ABSTRACT

          In
this paper, we introduce the notions of R*-LC-sets, R*-
I-LC-sets,  I*R-closed
sets,  Rt-sets, and I*-Rt-sets. Also
defined the notions of R*- I-LC-continuous
maps, I*R-
continuous maps, g­-R*- continuous maps and obtained decompositions
of  *-continuity.

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Keywords:R*-LC-sets, R*-I-LC-sets,  I*R-closed
sets,  Rt-sets,  I*-Rt-sets
and g-R*- sets.

AMS subject classifications: 54A05, 54A10.

 

                1.INTRODUCTION
AND PRELIMINARIES

                      The
triplet (X,t, I)
usually represents an ideal topological space where (X,t) is a topological space
and an ideal I is a nonempty collection of subsets of X which satisfies the
properties heredity and finite additivity. 
For a subset AÍX, A*( I)={xÎX  : UÇAÏ I for every UÎt(x)},
is called the local function   8 of A
with respect to I and t. Whenever no confusion about the ideal I
, we write just A* instead of A*( I) . The closure
operator cl*(.) defined as cl*(A)=AÈ A*11
satisfies the conditions of being a kuratowski closure operator  for a
topology t*(I)
called the *-topology  which is finer then t . The
notions cl(A) and Int(A) are  used to represents the closure and the
interior of any subset A of X in a topological space (X,t) respectively. A subset A
of an ideal topological space (X,t, I) is said to be R*-perfect
if A* A Î I9.
The collection of all R*-perfect sets form a basis for a topology
called R*-topology. The members of R*-topology are called
R*-open sets. A subset A of an ideal
topological space (X,t, I) is said to be R*-closed
if it is a complement of an R*-open set. A subset A of an ideal
space (X,t, I)
is *-closed 6 (resp. *-perfect
4) if A*ÌA (resp. A= A*).

 

DEFINITION 1.1 Let  (X,t) be a
topological space and S Í X then S is called

(1)    
Locally closed (or  LC) if S is intersection of an  open set  and a closed set 2.

(2)    
t-set if int(A)
= int(cl(A)) 10.

 

DEFINITION 1.2 Let (X,t, I)
be an ideal topological space and S Í X then

(1)  
 S is a t-I-set if int(S)= int (cl*(S)), 3

(2)  
S is a ?*-I-set if int (cl*(int(S)))= int(S),3

(3)  
S is a I-LC
set if S=CÇD,
where CÎt and
D is *-perfect, 1

(4)   S
is a Weakly- I-LC set if S is
intersection of aopen set and *-closed set.7.

 

DEFINITION
1.3 5 A *-continuous function f is defined as a
function from an ideal space (X,t, I)  to topological space (Y
,s)
such that  f–1(S) is *-closed in (X,t, I) for every closed set S in (Y ,s) .

              

          2. R*-I-LC-SETS

 

                In
this section, we introduce and study about R*-LC sets, R*t-
sets, R*-CIsets
, R*-I-LC sets, I*R-closed sets and I*-Rt setsand the relation between  these sets.

 

DEFINITION 2.1(1)R*-LCset
if S is the intersection of  R*-open
set  and a closed set.

(2)   R*t-set
if S is the intersection of  R*-open
set  and 
a t-set.

(3)   R*-CI-set if S is the
intersection of   R*- open set and  a ?*-I-set.

 

DEFINITION 2.2  An
R*-LC-continuous
function(resp.R*t-continuous ) f is defined as a function from
an ideal space (X,t, I) 
to topological space (Y ,s) such that  f–1(S) is R*-LC-set
(resp. R*t-set)  in (X,t, I) for every closed set S in (Y ,s) .

 

 

DEFINITION
2.3A subset S of an
ideal topological space (X,t, I) is called an R*-I-LC-set if S=CÇD, where C is R*-open
and D is *-closed.

 

PROPOSITION
2.4Let (X,t, I)
be an ideal space and SÌX. Then the following hold.

