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Int. J. of Mathematical Sciences and Applications,
Vol. 1, No. 3, September 2011
Copyright  Mind Reader Publications
www.journalshub.com
On Slightly continuous functions via g~ -sets
*
*
M. Lellis Thivagar and ** Nirmala Rebecca Paul
School of Mathematics, Madurai Kamaraj University
Madurai-625021, Tamil Nadu, India
e.mail:[email protected]
**
Department of Mathematics, Lady Doak College,
Madurai-625002, Tamil Nadu, India
e.mail:[email protected]
Abstract
The aim of this paper is to define a new class of functions namely slightly
g~ -continuous
~ -connected, g~ -normal and g~ -compact
functions and study its properties. We introduce g



~
spaces and study their properties in terms of slightly g  -continuous functions. We prove that
~ -continuous function is weaker than g~ -continuous functions.
the slightly g


2010 Mathematics Subject Classification : 57N05
Keywords and phrases : # gs-closed sets,# gs-open sets,
function.
g~ -closed sets, g~ -open sets and g~ -continuous
1 Introduction
The generalized closed sets has been extended to study the characterization of continuity, compactness,
connectedness etc. Slightly continuity was introduced by Jain[4] and has been applied for semi-open and
~ -closed sets in
pre-open sets by Nour[5] and Baker[1] respectively. Jafari et al. [3] have introduced g

g~ -continuous functions has been
~ -continuous functions. We
defined and proved that it includes the class of slightly continuous and g

~ -continuous functions in terms of composition and restriction. The
investigated the properties of slightly g

~ -compactness and g~ -connectedness is studied under slightly g~ -continuous functions.
behavior of g



topology and proved that it forms a topology. In this paper slightly
2 Preliminaries
We list some definitions in a topological space (X, τ ) which are useful in the following sections. The
interior and the closure of a subset A of (X, τ) are denoted by Int(A) and Cl(A), respectively. Throughout the
present paper (X, τ) and (Y, σ)(or X and Y ) represent topological spaces on which no separation axiom is
defined, unless otherwise mentioned.
Definition 2.1 A subset A of a topological space (X, τ ) is called
(i)
(ii)
(iii)
(iv)
an ω-closed set [7 ] (= ĝ − closed) if Cl(A)  U whenever A  U and U is semi-open in (X, τ ),
a *g-closed set[10] if Cl(A)  U whenever A  U and U
is ω-open in (X, τ ) ,
a #g-semi-closed set(briefly #gs-closed)[11] if sCl(A)  U whenever A  U and U is *g-open in (X, τ )
and
~ -closed set[3] if αCl(A)  U whenever A  U and U is #gs-open in (X, τ )
a g

The complement of ω-closed(resp *g-closed, # gs-closed,
g~ -closed)set is said to be ω-open (resp *g-open,
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M. Lellis Thivagar and Nirmala Rebecca Paul
#
gs-open,
g~ -open)
Definition 2.2 A function f: ( X , )  (Y ,  ) is called
g~ -continuous[8 ] if the inverse image of every closed set in Y is g~ -closed in X.
~ -irresolute[8] if the inverse image of every g~ -closed set in Y is g~ -closed in
(ii) g



(i)
X.
g~ -closed[9] if the image of every closed set in X is g~ -closed in Y.
~ -closed[ 9] if the image of every g~ -closed set in X is g~ -closed in
(iv) strongly g



(iii)
Y.
(v) slightly continuous[4] if the inverse image of every clopen set in Y is open in X.
Definition 2.3[3] Let (X, τ) be a topological space and E be a subset of X. We define the
g~ -Interior of E denoted by g~ -Int(E) to be the union of all g~ -open sets contained in E. Similarly the
g~ -closure of E denoted as g~ -Cl(E) is defined to be the intersection of all g~ -closed sets containing E.
~
~ -closed ( g~ -open sets) are denoted by G
Remark 2.4 The collection of all g
 C(X)


