ON I-OPEN SETS AND I-CONTINUOUS FUNCTIONS

(1)

Kyungpook Mathematical Journal Volume ³²

,

Number ¹

,

June

,

¹⁹⁹²

ON I-OPEN SETS AND I-CONTINUOUS FUNCTIONS

M.E. Abd EI-Monsef

,

E.F. Lashien and A.A. Nasef

In 1990

,

D. Jankovic

’

and T.R. Hamlett have introduced the notion of I-open sets in topological spaces. The aim of this paper is to introduce more new properties of I-openness. Also

,

we introduce and study new topological notions via ideals

,

namely

,

I-c\osed sets

,

I-continuous functions

,

I-open (c1osed) functions. Relationships between these c\asses and other relevant c1asses are investigated

1.

Introduction

Throughout the present paper

,

(X

, T)

and (Y

, (7)

(or simply X and Y) denote topological spac얹 on wl삐1 no separation axioms are assumed

unl얹s explicitly stated. Let A be a subset of (X

,

T). The c\osure of A and the interior of A are denoted by Cl(A) and int(A)

,

respectively. Recall that A is said to be regular closed if Cl(intA) = A. A is said to be 0^•open (0.0.) [8] (resp. semi-open (5

.0.)

[5]

,

preopen

(P.

O.) [6

],

β-open (β 0.) [1]) if A

c

ir벼 CI(int(A))) (resp. A C Cl(int(A))

,

A

c

^int(Cl(A))

,

A C CI(int(CI(A))). The complement of an o-open (resp. serni-open

,

preopen

,

β-open) set is called o-c\osed (resp. semi-closed

,

pre-closed

,

β

c1osed). The family of all o-open (resp. sem야pen ， preopen

,

ß-open) sets of (X

, T)

is denoted by

T

^Q (resp. SO(X

, T) ,

PO(X

, T) ,

(더O(X， T))

lt is shown in [8] that

T'"

is a topology on X and

T

C

T

^Q^•

An ideal on a nonempty set X is a collection 1 of subsets of X which is closed uIldel the opeI

a t k m d i t i v l t y ) [lO] We denote by (X

, T ,

I) a topological space (X

, T)

and an

R.eceived J anuary 4, 1991

Key Words and Phrases. Ideal

,

regular open set

,

a-open set

,

semi-open set

,

preopen set

,

β-open set

,

1←。pen set

,

/-c1osed set

,

pre-continuous function

,

M-pre-continuous function

,

I-open function

,

I-closed function

,

I-continuous function

1980 AMS Subject Classification Codes Primary: 54C10;Secondary; 54D25, 54D30 21

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22 M.E. Abd EI-Monsef. E.F. Lashien and A.A. Nasef

ideal 1 on X. Given a space (X

,

^T

,1)

and a subset A 드 X

,

we denote by A*(I)

=

{x E X : U n A

rf.

1 for every (open) neighborhood U of x}

,

wri t ten simply as A* when there is no chance for confusion; CI*(A)

=

A U A‘

defines a Kuratowski closure operator [10] for a topology T*(

I)

(also denoted T* when there is no chance for confusion) finer than T. The topol ogy T* has as a basis 더 (I， T) = {U \ E : U E T, E E

I}

[9]. Recall that A

c

^(X

,

^T

, 1)

is called *-dense-in-itself [2] (resp. ^T‘-closed [3]

,

*-perfect [2]) iff A

c

A* (resp. A*

c

A

,

A = A*)

A function

f :

(X

, T)

• (Y

, (7)

is said to be pre-continuous [6] (resp M-pre-continuous [7]) if for each V E σ (resp. V E PO(Y))

,

f-l(V) E PO(X). f is called preopen [6] (resp. preclosed [6]) if the image of each open (resp.closed) set in X is preopen (resp. preclosed).

