? ?

1 ,

G ) and

( Y

2,

? ?

2 ,

G ) be grill topological areas. Let ( X

1,

?

1,

G ) be product

associated to ( X

2,

?

2,

G ) and let ( Y

1,

? ?

1 ,

G ) be product associated 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 considerably 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 ) can also be considerably

G -semicontinuous mappings.

Proof. LetG=

i,j (

M

i?

N

j) be an open set on

Y

1 ?

Y

2 the place

M i?

=

Y1 and

N

j?

=

Y2 are open units 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 considerably

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 considerably

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. Therefore

i,j (

M

i?

N

j)

?

=

X 1 ?