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.

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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