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In this paper, we introduce the concepts of g and b approximations via general ordered topological approximation spaces. Also, increasing (decreasing) g , b boundary, positive and negative regions are given in general ordered topological approximation spaces (GOTAS, for short). Some important properties of them were investigated. From this study, we can say that studying any properties of rough set concepts via GOTAS is a generalization of Pawlak approximation spaces and general approximation spaces.

Rough set theory was first proposed by Pawlak for dealing with vagueness and granularity in information systems. Various generalizations of Pawlak s rough set have been made by replacing equivalence relations with kinds of binary relations and many results about generalized rough set with the universe being finite were obtained [

In this section, we give an account of the basic definitions and preliminaries to be used in the paper.

Definition 2.1 [

Definition 2.2 [

Definition 2.3 [

Definition 2.4 [

Definition 2.5 [

(1)

(2)

(3)

(4)

(5)

Definition 2.6 [

(1)

(2)

(3)

(4)

A is R- increasing (resp. decreasing) semi exact if

Proposition 2.7 [

(1)

(2)

In this section, we introduce some definitions and propositions about near approximations, near boundary regions via GOTAS which is essential for a present study.

Definition 3.1. Let

(1)

(2)

(3)

(4)

A is R-increasing (resp. R-decreasing)

Proposition 3.2. Let

(1)

(2)

(3)

Proof.

(1) Omitted.

One can prove the case between parentheses.

Proposition 3.3. Let

(1)

(2)

(3)

Proof.

(1) Easy.

One can prove the case between parentheses.

Proposition 3.4. Let

Proof.

Let A be R-increasing exact. Then

One can prove the case between parentheses.

R-increasing (resp. decreasing) exact

Proposition 3.5. Let

Proof.

Since

fore,

One can prove the case between parentheses.

Proposition 3.6. Let

Proof. Since

Therefore

Proposition 3.7. Let

Proof. Let

Thus

One can prove the case between parentheses.

Proposition 3.8. Let

Proof.

Let

Thus

One can prove the case between parentheses.

Proposition 3.9. Let

Proof.

Let

Thus

Proposition 3.10. Let

Proof. Omitted.

Definition 3.11. Let

(1)

(2)

(3)

(4)

A is R-increasing (decreasing)

Proposition 3.12. Let

(1)

(2)

(3)

Proof.

(1) Omitted.

One can prove the case between parentheses.

Proposition 3.13. Let

(1)

(2)

(3)

Proof.

(1) Easy.

One can prove the case between parentheses.

Proposition 3.14. Let

Proof.

Let A be R-increasing exact. Then

One can prove the case between parentheses.

Proposition 3.15. Let

Proof.

Since

Therefore

One can prove the case between parentheses.

Proposition 3.16. Let

Proof. Since

Therefore

Definition 3.17. Let

(1)

(2)

(3)

Proposition 3.18. Let

(1)

(2)

Proof.

One can prove the case between parentheses.

Proposition 3.19. Let

(1)

(2)

Proof.

One can prove the case between parentheses.

Proposition 3.20. Let

Proof.

Let

and thus

Hence

Since

Thus

From (1) and (2) we have,

One can prove the case between parentheses.

Proposition 3.21. Let

Proof.

Let

Thus

Since

From (1) and (2) we have,

One can prove the case between parentheses.

Definition 3.22. Let

Proposition 3.23. Let

Proof. Omitted.

In the following example we illustrate most of the properties that have been proved in the previous propositions.

Example 3.24. Let

For

Proposition 3.25. Let

Proof. Omitted.

Remark 3.26.

Remark 3.27.

Proposition 3.28. Let

Proof. Omitted.

Proposition 3.28. Let

Proof. Let

Therefore

thus

One can prove the case between parentheses.

In this paper, we generalize rough set theory in the framework of topological spaces. Our results in this paper became the results about of

Mohamed Abo-Elhamayel, (2016) γ and β Approximations via General Ordered Topological Spaces. Applied Mathematics,07,1580-1588. doi: 10.4236/am.2016.714136