2. On I-open and I-closed sets

Definition 2.1 [4]. Given a space (X

,

T

, 1)

and A 드 X

,

A is said to be 1-open if

...1

드 int(A*)

We denote by IO(X

, T)

=

{...1

드 X:A 드 int(A

‘)}

or simply write 10 forIO(X

, T)

、애en there is no chance for conf뼈 on

Remark 2.1. It is clear that

,

1-openness and openness are independent concepts (Examples 2.1

,

2.2)

Example 2.1. Let X ⁼ {a

,

b

,

c

,

d} with a topologyr = {X

, ø ^,

^{c}

^,

^{a

^,

^b}

^,

^{a

^,

^b

^,

^c}}

and 1 ⁼

{</>,

{a}}. Then {b

,

c

,

d} E IO( X

, T)

but {b

,

c

,

d}

rf. T

Example 2.2. Let

X

be as in Example 2

‘ 1 ,

T =

{X , ø ^, {d} , {a ,

c}

, {a ,

c

, d} }

and 1

= {ø ,

{c}

,

{d}

,

{c

,

d}}. It is clear that {a

,

c

,

d} E ^T

,

but {a

,

c

,

d}

rf.

IO(X

, T)

Remark 2.2. One can deduce that: I-open set =수 preopen set

,

and the converse is not true

,

in general

,

as shown by the following example.

Example 2.3. Let X

,

^T and 1 be as in Example (2.2). Then

,

we can easily deduce that {d} E PO(X

, T) ,

but {d}

rf.

IO(X

, T).

Remark 2.3. The intersection of two I-open sets need not be 1-open 잃 IS

illustrated by the following example.

Example 2.4. Let X ⁼ {a

,

b

,

c

,

d}

,

^T ⁼ {X

, ø ^,

^{a

^,

^b}, ^{a

^,

^b

^,

^c}}^and¹⁼

{</>}‘

Then {a

,

c}, {b

,

c

,

d} E IO(X

, T) ,

but {a

,

c} n {b

,

c

,

d}

rf.

IO(X

, T).

Theorem 2.1. For a space (X

,

T

, 1)

and

...1 c

X

,

ψe have:

(i)

1f

1

=

{ø

}, then

...1(1)*

= CI(A)

,

and hence each of I-open set and

(3)

On I-open sets and /-continuous functions 23

preopeη

set are

coiη cide.

(ii) /1/=

P(X) , then A* (1) =

^</>

and hence A is /-open iJJ A =

</>.

Theorem 2.2. For aηy

1 -open set A 01

a space

(X ,

^T

, 1), we

have

A*

⁼

(int(A)).

Definition 2.2. A subset

F

드

(X ,

^T

, 1)

is called /-closed if its complement is /-open.

Remark

2.4. The concept of /-closeness makes a very important deviation from the closeness for the topology in ordinary sense.

Theorem 2.3. For

A

ζ

(X ,

T

,I)

ψe haνe

((int(A)))C*

카

int((AC)) in*

geη eral

(Example

2.5) 뼈eπ

AC

denotes

the

complemeηt

01 A.

Example 2.5. If

X = {a , b , c , d

}, T

= {X ,</>, {a} , {a , b} ,

{a

,

b

,

c}} and / =

{</> , {a}}.

Then it is clear that if

A

=

{a ,

b}

,

then : ((

int(

A))γ

= {a

}, but

int((Ac))* =

^</>

Theorem 2.4.

/1 A

드 (X

,

^T

,I) is

^/-c/

osed , then

A ^::J

(intA))

‘

Proof

Follows from the definition of /-closed sets and Theorem 2.3(c) [3]

Theorem 2.5.

Let A

드 (X

,

^T

, 1) and (X \ (int(A)))* = int((X\ A) ‘)

Theη

A is /

^-c/osed

iJJ A

그

(int(A))

‘

Proof

Obvious

Theorem 2.6.

Let (X ,

^T

, 1) be a space and A , B ç X. Then:

(i) /1 {U

o : a ^E,",} 드

IO(X , T) ,

theη U

{u

o:

a

^E ,",} E

IO(X , r)

μ].

(ii)

/J A

^E

IO(X , T) and B

^E

T , then A n B

^E

IO(X , T)

μl (i

끼11페1

(iω비v끼) IIAEIO(X

, T) andBESO(X , T) , thenAnBESO(A).

