Vol 2, No 9 (2011)

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#

_{$$$% &}

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!

On

ω

I

-continuous functions in Ideal topological spaces

*N. Chandramathi

1

and K. Bhuvaneswari

2

1

Department of Mathematics, Hindusthan college of Engineering and Technology, Coimbatore-32 *E-mail: [email protected]

2

Department of Mathematics, Mother Teresa Women’s University, Kodaikanal, Tamilnadu, India E-mail: [email protected]

(Received on: 25-08-11; Accepted on: 09-09-11)

--- ABSTRACT

I

n this paper,

ω

I -closed sets and,

ω

I –open sets are used to define and investigate a new class of functions and relationship between this new class and other classes of functions are

established.

2000 Mathematics subject classification: 54A05, 54A10

Key words: ideal, local function, semi-I-open set,

ω

-closed,

ω

I-closed sets and

ω

I -continuity.

---

1. Introduction and preliminaries:

Topological ideal plays an important role in topology for several years. In 1992, Jankovic and Hamlett [6] introduced the notion of I-open sets in topological spaces .El-monsef et.al [6] investigated I-open sets and I-continuous functions introduced and investigated the notion of

ω

I-closed sets. Quite recently, Hatir and Noiri [7] have introduced the notion of semi-I-open sets and semi-I-continuous function to obtain a decomposition of continuity via ideals.

In this paper, by using

ω

I-closed sets due to N. chandramathi et. al [3], we introduce the notion of

ω

I

-continuous functions in ideal topological spaces and obtain several properties of

ω

I

-continuity and the relationship between this function and other related functions .

An ideal

I

on a topological space

(

X

,

τ

)

is a Collection of subsets of X which satisfies

(i)

A

∈

I

and

B

⊂

A

implies

B

∈

I

and

(ii)

A

∈

I

and

B

∈

I

implies

A

B

∈

I

.

Given a topological space

(

X

,

τ

,

)

with an ideal

I

on X and if (X) is the set of all subsets of X, a set operator

(.) * : (X) (X), called a local function [7] of A with respect to and

I

is defined as follows: for A X , A *(I, )=

{

x

∈

X

/

U

A

∉

I

for every U

∈

(x)} where (x) = U | x U . We will make use of the basic facts about the local functions without mentioning it explicitly. A Kuratowski closure operator cl*_{(.) for a topology}

*

(I , ), called the * − topology, finer than and is defined by

cl

*

( )

A

=

A

*

(

I

,

τ

)

.When there is no chance for confusion we simply write

A

*instead

A

*

(

I

,

τ

)

and

τ

*or

τ

*

( )

I

for

A

*

(

I

,

τ

)

. X* is often a proper subset of X. The hypothesis X = X* is equivalent to the hypothesis I =

φ

For every ideal topological space (X, , I), there exists a topology * (I), finer than , generated by (I, ) = { U\I : U and I I}, but in general (I, ) is not always a topology[10] .If is

I

an ideal on X, Then

(

X

,

τ

,

I

)

is called ideal space. By an ideal space we always mean

an ideal topological space

(

X

,

τ

,

I

)

with no separation properties assumed. If

A

⊂

X

,

cl

( )

A

and

int

( )

A

will

respectively denote the closure and interior of A in

(

X

,

τ

,

)

and

cl

*

( )

A

and

int

*

( )

A

will respectively denote the closure and interior of A in

(

X

,

τ

,

I

)

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Definition: 1.1

A subset A of a space

(

X

,

τ

)

is called

(i) Semi open [7] if A cl (int (A)) and semi closed if int(cl (A)) A. (ii) Pre-open [7] if A int (cl (A)) and pre-closed if cl (int (A)) A.

(iii)

ω

-closed if [7] Cl(A) U whenever A U and U is semi-open in X,

ω

-open if

X − A is

ω

– Closed.

A subset A of a space

(

X

,

τ

,

I

)

is called

(iv) Semi - I -open [7] if A

⊆

Cl*(Int (A)) semi–I-Closed if int (cl* (A)) A (v) -I-open [7] if A Int (Cl*(Int (A).

2.

ω

I – closed sets:

Definition: 2.1 A subset

A

of an ideal topological space

(

X

,

τ

,

I

)

is called

ω

I – closed[3] if

cl

*

( )

A

⊆

U

whenever A U and U is semi- I- open in

(

X

,

τ

,

I

)

. The complement of

ω

I -closed set is called

ω

I- open if X − A is

ω

I – closed. We denote the family of all

ω

I – closed sets by

ω

IC

(

X

,

τ

,

I

)

.

