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 Kuratowskis 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)
?
=