(v)

/J A

E

IO(X , T)

aη d

B

E

T ,

theη AnB 드

int(B n (B n A)).*

Proof

(i) Since

{U

^O ^:

a

E ,",} 드

IO(X , T) ,

then Uo

드

^{int(U~) ，}for every

a E ,"" thus

,

UU" 드 U(in tU~)) 드 int(UU~) 드

int(UU

o

)'‘

for every

a

E ,",.

(ii)

A n

B 드 1ηt(A*)

n B

=

int(A* n B)

(since

B

E

T) ,

from Theorem 2.3 (g) [3], we have :

A n B ç int(A n B).*

(iii) Obvious

,

since A

*(1)

is closed and A* 드 Cl(A).

(iv) Follows from Theorem (2.3) (c) [3].

(v) Follows directly [rom Theorem 2.3 (g) [3]

Corollary 2.1. (i)

The union 01 /-closed set and

c/

osed set is /-closed.

(4)

24 M.E. Abd 티 Monsef. E.F. Lashien and A.A. Nasef

(ii)

Th

e union of 1 -c/

osed set and an a-closed

set is preclosed

Theorem 2.7. If A

드

(

X ,

^T,I) is 1 ~ope

n and

se

m

i-close

d , then A

=

int(A-

).

Proof Follows (rom Theorem 2.3(c) [3].

Theorem 2.8.

Let A

E

IO( X

) and B E

IO

(Y)

, th

en A x

B

E

IO

(X x

Y

)

ν A- x

B-

⁼ (A x

B

)"

,

ψheπ

X

x

Y

is

the product spa

ce

Proof

AxB

드

int

(

A")

x

int(B"

) =

int

(A‘ x

B-

)

,

from hypothesis

,

=

int

(A x

B)".

Therefore

, A

x

B

E

IO

(

X

x y)

Theorem 2.9. If

A c W

C

CI( A) and A

E IO( χ T)，

then

W

is

ß

-open

.

Proof Follows directly from Theorem 2.3 (c) [3J.

Theorem 2.10. If(X

, T

,I) is a

space and

W E IO(X

, T) , then CI(V) n

W

c (V n

W)*

,

for

every V

E

SO(X).

Proof Let

V

E

SO(X) ,

then:

CI(V)

=

CI(int(V)) ,

since W E IO(X)

,

then

CI(V) n

W C

CI(int(V)) n int(W")

C

CI(int(V) n W)*

c CI(V n

W)"

,

by using Theorem 2.3 (c) [3]

=

(vnw)".

Theorem 2.11. If(X

,

T

,I) is a space , A

E T

and

B E IO(X

,

T)

,

then

there exists an open subset G of X such that AnG = 4>,

implies AnB

= 4>

Proof Since

B

E IO(X

, T) ,

then B 드

int(B-) ,

by talcing G

= int( B")

to be an open set such that B

c

G

,

but

A n

G =

4>,

then G

ç

X\

A

implies that C

I(G)

드 (X\

A

). Hence B 드 (X\

A

) and this completes the proof.

Theorem 2.12. If (X

,

T,If) is a 되 -space

and A

E IO(X)

, then A

드

int(A

^d⁾

,

ψheπ

A

^d

denotes the

d따erived

set of A

and If

denotes the ideal of finit

e subsels.

Proof Follows directly from the definition of I-open set and the fact that

A* (I f)

=

A

^din a T1-space [3].

(5)

On I-open sets and I-continuous functions 25

Theorem 2_13. Let {X" : a E

l'-}

be a family of spaces

,

X

== n

^X。

be the prodπct space and A

== II ’‘

^A"^x

II

^x，β^{a non}^empty^subset^{of X}

,

0=1 o :J:.β

where n is a positive integer and A"

c

X". Then

,

A" E IO(X,,) for each (1 ~ a ~ n) iJJ A E IO(X).

Proof (Nece5sity): Suppose A" E IO(X,,) for each (1 ~ ^(l'~ n). Since

a

A

== II

^A"^x

II

^Xß

드

înt(A')^‘ ^T^henÂÊÎO(X).