Theorem: 2.1 Every open set is

ω

I –open.

Proof: Let U be an open set. We need to show that U is

ω

I –open. For this we show that X-U is

ω

I –closed. Let X-U

G

⊂

where G is semi-I-Open in X. Since X-U is closed. So by [8, Theorem2.3] or

cl

∗

(

X

−

U

)

⊆

cl X

(

−

U

)

⊆

G

.this proves that X-U is

ω

I –closed or U is

ω

I –open.

Theorem: 2.2 A set A is

ω

I –open iff

F

⊆

int ( )

∗

A

whenever F is semi –I-closed and

F

⊆

A

Proof: Suppose that

int ( )

F

⊆

∗

A

, where F is semi-I-closed and

F

⊆

A

.Let

A

c

⊆

U

,

where U is semi-I-open .Then

U

c

⊆

A

and

c

U

is semi-I-closed. Therefore,

U

c

⊆

int ( )

∗

A

.Since

U

c

⊆

int ( )

∗

A

, we have (int ( ))∗ A c ⊆U ,i.e

(

c

)

cl

∗

A

⊆

U

, since

cl

∗

(

A

c

)

=

(int ( ))

∗

A

c .Thus

A

cis

ω

I –closed, i.e. A is

ω

I –open.

Theorem: 2.3 If A is an

ω

I –open set of

(

X

,

τ

,

I

)

such that

int ( )

∗

A

⊆ B⊆ A then B is also an

ω

I –open set of

(

X

,

τ

,

I

)

Proof: Let U be a semi-I-closed set of

(

X

,

τ

,

I

)

such that

U

⊂

B

.Then,

U

⊂

A

.since A is

ω

I –open set, we have

int ( )

F

⊆

∗

A

.but

int ( )

∗

A

⊆

int ( )

∗

B

, implies

F

⊆

int ( )

∗

B

.Therefore by theorem 2.2, B is also an

ω

I –open set of

(

X

,

τ

,

I

)

.

Theorem: 2.4 Let

(

X

,

τ

,

I

)

be an ideal space and A a non empty subset of X .Then A is

ω

I –closed if and only if

(

( ))

A

X

−

cl

∗

A

is

ω

I –closed.

Proof: Suppose A is

ω

I –closed. Let U be a semi-I-open set such that

A

(

X

−

cl

∗

( ))

A

⊂

U

. Then

X

−

U

⊂

X

−

(

A

(

X

−

cl

∗

( ))

A

=

cl

∗

( )

A

−

A

.

Since A is

ω

I –closed by [4, theorem 2.10]

X

−

U

=

φ

and hence

X

=

U

.Thus X is the only set containing

(

( ))

A

X

−

cl

∗

A

.This gives,

[

A

(

X

−

cl

∗

( ))]

A

∗

⊂

X

.

This proves,

A

(

X

−

cl

∗

( ))

A

is

ω

I –closed.

Conversely let F be any semi-I-closed set such that

F

⊂

cl

∗

( )

A

−

A

. Since

cl

∗

( )

A

−

A

=

X

−

(

A

(

X cl A

−

∗

( ))

.

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2011, Page: 1628-1635

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and hence

F

⊂

X

−

cl

∗

( )

A

.since

F

⊂

cl

∗

( )

A

−

A

it proves that

F

=

φ

and hence

cl

∗

( )

A

⊂

X

−

F

and

X

−

F

is semi –I-closed. This proves that A is

ω

I –closed.

Theorem: 2.5 Let

(

X

,

τ

,

I

)

be an ideal space and

A

⊆

X

.Then

A

(

X

−

cl

∗

( ))

A

is

ω

( )

cl

∗

A

−

A

is

ω

I –open.

Proof: Let

A

(

X

−

cl

∗

( ))

A

be

ω

I –closed. We show that

X

−

(

cl

∗

( )

A

−

A

)

is

ω

I –closed. Let U be a

semi-I-open set containing

X

−

(

cl

∗

( )

A

−

A

)

Then

X

−

U

⊆

(

cl

∗

( )

A

−

A

)

.By theorem [4, theorem 2.10,]

X

−

U

=

φ

.