(1)    If S is R*-open, then A is an R*-I-LC-set;

(2)    If S is *-closed, then A is an R*-I-LC-set;

(3)    If S is weakly-I-LC-set, then A is an  R*-I-LC-set.

Proof

(1)   Let
S be an R*-open. Since S = S Ç X, where S is R*-open
andX is *-closed, S is anR*-I-LC-set.

(2)    Let S be a *-closed. Since S = X Ç S, where X is R*-open
and S is *-closed, S is an R*-I-LC-set.

(3)   Since
every open set is R*-open, every weakly-I-LC-set is an
R*-I-LC-set.

The
converse of the Proposition 2.4 need not be true as seen from the following
examples.

 

 

EXAMPLE 2.5Let X = {1, 2, 3, 4}, t={Æ, X,
{1, 3}, {4}, {1, 3, 4}} and I ={Æ, {3}, {4},
{3, 4}}.

(1)  
 S = {2, 3} is an R*-I-LC-set but not
an R*-open set.

(2)  
 S = {1} is an R*-I-LC-set but not a *-closed set.

(3)  
 S= {1} is anR*-I-LC-set but not a weakly-I-LC-set.

THEOREM 2.6Let (X,t, I)
be an ideal space and S be an R*-I-LC-subset of X. Then

(1)    For any *-closed set T , S ÇT is an R*-I-LC-set;

(2)    For any R*-open set T , S ÇT is an R*-I-LC-set;

(3)    For any R*-I-LC-set T , S ÇT is an  R*-I-LC-set.

 

Proof is obvious
from the definition

                                                                     

REMARK2.7The union of any two R*-I-LC-sets need not be an R*-I-LC-set.

                  

EXAMPLE 2.8Let X = {1, 2, 3, 4}, t={Æ, X,
{1, 3}, {4}, {1, 3, 4}} and I ={Æ, {3}, {4},
{3, 4}}.

Then, S = {1}
and T = {2} are R*-I-LC-sets but S È T = {1, 2} is not an R*-I-LC-set.

 

DEFINITION 2.9 Let (X,t, I)
) be an ideal topological space and S Í X  is called I*R-closed
if S*Ì
U whenever S Ì
U and U is R*-open in X.

 

DEFINITION 2.10A subset S of an ideal
topological space (X,t, I) is said to be I*-Rt-set if S is
the intersection of R*-open
set  and 
t-I-set.

 

PROPOSITION 2.11Let (X,t, I)
be an ideal topological space and S be a subset of X. If S is an R*-I-LC-set, then S is anI*-Rt-set.

Proof                 

            Let S be an R*-I-LC-set. Then S is the intersection of R*-open set and a *-closed set. Since every *-closed set is a t-I-set, S is the intersection of R*-open set  and  a
t-I-set. Hence S is an I*-Rt-set.

The converse of
proposition need not be true as seen from the following example.

 

EXAMPLE 2.12Let X = {1, 2, 3, 4}, t={Æ, X,
{1, 3}, {4}, {1, 3, 4}} and I ={Æ, {3}, {4},
{3, 4}}.

Then A = {1, 2}
is an I*-Rt-set
but not R*-I-LC-set.

 

 

PROPOSITION 2.13Let (X,t, I)
be an ideal topological  space and S be a
subset of X. If S is an I*-Rt-set,
then S is a  R*- CI
–set.

Proof

            Let S be anI*-Rt-set, This implies S = C Ç D, where C is R*-open
and D is a t-I-set. Since D is a t-I-set, implies  int(S)= int (cl*(S)), impliesint (cl*(int(S)))
Íint(S). Also, int(S)Ícl*(int S)),
implies int(S)Íint
(cl*(int(S))). Thereforeint (cl*(int(S)))= int(S) and therefore D is a?*-I-set. Hence S is a R*- CI
–set.

The converse of
proposition need not be true as seen from the following example.