0=1 0#β

(Sufficiency): AS5ume that A E 1 O( X). So A 드 int(A')

== n := ,

A: x

n. ，βXβ. Since A

fo

rþ and A E IO(X) then int(A')

fo

rþ and hence int(A:) 폼 rþ

,

for each (1

<

^(l' ~ η). Therefore

,

^A"드 int( A:) and 50

,

A" E IO(X) for each (1 ~ ^(l'~ n).

Theorem 2.14. For a subset A 드 (X

,

^r

,1)

ψe have:

(i) If A is r' -c/osed and A E IO(X)

,

theπ， int(A)

==

int(A*) (ii) A is r' ^-^c/osed íJJ A is open and 1 ^-c/osed.

(iii) If A 생 *-perfecl

,

^then^A

==

int(A')

,

^f^{or every}^AE IO(X

,

^r)

(iv) Jf A is regular closed and I-open

,

then A'

(J

n)

==

int(A'Un)) ψhe1'e

In is the ideal of nowhere dense sets.

(T

n

==

{A

c

^X ^:^int(C^α^I^(A^꺼))

==

rþ}ηn ) ^.

.

p좌roof (i띠i) ^,’ (ii) and (ii폐l

(iv비v) Follows from the definition of J-open and the fact that A is regular closed iff A

==

A*

(J

n) [3]

3. I-continuous functions

Definition 3.1. A function

f (X ,

r

, 1)

• (κ

0")

is 5aid to be

1-

continuous iffor every V E

O",

f-'(V) E IO(X

,

r).

From the above definition one may notice that

1 -continuity =추 precontinuity[6]

and the converse is not true as 5hown by the following example.

Example 3.1. Let X

==

Y

== {a ,

b

, c ,

d}

,

r is the in띠1κ띠d미liscre c

the discrete topology and 1

==

{rþ

,

{c}} on X. Then the idcntity function

f :

^(X

,

r

,1)

• ()이

0")

is precontinuou5 but not I-continuous

,

because

{c} E

0",

but f-'({c})

==

{c}

f/.

IO(X).

The following two examples show that the concept of continuity and I-continuity are independent

(6)

26 M.E. Abd EI-Monsef, E.F. Lashien and A.A. Nasef

Example 3.2. Let X

=

Y

=

{a

,

b

, c} ,

T

= {X , 4>,

{a}

, {c} , {a , b}, {a , c}} ,

1 = {4>, {b

}, {c}, {b

,

c}} on X and u

=

{Y

, 4>, {a

}, {c}, {a

, c}}.

Then the identity function

f :

(X

,

^T

,

1) • (Y

,

u) is continuous but not I-continuous because

{c}

E

u ,

but

f-l({C}) = {c} rf.

JO(X)

Example 3.3. Let X

=

^Y

= {a ,

b

, c},

T

=

^U

=

^{X

,

rþ

,

{a}} and 1 =

{4>, {b}}

on X. Then

f: (X ,

T

,I)

• (κ u) which is defined by:

f(a)

=

a = f(b)

and

f(c) =

c is I-continuous but not continuous.

Theorem 3.^1.

For a funclion f

:

(X ,

T

,I)

• (Y

, u) the

follouηng

are

equivalen

t:

(i) f is

1 -contin.LOUS.

(ii)

For each x

E X

and each V

ε

u containing f( x) , there exists W

E I

O(X) containing x such that f(W)

C

V.

(iii)

For each x

E X

and each V

E

u containing f(x) , (J -l(V )t is a neighborhood of x.

Proof

(i) ^=}(ii) : Since

V

E u containing f

(x) ,

then by (i)

,

f-l(V

)

E

IO(X) ,

by putting

W

=

f

-l

(V)

which containing

x.

Therefore

f(W)

C

v

(ii) =추 (iii): Since V E u containing f(

x) ,

then by (ii)

,

there exisls

W

E

IO(X)

containing x such that f(W

)

C V. So

, x

E W 드

int(W ‘)

드

int (J

-l(V

))' ç U-

¹

(V))*

. Hence

U-

¹

(V))

‘ is a neighborhood of

x

(iii) =추 (i) : Obvious

Theorem 3.2.