Therefore, X is the only semi-I-open set which contains

X

−

(

cl

∗

( )

A

−

A

)

and hence

(

X

−

(

cl

∗

( )

A

−

A

))

∗

⊆

X

.

this proves

X

−

(

cl

∗

( )

A

−

A

)

is

ω

I –closed or

cl

∗

( )

A

−

A

is

ω

I –open.

Conversely, let

cl

∗

( )

A

−

A

is

ω

I –open. Then

X

−

(

cl

∗

( )

A

−

A

)

=

A

∪

(

X

−

cl

∗

( ))

A

is

ω

I –closed.

Corollary: 2.1 Let

(

X

,

τ

,

I

)

be an ideal space and

A

⊆

X

.Then A is

ω

cl

∗

( )

A

−

A

is

ω

I –open.

Theorem: 2.6 For a subset

A

⊆

X

the following are equivalent: (i) A is

ω

I

-closed

(ii)

cl

*

( )

A

−

A

contains no non empty semi-I-closed set

(iii)

cl

*

( )

A

−

A

_is

ω

I-open.

Proof: (i)

⇔

(ii) by [4, theorem2.10], and (i)

⇔

(iii) by corollary above.

Theorem: 2.7 Let

(

X

,

τ

,

I

)

be an ideal space .Then every subset of X is

ω

I –closed if and only if every semi –I-open set is *-closed

Proof: Suppose every subset of X is

ω

I –closed .Let U be semi-I-open set then U is

ω

I –closed and

cl A

*

( )

⊆

U

implies

U

∗

⊆

U

.Hence U is *-closed.

Conversely, suppose that every semi –I-open set is *-closed. Let U be a non empty subset of X contained in a

semi-I-open set U. Then

A

∗

⊆

U

∗ implies

A

∗

⊆

U

.This proves that A is

ω

I –closed.

3.

ω

I

-continuous functions:

Definition: 3.1 A function f:

(

X

,

τ

,

I

)

→

(

Y

,

σ

)

is called

ω

I

-continuous if for every closed set

V

of

( , )

Y

σ

,

( )

(

,

)

1

I

X

IC

V

f

−

∈

ω

τ

Definition: 3.2 An ideal topological space

(

X

,

τ

,

I

)

is said to be T-dense [14] if every subset of X is *-dense in itself.

Remark: 3.1 Every continuous function is

ω

I

-continuous but not conversely as seen from the following example.

Example: 3.1 Let

X

=

Y

=

{

a b c d

, , ,

}

,

τ

=

σ

=

{

X

, ,

φ

{

a b

,

}

I

=

{

φ

,

{ }

a

}

(4)

Definition: 3.3 A space is

(

X

,

τ

,

I

)

called a

T

_ω* space if every

ω

I

-closed set in it is closed.

Theorem: 3.1 Let

(

X

,

τ

,

I

)

be an ideal topological space,

( , )

Z

η

be topological space and

(

Y

, ,

σ

J

)

be a

T

_ω*space. Then the composition

g

f

: ( , , )

X

τ

I

→

( , )

Z

η

of the

ω

I

-continuous maps f:

(

X

,

τ

,

I

)

→

(

Y

, ,

σ

J

)

and g:

(

Y

, ,

σ

J

)

→

( , )

Z

η

is

ω

I

-continuous.

Proof: Let F be any closed set of

( , )

Z

η

. Then

g

−1

( )

F

is closed in closed in

(

Y

, ,

σ

J

)

. Since g is

ω

I

-continuous and

(

Y

, ,

σ

J

)

is a

T

_ω*space. Since

g

−1

( )

F

is closed in

(

Y

, ,

σ

J

)

and f is

ω

I

-continuous,

f

−1

(

g

−1

( ))

F

is

ω

I

-closed in

(

X

,

τ

,

I

)

. But

f

−1

(

g

−1

( ))

F

=

(

g

f

) ( )

−1

F

and so

g

f

is

ω

I

-continuous.

Theorem: 3.2 Let f:

(

X

,

τ

,

I

)

→

(

Y

,

σ

)

and g:

(

Y

,

σ

,

J

)

→

(Z,µ)be two functions where I and J are ideals on X and Y ,respectively. Then

g

f

is

ω

I

-continuous if f is

ω

I

-continuous and g is continuous

Proof:Let

w

∈

µ. Then

(

g

f

)

−1=

(

f

−1

g

−1

)

( )

w

=

f

−1

g

−1

( )

w

since

g

−1

( )

w

is closed ad g is

ω

I

-continuous. Now since f is

ω

I

-continuous. So

f

−1

g

−1

( )

w

is

ω

I

-closed. Hence

g

f

is

ω

I

-continuous.