~
~
~
( G O(X)) respectively. Similarly the g  –clopen subsets of X is denoted by G CO(X) and the clopen
subsets of X is denoted as CO(X).
~ -continuous function
3 Slightly g

g~ -continuous at a point x  X if for
Definition 3.1 A function f: ( X , )  (Y ,  ) is said to be slightly
g~ -open subset U in X containing x such that
~ continuous if f is slightly g~ -continuous at each of its
f(U)  V. T he function f is said to be slightly g


each clopen subset V of Y containing f(x), there exists a
points.
Definition 3.2 A function f: ( X , )  (Y ,  ) is said to be slightly
g~ -continuous if the inverse image of
g~ -clopen in X.
Example 3.3 Let X={a,b,c}=Y,  ={  , X,{a},{b},{a,c},{a,b}},  ={  ,Y,{a},{b,c}}.
~
G O(X)={  ,X,{a},[b},{a,c},{a,b}}. The function f is defined as f(a)=c,f(b)=a,f(c)=b. The function f is
~ -continuous .
slightly g

every clopen set in Y is
Proposition 3.4 The definitions 3.1 and 3.2 are equivalent.
Proof. Suppose that definition 3.1 holds. Let V be a clopen set in Y and let x 
1
(V). Then f(x)  V and
1
~
thus there exists a g  -open set U x such that x  U x and f( U x )  V. Now x  U x  f (V) and
f 1 (V)= 
U . Since arbitrary union of g~ -open sets is g~ -open, f 1 (V) is g~ -open in X and
x f 1 (V )

x
f


~ -continuous.
therefore f is slightly g

~ -continuous, we
Let definition 3.2 holds and f(x)  V where V is a clopen set in Y. Since f is slightly g

1
1
1
~
~
x  f (V) where f (V) is g -open . Let U= f (V) . Then U is g -open in X, x  U and f(U)  V.


Theorem 3.5 Let f: ( X , )  (Y ,  ) be a function then the following are equivalent.
g~ -continuous.
(i)
f is slightly
(ii)
The inverse image of every clopen set V of Y is
g~ -open in X.
~ -closed in X.
The inverse image of every clopen set V of Y is g

~ -clopen in X.
The inverse image of every clopen set V of Y is g

(iii)
(iv)
Proof. (i)  (ii) Follows from the Proposition 3.4.
(ii)  (iii) Let V be clopen in Y which implies that Vc is clopen in Y. By (ii)
Since (
f
1
(V))c=
f
1
(Vc ),
(iii)  (iv) By (ii) and (iii)
f
f
1
1
(V) is
(V) is
g~ -closed.
g~ -clopen in X.
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f
1
(Vc) is
g~ -open in X.
On Slightly continuous functions via
(iv)  (i) Let V be a clopen subset of Y containing f(x). By (iv)
U=
f
1
(V) then f(U)  V. Hence f is slightly
f
1
f
1
(V) is
g~ -continuous.
4 Comparison
Proposition 4.1 Every slightly continuous function is slightly
Proof. Let U be a clopen set in Y then
g~ -sets
g~ -continuous.
(U) is open in X. Since every open set is
g~ -open. Hence f is slightly g~ -continuous.
Remark 4.2 The converse of the proposition 4.1 need not be true.
Example 4.3 Let X={a,b,c,d},Y={p,q,r},
 ={  , X,{b,c},{b,c,d},{a,b,c}},  ={  ,Y,{p},{q,r}}.
The function f is defined as f(a)=p,f(b)=q,f(c)=f(d)=r. The function f is slightly
g~ -clopen in X. Put
g~ -open, f
1
(U) is
g~ -continuous but not
slightly continuous.
Proposition 4.4 Every
g~ -continuous function is slightly g~ -continuous.
1
~ -open in X. Hence f is slightly g~ -continuous.
Proof. Let U be a clopen set in Y then f (U) is g


Remark 4.5 The converse of the Proposition 4.4 need not be true.
Example 4.6 Let X={a,b,c},Y={p,q},
 ={  , X,{a},{b},{a,b}},  ={  ,Y,{p}}.
g~ -continuous but not g~ -
The function f is defined as f(a)=q,f(b)=f(c)=p. The function f is slightly
continuous. Since
f
1
({p})={b,c} is not
g~ -open in X.
Definition 4.7 A space (X, τ) is called
(i)
a locally indiscrete space if every open subset is closed[2].
Tg~ -space if every g~ -closed subset in it is closed[3].
~ -continuous and (Y, σ) is a locally
Theorem 4.8 If the function f: ( X , )  (Y ,  ) is slightly g

~
indiscrete space then f is g  -continuous.
~ Proof. Let U be an open subset of Y. Since Y is locally indiscrete U is closed in Y. Since f is slightly g
(ii)
a
#

g~ -open in X. Hence f is g~ -continuous.
f
~ -continuous and (X, τ) is a #T ~ -space
Theorem 4.9 If the function f: ( X , )  (Y ,  ) is slightly g
g