For

f : (X

,

^T

, 1)

•

(Y , u) the following

are

equivalenl:

(i) 1 is 1

-contin uous.

(ii)

The inverse image of each closed set

in

Y is

1 -closed.

(iii)

(int (J -

l

(M)))‘ c f-l(M

‘), for

each *-dense-in-ilself su

b

set M

C Y

(iv) 1((int(U)))*

C

(J (U))' ,

for

each U c

^X

, and

for

each -perfect subset* 01 Y

Proof

(i) ^=} (ii): Let

F

C

Y

be c1osed

,

then

Y \ F

is open

,

by (i)

,

1

-¹(y \

F)

= X\

f-l(F)

is I

-open

. Thus

, 1-

¹

(F)

is I-c1

osed

(ii) =송 (iii): Let

M

C

Y ,

since

M"

is c1osed

,

then by

(ü) f-l(M')

is I-c1

osed.

Thus

,

by using Theorem (2

.4)

f-

l(M")

::l

(int (J

-l

(M ’)))",

since

M

is *-dense-in-itself

,

then f

-

l

(M")

그

(int (J -l (M")))"

그

(int (J -

l

(M)))".

(iii) ^=}(iv): Let

U

C X and

W =

I

(U) ,

then by (iii)

, f-l(W')

그

(int (J -l(W)))*

::l

(int(U)).*

Hence

,

f

((int(U))')

C

W'

=

(J (U))"

. (iv) ^=}(i): Let

V

E u

,

W

= Y \ V

and

U =

f-l(W

) ,

then I(

U) c W

and by (iv)

,

1

((int(U)))*

C

(J (U))'

C W" (by using Theorem 2.3(a) [3])

(7)

On I-open sets and I-continuous functions 27

=

W

(because

W

is *-perfect).

Thus

,

f-l(W) 그 (int(U))*

=

(int

(J

-l(W)))*

,

and therefore

,

f-1(W)

=

f-l(y \ V) is I-closed. Hence

,

f-l(V) is I-open in X and f is I-continuous

Theorem 3.3. The function f : (X

,

^T

, I)

• (Y

,

a) is 1 -continuous 퍼 the graph function 9 : X • X x Y is 1 -continuous

Proof (Necessity): Let f be I-continuous. Now let x E

X

and let V be any open set in X x Y containing g(x) = (x

,

f(x)). Then there exists a basic open set U x W such that g(x) E U x W C V. Since

f

is 1 continuous

,

there exists I-open set U!, in X sucht that x E U1 C X and f(U1) C W. Since U1

n

^Uis I-open set in X and U1

n

^U

c

^U

,

then g(U1

n

U)

c

U x W C V showing that 9 is I-continuous.

(Sufficiency): Let 9 : X • X x Y be I-continuous and let V be open containing f (x). Then X x V is open in X x Y and the I-continuity of 9 1r때lies there exists I-open set W such that g(W) C X x V. But this implies f(W) C V. Therefore

,

f is I-continuous.

Theorem 3.4. Let f : (X

,

T

, I)

• (Y

,

a) be an 1 -contínuous and U E ^T.

Then the ^Testriction flU is an 1 -continuous.

Proof Let V E a. Then f-1(V)

ç

int

(J

-1(V))* and so

,

U

n

f-l(V) 드

U

n

int

U-

1(V))*. Thus (JlUt 1(V) 드 U

n

int(J-1(V))“ since U E ^T

,

then

(J

lut1(V) = int[U

n (r

¹(V))*]

,

by using Theorem 2.3 (g) [3]

C

i떠 [U n r

¹

(V) ]

^‘

int[(JIUt1(V)]*. Therefore

,

flU is 1 - continuous Theorem 3.5. Let f : (X

,

^T

, I)

• (Y

,

a

, J)

be a function and {U^{", :} Q E

6 }

be an open cover of X. If the restriction funciion f lU '" 양 1 -continuous

,

for each Q E

6 ,

then f is 1 -continuous.

Proof Similar to Theorem 3.4.

The folJowing results are immediate and the obvious proofs are omit- ted.