 

EXAMPLE 2.14Let X = {1, 2, 3, 4} t={Æ, X,
{2}, {1, 4}, {1, 2, 4}} and I ={Æ, {2}}.

Then S = {1} is a
CI –set but not an I*-Rt-set.

 

THEOREM 2.15 Let (X,t, I)
be an ideal topological space and S be an I*-Rt-subset
of X. Then

(1)    For any t-I-set
T, S Ç T is an I*-Rt-set;

(2)    For any R*-open set T, S Ç T is an I*-Rt-set;

(3)    For any I*-Rt-set
T, S Ç T is an I*-Rt-set.

 

Proof directly
follows from the definition

 

REMARK
2.16   The union of any two I*-Rt-sets need
not be an I*-Rt-set.

 

EXAMPLE 2.17Let X = {1, 2, 3, 4}, t={Æ, X,
{2}, {2, 3, 4}}, I = {Æ, {2}}.

Then, S = {2} and
T= {3} are I*-Rt-sets
but S È
T = {2, 3} is not an I*-Rt-set.

 

THEOREM 2.18 Let (X,t, I)
be an ideal topological space and S be subset of X. Then the properties
mentioned below are equivalent:

(1)    S is *-closed;

(2)   S
is weakly-I-LC-set and  an I*R-closed
set;

(3)    S is an R*-I-LC-set and  anI*R-closed
set.

Proof

Since every *-closed set is an R*-I-LC-set, it is easy to prove (1) Þ (2).

(2) Þ (3):
Since every *-closed set is a
weakly-I-LC-set and t-I set, (3) follows from (2).

(3) Þ (1):
Since every open set is an R*-open set, every weakly-I-LC-set is an R*-I-LC-set.

 

REMARK
2.19The notions of R-I-LC*-sets and  anI*R-closed sets
are independent.

 

 

EXAMPLE 2.20Let X = {1, 2, 3, 4}, t={Æ, X,
{2}, {1, 4}, {1, 2, 4}}, I = {Æ, {2}}.

(1)    S = {2} is both R*-I-LC-set andI*-Rt-set
but not an I*R-closed
set.

(2)    S = {2, 3. 4} is an I*R-closed set but not both R*-I-LC-set and I*-Rt-set
.

 

            3.
DECOMPOSITIONS OF *-CONTINUITY

 

            In
this section, we introduce and study about I*R- continuous, R*-I-LC-continuous andI*-Rt-
continuous functions and also we obtain a new decomposition of *-continuity.

 

DEFINITION
3.1A function f:
(X,t, I)
®
(Y ,s)
is said to be I*R-
continuous (resp. R*-I-LC- continuous, I*-Rt- continuous) if f–1(V) is I*R-closed (resp.
R*-I-LC- set, I*-Rt- set) in (X,t, I) for every closed set
V in (Y ,s).

 

REMARK 3.2(1) Every *-continuous function is weakly-I-LC-continuous.

(2) Every weakly-I-LC-continuous function is R*-I-LC- continuous.

 

PROPOSITION 3.3(1) Let f: (X,t, I)
®
(Y ,s,J)
is said to be I*R-
continuous and g: (Y,s, J) ® (Z ,h) be
continuous. Then g? f: (X,t, I) ® (Z ,h) is I*R- continuous.

(2) Let f: (X,t, I)
®
(Y ,s,
J)
be I*R-
continuous and g: (Y,s, J) ® (Z ,h) be *-continuous. Then g ?
f: (X,t, I)
®
(Z ,h)
is I*R- continuous.

 

THEOREM 3.4For a functionf: (X,t, I)
®
(Y ,s),
the following are equivalent:

(1)    f is continuous;

(2)   f
is weakly-I-LC-continuity and  an I*R-continuity;

(3)   f
is R*-I-LC- continuous and I*R- continuous.

Proof

            Immediately follows from theorem
2.18.

 

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