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In many papers, new classes of sets had been studied in topological space, then the notion of continuity between any two topological spaces (a function from
*X* to
*Y* is continuous if the inverse image of each open set of
*Y* is open in
*X*) is studied via this new classes of sets. Here the authors also introduce new classes of sets called
*pj*-
*b*-preopen,
*pj*-
*b*-
*B* set,
*pj*-
*b*-
*t* set,
*pj*-
*b*-semi-open and
*pj*-
*sb*-generalized closed set in bitopological space [1] which is a set with two topologies defined on it, then they study the notion of continuity via this set and introduce some of the theories which are studying the decomposition of continuity via this set in bitopological space.

In topological space, there are many classes of generalized open sets given by [

Definition 1.1. Let

1) b-t-set [

2) b-B-set [

3) Locally b-closed [

4) b-preopen [

5) b-semiopen [

Definition 1.2. Let

Definition 1.3. A subset

1) pj-b-open [

2) pj-b-closed [

3) pj-semiopen [

4) pj-preopen [

5) pj-t-set [

6) pj-B-set [

7) jp-regular open [

In this section, we investigated our new classes of sets pj-b-preopen, pj-b- semiopen, pj-b-t set, pj-b-B set and pj-sb-generalized closed set and study some of its fundamental properties and examples also we introduce some of important theories which is useful to study the decomposition of continuity via our new classes of sets.

Definition 2.1. A subset

1) pj-b-t-set if

2) pj-b-B-set if

3) pj-b-semiopen if

4) pj-b-preopen if

Example 2.2. Let

Example 2.3. Let

Example 2.4. Let

Proposition 2.5. If

1)

2) If

3) If

proof. 1) Let

2) Let

3) Let

The following example shows that the converse of (2) is not true in general.

Example 2.6. From example 2.2 it is clear that

Lemma 2.7. Let

proof. Let

Proposition 2.8. Let

1) If

2) If

3) If

proof. 1) Let

2) Let

3) Let

Theorem 2.9. Let

1)

2)

proof. (1) Þ (2) Let

(2) Þ (1)

Hence,

Therefore

The following examples show that pj-b-preopen sets and pj-b-B-sets are independent.

Example 2.10. From example 2.3 it is clear that

Example 2.11. From example 2.4 it is clear that

Corollary 2.12. A subset

Proposition 2.13. Let

1)

2)

3)

proof. (1) Þ (2) Let

(2) Þ (3) This is obvious.

(3) Þ (1) Let

Definition 2.14. A subset

Definition 2.15. pj-

Theorem 2.16. Let

1)

2)

proof. (1) Þ (2) Let

(2) Þ (1) Let

Corollary 2.17. A subset

After we had been defined and studied the propriety of our new classes of sets we are ready to study the concept of continuity between any two bitopological spaces via our new classes of sets.

Definition 3.1. A function

Theorem 3.2. A function

proof. It is following from lemma 3.4 in [

Definition 3.3. Afunction

Theorem 3.4. A function

proof. It is follows from theorem 2.1.

Theorem 3.5. A function

proof. It is follows from corollary 2.1.

Definition 3.6. Afunction

Theorem 3.7. A function

proof. It is follows from proposition 2.3.

Theorem 3.8 A function

proof. It is follows from theorem 2.2.

Theorem 3.9 A function

proof. It is follows from corollary 2.2.

Al-Malki, H. and Al-Blowi, S. (2017) On Decomposition of New Kinds of Continuity in Bitopological Space. International Journal of Modern Non- linear Theory and Application, 6, 98-103. https://doi.org/10.4236/ijmnta.2017.63009