Definition: 3.4[1] If

(

X

,

τ

,

I

)

is an ideal topological space and A is a subset of X, we denote

τ

|

_A_.the relative topology on A and

I

_A

=

{

A

∩

I I

:

∈

I

}

is obviously an ideal on A.

Theorem: 3.3Let f:

(

X

,

τ

,

I

)

→

(

Y

,

σ

)

be a

ω

I

_{-continuous function and let A be * - closed, then the restriction}

(

A A

)

A

I

f

|

:

,

τ

|

,

|

→

(

Y

,

σ

)

is

ω

I

-continuous.

Proof: Let V be any closed subset of

(

Y

,

σ

)

since f is

ω

I

-continuous we have

f

−1

( )

V

is

ω

I

–closed. Also,

(

f

|

_A

)

−1

=

f

−1

( )

V

A

.Since A is * -closed, then by [4, theorem 2.6]

f

−1

( )

V

A

∈

ω

IC

(

X

,

τ

)

.On the other

hand

(

f

A

)

A

f

( )

V

1 1

|

−

=

− and

(

f

|

A

)

−1

∈

ω

IC

(

A

,

τ

|

A

,

I

|

A

)

.This shows that

f

|

A

:

(

A

,

τ

|

A

,

I

|

A

)

is

ω

I

-continuous.

Theorem: 3.4 Let f:

(

X

,

τ

,

I

)

→

₍

Y

, ,

σ

J

₎

be a function and let

{

U

_α

:

α

∈ ∆

}

be an open cover of a T-dense space X. If the restriction function

f

|

_U_αis

ω

I

-continuous for each

α

∈ ∆

, then f is

ω

I

-continuous.

Proof: Let

V

be an arbitrary open set in

(

Y

, ,

σ

J

)

.Then for each

α

∈ ∆

, we have

(

f U

|

_α

) ( )

−1

V

=

(

f

−1

( )

V

U

_α

)

.Because

f

|

_U_αis

ω

I

-continuous, therefore

f

−1

( )

V

U

_α is

ω

I

_{-open in X for each}

α

∈ ∆

. Since for each

α

∈ ∆

,

U

_α is open in X,

f

−1

( )

V

=

α

(

f

1

( )

V

U

α

)

−

∈∆ is

ω

I

-open .by [4, theorem 2.5]

(

) ( )

1

|

f U

_α −

V

is

ω

I

-open.

Then by proposition

f

−1

( )

V

is

ω

I

-open. Hence f is

ω

I

-continuous.

Definition: 3.5 Let x be a point of

(

X

,

τ

,

I

)

and W be a subset of

(

X

,

τ

,

I

)

.Then W is called an

ω

I

–neighborhood of x in

(

X

,

τ

,

I

)

if there exists an

ω

I

–open set U of

(

X

,

τ

,

I

)

such that

x U

∈

⊆

W

.

Theorem: 3.5 Let

(

X

,

τ

,

I

)

be a T-dense. Then for a function f:

(

X

,

τ

,

I

)

→

(

Y

,

σ

)

_,the following are equivalent . (i) The function f is

ω

I

-continuous.

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2011, Page: 1628-1635

" # $ % # % & '

(iii) For each point x in

(

X

,

τ

,

I

)

and each open set V in

(

Y

,

σ

)

with

f x

( )

∈

V

,

there is an

ω

I

- Open set U in

(

X

,

τ

,

I

)

_{such that}

x

∈

U f U

, ( )

⊆

V

.

(iv) The inverse image of each closed set in

(

Y

,

σ

)

is

ω

I

-closed.

(v) For each x in

(

X

,

τ

,

I

)

, the inverse of every neighborhood of f(x) is an

ω

I

-neighborhood of x.

(vi) For each x in

(

X

,

τ

,

I

)

and each neighborhood N of f(x), there is an

ω

I

-neighborhood W of x such that

(

)

.

f W

⊆

N

(vii) For each subset A of

(

X

,

τ

,

I

)

,

f

(

ω

I

−

cl A

( ))

⊆

cl f A

( ( ))

(viii) For each subset B of

(

Y

,

σ

)

,

1 1

(

( ))

( ( )).

