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
?
[email protected]
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 KuratowskiÂ’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)
?
=