1
continuous
(U) is
then f is slightly continuous.
Proof. Let U be a clopen subset of Y. Since f is slightly
g~ -continuous f
1
(U) is
g~ -open in X. Since
Tg~ , f 1 (U) is open in X. Hence f is slightly continuous.
~ -continuous.
Hence f is g
X is a
#

Theorem 4.10 Let f: ( X , )  (Y ,  ) and g: (Y, σ)  (Z , ) be functions.
(i)
If f is
~ -continuous.
slightly g

(ii)
If
g~ -irresolute and g is slightly g~ -continuous thengof: (X, τ)  ( Z , ) is
f is
~ -continuous.
slightly g

(iii)
If f is
~ -continuous.
slightly g

g~ -irresolute and g is g~ -continuous then
g~ -irresolute and g is slightly continuous then gof: (X, τ)  ( Z , ) is
Proof. (i) Let U be a clopen set in (Z,  ). Then
g~ -irresolute f
gof: (X, τ)  ( Z , ) is
g 1 (U ) is g~ -open in Y. Since f is
( g 1 (U )) is g~ -open in X. Since ( gof ) 1 (U ) =
f 1 ( g 1 (U )) , gof is slightly g~ -continuous.
1
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M. Lellis Thivagar and Nirmala Rebecca Paul
(ii) Let U be a clopen set in (Z,  ). Then
g 1 (U ) is g~ -open in Y. Since f is g~ -
( g 1 (U )) is g~ -open in X. Since ( gof ) 1 (U ) = f
~ -continuous.
gof is slightly g

irresolute
f
1
1
( g 1 (U )) ,
(iii) Let U be a clopen set in (Z,  ). Then
g 1 (U ) is open in Y and any open set is
g~ -open, f 1 ( g 1 (U )) is g~ -open in X. Since ( gof ) 1 (U ) = f 1 ( g 1 (U )) ,
~ -continuous.
gof is slightly g

Theorem 4.11 Let f: ( X , )  (Y ,  ) and g: (Y, σ)  (Z , ) be functions. If f is surjective and strongly
g~ -open and gof: (X, τ)  ( Z , ) is slightly g~ -continuous then g is slightly g~ -continuous.
1
1
~ -open in X. Since f is surjective
Proof. Let U be a clopen set in (Z,  ) then f ( g (U )) is g

~ -open in Y. Hence g is slightly g~ -continuous.
f( f ( g (U )) = g (U ) is g


Theorem 4.12 Let f: ( X , )  (Y ,  ) and g: (Y, σ)  (Z , ) be functions. If f is surjective and strongly
g~ -open and g~ -irresolute then gof: (X, τ)  ( Z , ) is slightly g~ -continuous if and only if g is
~ -continuous.
slightly g

1
1
~ -open in X. Since f is surjective
Proof. Let U be a clopen set in (Z,  ) then f ( g (U )) is g
1
1
1

~ -open in Y. Hence g is slightly g~ -continuous.
f( f ( g (U )) = g (U ) is g


1
~
~ -open in Y.
Conversely let g be slightly g  -continuous and U be a clopen set in (Z,  ) then g (U ) is g

1
1
~
~
~
Since f is g -irresolute f ( g (U )) is g -open in X. Hence gof is slightly g -continuous.
1
1
1



5 Applications
~
G O(B) and B is open in X then A is g~ -open in X.
~ -continuous and A is an open subset of X then the
Theorem 5.2 If f: ( X , )  (Y ,  ) is slightly g

~
restriction f / A : ( A, A )  (Y ,  ) is slightly g  -continuous.
1
1
1
~ -open and
Proof. Let V be a clopen subset of Y. Then ( f ) (V )  f (V )  A . Since f (V) is g
Lemma 5.1[3] Let A  B 

/A
~ -open in the relative topology of A. Hence f is slightly g~ -continuous.
A is open ( f / A ) (V ) is g

/A

~
Theorem 5.3 Let f: ( X , )  (Y ,  ) be a function and   {U i : i  I } be a g  -open cover of X. If
f is slightly g~ -continuous for each i  I then f is a slightly g~ -continuous function.
1
/ Ui