Theorem 3.6. Let f : (X

,

^T

, I)

• (Y

,

a) be 1 -continuous and open func- tion

,

then the inverse image of each 1 -open set in Y ís preopen ín X.

Theorem 3.7. Let f: (X

,

T

,I)

• (Y

,

a) be I-continuous and f- 1(V*) C

(8)

28 M.E. Abd EI-Monsef. E.F. Lashien and A.A. Nasef

U-

¹(V)

t ^, ^Jor

^{each V}^CY. Then the inVel^‘'se image oJ each

1 -open

set is

I-open.

Remark

3.1. The composition of two I-continuous functions need not be I-continuous

,

in general

,

as shown by the following example.

Example 3.4. Let X

=

^Z

= {a , b , c} ,

Y

= {a , b , c , d}

with topologies r

=

^{X

^,Iþ,

^{a}^},¹⁷

= ^{Y ^,Iþ, ^{a ^, ^c}}

^and

^v =

^{Z

^,.p,

^{c^{}, {b}

^, ^c}}

^and^let

1 = ^{.p,

^{c}}^on^X ^and^J

= ^{Iþ,

^{a}}^on

^Y

^andlet the identity function

J :

(X

,

T

,I)

• (Y

,

σ) and 9 : (Y

,

17

,

J) •

(Z ,

v) defined as:

g(a)

=

a

,

g(b)

=

g(d)

= b

and

g(c) =

c. It is clear that both J and 9 are 1- continuous. However

,

the composition function 9 0

J

is not

I-continuou

s because {c} E v

,

but

(g

0

J)

-I({C}) = {c}

rt

IO(X).

Theorem 3.8.

The Jollowing hold Jor the Junctions: J: (X ,

^T

, 1)

• (Y

, (7) and

9 : (Y

,

17

,

J) • (Z

,

μ)

(i)

9 0

J

is

J-continuous ,

iJ J is J-continuous and 9 is

continuous

(ii)

go J

is precoη tinuous ν J is M-P-coη tinuous

and

9 is 1 -continuous (iii)

JJ J

is surjectioη，

J-l(B)* c [J -l(B)] Jor*

each

B c Y and both J and

9 are

J-continuous , then

9 0

J

is

also J-continuous

Proof

(i) This is obvious.

(ii) Follows from the fact that each

I-open

set is preopen set.

(iii) Is c1ear by using Theorem 3.7.

4. I-open and I-closed functions

Definition 4.1. A function

J: (X , T)

•

(Y ,

17

,

J) is called I-open (resp I-closed) if for each U E ^T (resp. U is closed)

,

J(U) E IO(Y) (resp. J(U) is

J-closed).

Remark 4

^.1.(i) l-open

(I

-closed) function

=>

preopen (preclosed) function and the converse is not true in general (Example 4.1)

(ii) Each of l-open function and open function are independent (Ex- amples 4.2

,

4.3).

Example 4.1. Let X

=

^Y

= {a ,

b

, c}

with two topologies ^T

=

^{X

^,Iþ {a} ,

{a , b} , {a , c}} ,

¹⁷

= {Y ,Iþ, {a} , {a , b}}

and

J = {Iþ, {a} , {b} , {a , b}}

on Y^‘ Then the identity function

J :

(X

, T)

• (Y

,

17

, J)

is preopen but not J- open

,

because

, {a}

E T

,

but

J({a})

=

{a} rt

IO(Y).

Example 4.2. lf

X = {a , b ,

c

,

d}

= Y ,

T

= {X ,Iþ, {a , b} , {a , b , d}} ,

17

= ^{Y ^,.p, ^{a ^, ^b}, ^{a ^, ^b ^,

^c}}^and

^J = ^{Iþ, ^{c} ^, ^{d

^},

^{c ^, ^d}}

^on

^Y.

^{Then the}

(9)

On 1-open sets and l-continuous functions 29

identity function f: (X

,

^T)• (Y

,

¹⁷

,

J) is 1-open function but not open function.