I

cl f

B

f

cl B

ω

− −

−

⊆

Proof: The implications

( )

i

⇔

( )

ii

: This follows from Theorem 3.1.

( )

i

⇔

( )

iii

Suppose that (iii) holds and let V be an open set in

(

Y

,

σ

)

and let

x

∈

f

−1

( )

V

.Then

f x

( )

∈

V

and

thus there exists an

ω

I

-open set U, such that

x U

∈

_x and

f U

(

_x

)

⊆

V

, Now,

x

∈

U

_x

⊆

f

−1

( )

V

. Hence

1

( )

_{x f} _V _x

f

V

−

U

−

∈

=

and so by theorem 2.6 [4] ,

f

−1

( )

V

_is

ω

I

-open in and

(

X

,

τ

,

I

)

therefore, f is

ω

I

-continuous.

Conversely, suppose that (i) holds and let

f x

( )

∈

V

.

since f is

ω

I

-continuous,

f

−1

( )

V

is

ω

I

-open in X. By

putting

U

=

f

−1

( ),

V

we have

x U

∈

and

( )

ii

⇔

( )

iv

This result follows from the fact that if A is a subset of

(

Y

,

σ

)

, then

f

−1

(

A

c

)

=

(

f

−1

( )) .

A

c

( )

ii

→

( )

v

: For x in

(

X

,

τ

,

I

)

let N be a neighborhood of f(x) .Then there exists an open set U in

(

Y

,

σ

)

such that

( )

.

f x

∈

U

⊆

N

Consequently,

f

−1

( )

U

is an

ω

I

-open set in

(

X

,

τ

,

I

)

and

f

−1

( ( ))

f x

∈

f

−1

( )

U

⊆

f

−1

( ).

N

that is

x

∈

f

−1

( )

U

⊆

f

−1

( ).

N

Thus

f

−1

( )

N

_{is an neighborhood of x.}

( )

v

→

( )

vi

.Let

x

∈

X

and N be a neighborhood of f(x).Then by assumption,

W

=

f

−1

( )

N

is an

ω

I

-neighborhood of x and

f W

(

)

=

f f

(

−1

( ))

N

⊆

N

.

( )

vi

→

( )

iii

. For x in

(

X

,

τ

,

I

)

,Let V be an open set containing f(x).Then V is a neighborhood of f(x), so by assumption, there exists an

ω

I

- neighborhood W of x such that

f W

(

)

⊆

V

.

Hence there exists an

ω

I

- open set U in

(

X

,

τ

,

I

)

such that

x U

∈

⊆

W

and so

f U

( )

⊆

f W

(

)

⊆

V

.

(

vii

)

⇔

( )

iv

: Suppose that (iv) holds and let A be a subset of

(

X

,

τ

,

I

)

.Since

A

⊆

f

−1

( ( )),

f A

we have

1

( ( ( )).

A

⊆

f

−

cl f A

Since

cl f A

( ( )

is a closed set in

(

Y

,

σ

)

, by assumption

f

−1

( ( ( )))

cl f A

is an

ω

I

-closed set

containing A. Consequently,

ω

I

₋

cl A

( )

_⊆

f

−1

( ( ( ))).

cl f A

Thus

f

(

ω

I cl A

₋

( ))

_⊆

f f

(

−1

( ( ( ))))

cl f A

_⊆

cl f A

( ( )).

Conversely, suppose that (vii) holds for any subset A of

(

X

,

τ

,

I

)

. Let F be a closed subset of

(

Y

,

σ

)

.Then by assumption, i.e.

f

(

ω

I

₋

cl f

(

−1

( ))

F

_⊆

cl f f

( (

−1

( )))

F

_⊆

cl F

( )

₌

F

.

₍

_I

_{cl f}

₍

1

_{( ))}

_F

_f

1

_{( )}

_F

ω

₋

−

_⊆

−

and so

1

( )

f

−

F

is

ω

I

-closed.

