_{1}

^{*}

The aim of this work is to introduce some weak forms of continuity in bitopological spaces. Then we use these new forms of weak continuity to give many decompositions of
*i*-continuity and pairwise continuity.

The concept of bitopological spaces has been introduced by Kelly [

Throughout this paper

We recall some known definitions

Definition 1 ([

Definition 2 ([

Definition 3 ([

Definition 4 ([

Definition 5 ([

Definition 6 ([

Definition 7 ([

The relations of the above weak forms of continuity are as follows:

[Diagram 1]

Lemma 1 For a subset

1)

2)

3)

4)

Proof (1) and (2) are obvious. (3) Since

Proposition 1 Let

1)

2)

3) f is

It is known [

Definition 8 Let

1)

2)

It is well known [

Definition 9 Let

1)

2) f is ij-α-almost continuous if and only if

mopen set V in Y.

Lemma 2 A function

Proof Let

Since

Conversely, assume that

In view of the above lemma we define the following:

Definition 10 Let

1)

2)

Definition 11 Let

1)

in Y;

2)

3)

The following diagram gives the relations between all the weak forms of continuity

[Diagram 2]

Proof (Proof of some relations in Diagram 2).

1)

Let

for each

2)

Let

Since

Hence, f is ij-pre weak continuous.

We could use similar ways to prove other relations in Diagram 2.

Let

Definition 12 Let

1)

2) f is ij-pre

3) f is

Proposition 2 Any function f is an i^{#}-continuous function.

Proof Let

Definition 13 Let

1) f is

2) f is ij-pre weak

3)

Definition 14 Let

1) f is ^{#}continuous if and only if

2)

3)

Definition 15 Let

1)

2)

3)

Lemma 3 Let

Proof Let

can write

Conversely, assume that

Lemma 4 If

Proof The proof follows immediately from

The following diagram gives the relations between all the weak forms of continuity

[Diagram 3]

Proof (Proof of some relations in Diagram 2).

1)

Let ^{#} continuous. Then

subspace

obtain

2)

Let

^{#} continuous.

3) ^{#}continuity ^{#} continuity;

Let

subspace

and Lemma 4.7, we obtain

^{#} continuous.

4) ^{#} continuity ^{#} continuity;

Let ^{#} continuous function. Then

in the subspace

5) ^{#}continuity ^{#}continuity;

Let

We could also use the similar ways to prove other relations in Diagram 3.

The following examples show that the reverse implications of Diagram 3 is not true.

Example 1 Let

Example 2 Let

Then the map f is 12-pre weak^{#} continuous but not 12-pre weak semi^{#} continuous, because

which is not 1-open set in the subspace

Example 3 Let

and

^{#} continuous but not 21-pre almost semi^{#} continuous because

For a property

In this section we will give eight decompositions of

Lemma 5 Let

mapping with

1)

2) There is an

Then

Proof Since

proved that

Now we turn to the decomposition of

Theorem 1 Let

1)

2)

3)

4)

5)

6)

7)

8)

Proof

1) Since f is ij-pre continuous,^{#}-continuous,

2) Since

3) Since

4) Since

continuous,

5) Since

weak^{#}-continuous,

6) Since

continuous, where

7) Since

5.1,

8) Since

By Lemma 5.1,

Corollary 1 Let

1)

2)

3)

4)

5)

6)

7)

8)

Proof The proof follows immediately from Theorem 5.3.

This work was funded by the Deanship of Scientific Research (DSR), king Abdulaziz University, Jeddah, under grat No. (363-006-D1433). The author, therefore, acknowledge with thanks DSR technical and financial support.