Exar매 le 4.3. If X

=

Y

=

{ι b， c}

,

^T

=

{X

, 1>,

{a}}

,

¹⁷

=

{Y

, 1>,

{a}

,

{a

,

b}}

and J =

{1>,

{a}}

,

then the identity function

f :

(X

, T)

• (Y

,

¹⁷

,

J) is open but not 1-open because {a} E T

,

but f({a}) = {a} (j 10(Y).

Theorem 4.1. Let f: (X

, T)

• (Y

,

σ， J) be a function. Then the folloψ

ing are equivalent:

(i) f is 1 -open function

(ii) For each x E X and each neighborhood U of x

,

there exists an 1-open set W C Y containing f(x) such that W

c

f(U)

Proof Immediate.

Theorem 4.2. Let f: (X

, T)

• (Y

,

¹⁷

,

J) be an 1 -opeη (resp. 1 -closed)

funclioη， if W C Y and F

c

X is a closed (resp. ope띠 set containmg f-1 (W)

,

then there exists aη 1-closed(resp. 1-open) set H

c

^Y containiηg W such that f-1(H) C F.

Proof This is obvious.

Theorem 4.3. If f : (X

, T)

• (Y

,

¹⁷

,

J) is 1 -open

,

theη f-1(int(B

)t

C

(f

-1(B))* such that f-l(B) is *-dense-in-itselj, for every B C Y Proof Obvious by using Theorem (4.2)

Theorem 4.4. For any bijeclive function f (X

, T)

• (κ 17

, J)

the following are equivalent:

(i) f-1 : (Y

,

σ，

J)

• (X

, T)

is 1-contiηuous

(ii) f is 1-open.

(iii) f is 1 ^-c/osed.

Theorem 4.5. 1f f : (X

, T)

• (Y

,

σ，

J)

is 1 -open and for each A

c

X

,

f(A

‘) c [J

(A)]"

,

then the image of each 1-opeη 8et is J -open

Theorem 4.6. Let f : (X

,

^T

, 1)

• (Y

,

¹⁷

,

J) and 9 : (Y

,

¹⁷

,

J) • (Z

,

ι J() be two funclions

,

where 1

,

J and κ are ideals on X

,

Y and Z respeclively Then:

(i) 9 0 f is J-opeη， if f is open and 9 is J -open.

(ii) f is 1 -opeη if 9 0 f is opeη and 9 is 1 -contin uous iη:jective.

(“11패 l

each V

c

Y

,

then 9 0 f is 1-open.

Proof Obvious.

(10)

30 M.E. Abd EI-Monsef, E.F. Lashien and A.A. Nasef

References

[1] M.E. Abd 티 Mon따 5.N. EI-Deeb and R.A. Mahmoud, β-open sets and β

continuous mappings, Bull. Fac. 5ci, Assiut Univ., 12(1)(1983),77-90

[2] E. Hayashi, Topologies defìned by local properties, Math. Ann. 156(1964), 205-215 [3] D. Jankovic and T.R. Hamlett, New topologies from old via ideals, to appear in

Amer. Math. Monthly, 97 No .4(199이， 295-310

[4] D. Jankovic and T.R. Hamlctt, Compatible Extensions of ldeals (to appear in BolI U.M.I.)

[5] N. Levine

,

Semi-open sets and semi-continu씨 in topological spaces

,

Amer. Math Monthly, 70(1963), 36-41

[6] A.5. Mashhour, M.E. Abd EI-Monsef and 5.N. EI-Deeb, On preconlinuous and weak precontinuous mappings

,

Proc. Math. alld Phys. 50c. Egypt‘(53) (1982)

,

47- 53

[7] A.5. Mashhour, M.E. Abd EI-Monscf and I.A. Hasanein, 0" prelopological spaces, Bull. Math. 50c. R.5. Romania, 28(76)(19없)， 39-45

[8] O. Njastad, On some classes of nearly open sets, Pacific J. Math., 15(1965),961- 970

[9] P. Samuels

,

A topology formed from a given topology and an ideal

,

.J. London Math. 50c. (2)

,

16(1975)

,

409-416

[10] R. Vaidyanathaswamy, The localizalion theory in set-topolog.y, Proc. Indian Acad Sci., 20(1945),51-61

TANTA UNJVERS1TY

,

^EGYPT