Proof. Let V be any clopen subset of Y. Since f / U i
~ -continuous for each i  I, it follows that
is slightly g

( f / U I ) 1 (V ) is g~ -open in U i . We have f 1 (V )   iI ( f 1 (V )  U i )   iI ( f / U i ) 1 (V ) . Then
1
~ -open in X. Hence f is slightly g~ -continuous.
by lemma 5.1 f (V) is g


~ -connected if X is not the union of two disjoint non empty g~ Definition 5.4 A space (X,  ) is called g


open sets.
~ -continuous surjective function and X is g~ Theorem 5.5 If f: ( X , )  (Y ,  ) is a slightly g


connected then Y is connected.
Proof. Suppose Y is not connected. Then there exists non-empty disjoint open sets U and V such that Y=U
 V. Therefore U and V are clopen sets in Y. Since f is slightly g~ -continuous, f 1 (U), f 1 (V) are
~ -open sets in X. Also f 1 (Y)=X= f 1 (U)  f 1 (V). This shows that X is not
non empty disjoint g

~
g -connected, a contradiction and hence Y is connected.

Theorem 5.6 If f is a slightly
g~ -continuous function from a g~ -connected space (X,  ) onto a space (Y,
σ) then Y is not a discrete space.
Proof. Suppose that Y is a discrete space. Let A be a proper nonempty open subset of Y. Then
1458
f
1
(A) is a
On Slightly continuous functions via
g~ -sets
g~ -clopen subset of X, which is a contradiction to the assumption that X is g~ -
proper nonempty
connected.
Theorem 5.7 A space X is
g~ -connected if every slightly g~ -continuous function from X into any T0
space Y is constant.
Proof. Let every slightly
g~ -continuous function from a space X into Y be constant. If X is not g~ ~ -clopen subset A of X. Let (Y, σ) be such that Y={a,b},
connected there exists a proper nonempty g

  { , Y,{a},{b}} be a topology. Let f:X  Y be any function such that f(A)={a} and f(X-A)={b}. Then
~ -continuous function such that Y is T which is a contradiction. Hence
f is a non-constant and slightly g

0
~
X is g  -connected.
Theorem 5.8 If a function f:X 
Y is slightly g~ -continuous, then P o f:X  Y is slightly g~ -

continuous for each



   , where each P is the projection of
 Y
onto


Y .
V be any clopen subset of Y . Then P (V ) is clopen in  Y and hence ( P o
1
1
~ -open in X. Therefore P o f is slightly g~ -continuous.
f ) ( V )= f ( P (V ) ) is g



~
Theorem 5.9 If a function f:
X 
Y is slightly g -continuous then f : X  Y is
1
Proof. Let
1


~ -continuous for each    .
slightly g

Proof. Let




1
V be any clopen subset of Y . Then P (V ) is clopen in
 Y


and hence ( P o
1
1
( P (V ))  f  (V )   { X  :     { }} . Since f is slightly g~ -continuous
1
f 1 ( P (V ) ) is g~ -open in  X  and the projection P of  X  onto X  is open and
1
~ -open in X and hence f is slightly g~ -continuous.
( V ) is g
continuous, f
f)
1
( V )=
f
1






Definition 5.10 A space (X, τ) is said to be
(i)
mildly compact[6] if every clopen cover of X has a finite subcover.
g~ -compact[8] if every g~ -open cover of X has a finite subcover.
(ii)
(iii)
A subset A of a space X is said to be mildly compcat[6] relative to X if every cover of A
by clopen sets of X has a finite subcover.
~ -compact[8] relative to X if every cover of A by
(iv)
A subset A of a space X is said to be g

g~ -open sets of X has a finite subcover.
(v)
compact.
(vi)
A subset A of a space X is said to be mildly compact[6] if the subspace A is mildly
A subset A of a space X is said to be
g~ -compact[8] if the subspace A is g~ -compact.
Theorem 5.11 If a function f: ( X , )  (Y ,  ) is a slightly
g~ -continuous and A is g~ -compact relative
to X then f(A) is mildly compact in Y.
Proof. Let { H  :   I } be any cover of f(A) by clopen sets of the subspace f(A).. For each
  I there
 x  I such that
A of Y such that H   A  f ( A) . For each x  A, there exists
~
f(x)  A x and there exists U x  G O(X) containing x such that f (U x )  A x . Since the family
~ -open sets of X, there exists a finite subset A of A such that
{ U x :x  A} is a cover of A by g
0

exists a clopen set
A  {U x : x  A0 } . Therefore we get
: x  A0 } and f(A) is mildly compact.
~ -continuous, surjective and X is g~ -compact then
Corollary 5.12 If f: ( X , )  (Y ,  ) is a slightly g