(

vii

)

⇔

(

viii

)

: Suppose that (vii) holds and B be any subset of

(

Y

,

σ

)

.Then replacing A by in

f

−1

( )

B

in (vii), we

obtain

f

(

ω

I

₋

cl f

(

−1

( ))

B

_⊆

1

( (

( ))

( )

(6)

Conversely, suppose that (viii) holds. Let B=f(A) where A is a subset of

(

X

,

τ

,

I

)

. Then we have,

1 1

(

ω

I

₋

cl A

( ))

_⊆

(

ω

I

₋

cl f

(

−

( ))

B

_⊆

f

−

( ( ( )))

cl f A

and so

f

(

ω

I

−

cl A

( ))

⊆

( ( ( ))).

cl f A

Theorem: 3.6 If

(

X

,

τ

,

I

)

is a T-dense space and a function f:

(

X

,

τ

,

I

)

→

(

Y

, ,

σ

J

)

is

ω

I

-continuous, then the graph function

g X

:

→

X

×

Y

defined by

g x

( )

=

( , ( ))

x f x

for each

x

∈

X

is

ω

I

-continuous.

Proof: Let f be

ω

I

-continuous. Now let

x

∈

X

and let W be any open set in

X

×

Y

containing

g x

( )

=

( , ( ))

x f x

.Then there exists a basic open set

U V

×

such that

g x

( )

⊂

U V

×

⊂

W

.

Since f is

ω

I

-continuous, there exists a

I

ω

-open set

U

₁in X such that

x U

∈

₁

⊂

X

and

f U

(

₁

)

⊂

V

.

Then by [4,Theorem 2.5]s

U

₁

∩

U

∈

ω

IO X

( , )

τ

and

U

₁

∩

U

⊂

U

,then

g U

(

₁

∩

U

)

⊂

U V

×

⊂

W

.

This shows that g is

ω

I

-continuous.

Theorem: 3.7 A function f:

(

X

,

τ

)

→

₍

Y

, ,

σ

J

₎

is

ω

I

-continuous, if the graph function

g X

:

→

X

×

Y

defined by

g x

( )

=

( , ( ))

x f x

for each

x

∈

X

is

ω

I

-continuous.

Proof: Suppose that g is

ω

I

-continuous and let V be open set in Y containing f(x).Then

X V

×

is open set in

X

×

Y

and by

ω

I

-continuous of g, there exists a

ω

I

-open set U containing x such that

g U

( )

⊂

X V

×

.

Therefore, we obtain

f U

( )

⊂

V

.

This shows that f is

ω

I

-continuous.

Theorem: 3.8 Let

{

X

_α

:

α

∈ ∆

}

be any family of ideal topological spaces. If f:

(

X

, ,

τ

I

)

→ ∏

(

_α∈∆

X

_α

, )

σ

is a

I

ω

-continuous function, then

P

_α

f X

:

→

X

_αis

ω

I

-continuous for each

α

∈ ∆

, where

P

_α is the projection of

X

_α

∏

onto

X

_α

.

Proof: We will consider a fixed

α

∈ ∆

.Let

G

_α be an open set of

X

_α . Then

(

P

_α

) (

−1

G

_α

)

is open in

∏

X

_α. Since f is

ω

I

-continuous,

f

−1

((

P

_α

) (

−1

G

_α

))

=

(

P

_α

f

) (

−1

G

_α

)

is

ω

I

-open in X. Thus,

P

_α

f

is

ω

I

-continuous.

Theorem: 3.9 For any bijection f:

(

X

,

τ

)

→

( , , )

Y

σ

I

, the following statements are equivalent: (i)

f

−1

: ( , , )

Y

σ

I

_→

( , )

X

τ

is

ω

I

-continuous. (ii) f (U) is

ω

I

-open in Y for every open set U in X. (ii) f (U) is

ω

I

-closed in Y for every closed set U in X.

Proof: The proof is trivial.

Definition: 3.7 A collection

{

A

_α

:

αε

∇

}

of

ω

I

– open set in an ideal topological Space X is called a

ω

I

-open cover of a subset B of X is

B

⊂

{

A

_α

:

αε

∇

}

holds.

(

X

,

τ

,

I

)

is called

ω

I -compact if for every

ω

I -open cover

{

W

_α

:

α

∈ ∆

}

of

(

X

,

τ

,

I

)

there exists a finite subset

∆

of

∆

such that

(

X

− ∪

{

W

_α

:

α

∈ ∆

}

)

∈

I

.

Lemma: 3.1 [Newcomb, 1967]: For any function f:

(

X

,

τ

,

I

)

→

(

Y

,

τ

,

)

,

f

( )

I

is ideal on Y.

Theorem: 3.10 The image of a

ω

I

-compact space under a

ω

I

-continuous surjective function is

f I

( )

-compact.