{ A
x
: x  A0 } . Thus f(A)=
f ( A)  { f (U x : x  A0 } which is a subset of
Y is mildly compact.
Proof. Let { V : V
{ A
x
 CO(Y),   I} be a cover of Y. Since f is slightly g~ -continuous
1459
M. Lellis Thivagar and Nirmala Rebecca Paul
{
1
f
X=
 I} be a g~ -open cover of X. So there is a finite subset I0 of I such that
( V ): 

1
f
I 0
( V ). Then Y=

I 0
V ( since f is surjective). Thus Y is mildly compact.
Definition 5.13 A space X is said to be
(i)
mildly countably compact[6] if every countable clopen cover of X has a finite subcover.
(ii)
Mildly Lindelof[6] if every cover of X by clopen sets has a countable sub cover.
~ -compact if every g~ -open cover of X has a finite subcover.
(iii)
Countably g


(iv)
(v)
(vi)
finite subcover.
(vii)
g~ -closed compact if every g~ -closed cover of X has a finite subcover.
g~ -Lidelof if every g~ -open cover of X has a countable subcover.
~ -closed compact if every countable cover of X by g~ -closed sets has a
Countably g


g~ -closed Lindelof if every g~ -closed cover of X has a countable subcover.
Theorem 5.14 Let f: ( X , )  (Y ,  )
following hold.
(i)
(ii)
be a slightly
g~ -continuous surjective function. Then the
g~ -Lindelof then Y is mildly Lindelof.
~ -compact then Y is mildly countably compact.
If X is countably g

If X is
 I} be a clopen cover of Y. Since f is slightly g~ -continuous surjection then
1
~ -open cover of X. Since X is g~ -Lindelof there exists a countable subset I of I
{ f ( V ):   I} is a g
0


Proof. (i) Let { V : 
 I0 }. Thus Y=  { V :   I0 } and hence Y is mildly Lidelof.
~ -continuous surjection then
(ii)Let { V :   I} be a clopen cover of Y. Since f is slightly g

1
~
~ -Lindelof there exists a
{ f ( V ):   I} is a g -open countable cover of X. Since X is countably g
such that X=

{
f
( V ): 
1


finite subcover of {
f
1
( V ): 

n
 I} such that X=  f 1 (Vi ) . Thus Y=
i 1
n
V
i
and hence Y is mildly
i 1
countably compact.
Theorem 5.15 Let f: ( X , )  (Y ,  )
following hold.
(i)
(ii)
(iii)
be a slightly
g~ -continuous surjective function. Then the
g~ -closed compact, then Y is mildly compact.
~ -closed Lindelof then Y is mildly Lindelof.
If X is g

~ -closed compact, then Y is mildly countably compact.
If X is countably g

If X is
 I} be a clopen cover of Y. Since f is slightly g~ -continuous surjection then
1
~ -closed cover of X. Since X is g~ -closed compact there exists a finite sub
{ f ( V ):   I} is a g


Proof. (i)Let { V : 
cover of {
f
1
( V ): 
n
 I} such that X=  f 1 (Vi ) . Thus Y=
i 1
n
V
i
and hence Y is mildly compact.
i 1
 I} be a clopen cover of Y. Since f is slightly g~ -continuous surjection then
1
~ -closed cover of X. Since X is g~ -closed Lindelof there exists a countable
{ f ( V ):   I} is a g


(ii)Let { V : 
subset I0 of I such that X=

{
f
1
( V ): 
 I0 }. Thus Y=  { V :   I0 } and hence Y is mildly
Lidelof.
(iii)Let { V : 
 I} be a countable clopen cover of Y. Since f is slightly g~ -continuous surjection then
1
~ -closed cover of X. Since X is countably g~ -closed compact, there
{ f ( V ):   I} is a countable g


1460
On Slightly continuous functions via
exists a finite sub cover of {
f
1
( V ): 
n
g~ -sets
 I} such that X=  f 1 (Vi ) . Thus Y=
i 1
n
V
i
and hence Y is
i 1
mildly countably compact.
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~ -closed sets in topological
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
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