Proof: Let f:

(

X

,

τ

,

I

)

→

(

Y

,

τ

,

)

be an

ω

I

-continuous subjective function and

{

A

_α

:

αε

∇

}

be an open cover of Y. Then

{

f

−1

( )

A

_α

:

αε

∇

}

is an

ω

I

-open cover of X. From the assumption, there exists a finite subset

∇

₀ of

∇

such that

X

−

{

f

−

( )

A

∈

∇

}

∈

I

0 1

:

α

α .Therefore,

Y

−

{

A

α

:

α

∈

∇

0

}

∈

f

( )

I

which shows that

(7)

2011, Page: 1628-1635

" # $ % # % & '(

Definition: 3.9 An ideal topological space is said to be – connected if X cannot be written as the disjoint union of two non – empty – open sets. A subset of is – connected if it is – connected as a subspace.

(

X

,

τ

,

I

)

is called

ω

I

-normal if for every pair of disjoint

ω

I

-closed sets A and B of

(

X

,

τ

,

I

)

,there exist disjoint

ω

I

-open sets

U

,

V

⊆

X

such that

A

⊆

U

and

B

⊆

V

Theorem: 3.11 If f:

(

X

,

τ

,

I

)

→

( , )

Y

τ

is a

ω

I

-continuous, closed injection and Y is normal, then X is

ω

I

-normal.

Proof:Let

F

₁ and

F

₂ be disjoint

ω

I

- closed subsets of X.Since f is closed and injective, are disjoint closed subsets of Y. Since Y is

ω

I

- normal ,

f F

( )

₁ and

f F

(

₂

)

are separated by disjoint

ω

I

-closed sets

V

₁and

V

₂ respectively. Hence

F

₁

⊂

f

−1

( ),

V

₁

F

₂

⊂

f

−1

(

V

₂

)

,and

f

−1

( )

V

₁

f

−1

(

V

₂

)

φ

∩

=

.Since f is

ω

I

-continuous and

f

−1

( )

V

₁ and

1 2

(

)

f

−

V

are

ω

I

-open in

(

X

,

τ

,

I

)

, we have

(

X

,

τ

,

I

)

is

ω

I

-normal.

Theorem: 3.12 An

ω

I

–continuous image of

ω

I

–connected space is connected.

Proof: Let f:

(

X

,

τ

,

I

)

→

( , )

Y

τ

be a

ω

I

– continuous function of a

ω

I

–connected space X onto a topological space Y. If possible .Let Y be disconnected. Let A and B form a disconnected set of Y. Then A and B are clopen and

Y

=

A

∪

B

, where

A

∩

B

=

φ

.since f is

ω

I

–continuous,

X

=

f

−1

( )

Y

==

f

−1

(

A

∪

B

)

, where

f

−1

( )

A

and

1

_{( )}

f

−

B

are nonempty

ω

I

–closed sets in X. Also,

f

−1

( )

V

₁

f

−1

(

V

₂

)

φ

∩

=

.Hence X is non

ω

I

–connected, which

is a contradiction. Therefore, Y is connected.

Theorem: 3.12 f:

(

X

,

τ

,

I

)

→

( , , )

Y

σ

J

and g:

( , , )

Y

σ

J

→

( , )

Z

η

are functions. Then their composition

(

)

:

, ,

( , )

g

f

X

τ

I

→

Z

η

is

ω

I

- continuous if f is

ω

I

- continuous and g is continuous.

Proof: Let W be any closed set in

( , )

Z

η

.Since g is continuous,

g

−1

( )

W

is closed in

(

Y

,

σ

)

.Since f is

ω

I

-

continuous, then

(

g

f

) ( )

−1

W

=

f

−1

(

g

−1

( )

W

)

is

ω

I

-closed in

(

X

,

τ

,

I

)

and hence

(

g

f

)

is

ω

I

-continuous.

Acknowledgement:

We, the authors thank the referee for his help in improving the quality of this paper.

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α

−

I

–open sets and

α

−

I

–Continuous Functions Acta Math. Hungar, 105, (1-2) (2004), 27-37

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ω

I

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ω

I

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β

−

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open sets and a decomposition of Almost-I- continuity, Bull. Malays Math. Sci. Soc. (2)), 29(1) (2006) 119-124.

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X

,

τ

,

I

)

ω

I

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_{s g}_∗ -closed sets in ideal topological spaces, Global Journal of Pure and applied

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_{s g}_∗ -continuous functions in ideal topological spaces, European Journal of Pure and

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