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In this paper, we introduce the notion of intuitionistic fuzzy α -generalized closed sets in intuitionistic fuzzy minimal structure spaces and investigate some of their properties. Further, we introduce and study the concept of intuitionistic fuzzy α -generalized minimal continuous functions.

The concept of fuzzy sets was introduced by Zadeh [

Throughout this paper, by (X, τ) or simply by X we will denote the Coker’s intuitionistic fuzzy topological space (briefly, IFTS). For a subset A of a space (X, τ), cl(A), int(A) and

We introduce some basic notions and results that are used in the sequel.

Definition 2.1. [

Definition 2.2. [

Definition 2.3. [

Obviously, every fuzzy set A on a nonempty set X is an IFS of the following form

Definition 2.4. [

1)

2)

3)

4)

Definition 2.5. [

Definition 2.6. [

Definition 2.7. [

Definition 2.8. [

Definition 2.9. [

Definition 2.10. [

Definition 2.11. [

Definition 2.12. [

Definition 2.13. [

In this section the concept of IF α-generalized minimal open set is introduced and some of its properties are discussed. Lastly the IF topological structure obtained by the collection of this set is studied.

Definition 3.1. An IF set A is said to be an IF α-generalized minimal closed set, if there exist at least one IF Minimal Open Set U containing A such that

Example 3.2. Let

Theorem 3.3.

1) Let

2) If

Proof. a) Let

b) Since B is an IF α-generalized minimal closed set i.e.

Theorem 3.4. An IF set A is IF α-generalized minimal closed and IF α-minimal open set then A is an IF closed set. Conversely if A be an IF α-closed set and an IF minimal open set then A is an IF α-generalized minimal closed set.

Proof. Let if possible A be an IF α-generalized minimal closed set i.e. there exist an IF minimal open set U containing A such that

Conversely, Let A be an IF α- closed set and an IF minimal open set then from definition it is an IF α-generalized minimal closed set.

Theorem 3.5. Every IF α-generalized minimal closed set is either IF rare set or an IF minimal open set i.e. A is an IF rare set or the IF minimal open set containing A is an IF closed set i.e. A is an IF closed set.

Proof. Let if possible A be an IF α-generalized minimal closed set then there exist an IF minimal open set U containing A such that αcl(A) is contained in U. From Theorem 3.3., Let

Converse of the above theorem need not be true which follows from the following example.

Example 3.6. Let

Theorem 3.7. Every IF α-generalized minimal closed set is an IF α-generalized closed set.

Proof. Let A be an IF α-generalized minimal closed set then there exist an IF minimal open set U such that

Converse of the above theorem need not be true which follows from the following example.

Example 3.8. Let

Theorem 3.9. Let A be any IF α-generalized minimal closed set then

Proof. Let A be an IF α-generalized minimal closed set then

Theorem 3.10. Let A be any IF generalized minimal closed set then

Proof. Since A is an If α-generalized minimal closed set,

Theorem 3.11.

(i)

(ii) Arbitrary union of IF α-generalized minimal closed set is an IF α-generalized minimal closed set.

(iii) Arbitrary intersection of IF α-generalized minimal closed set is an IF α-generalized minimal closed set.

Proof. (i) is obvious.

To prove (ii) Let

To Prove (iii) Let

Definition 3.12. Let

Theorem 3.13. Let

Proof. It is obvious from Theorem 3.7.

Remark 3.14. Converse of the above theorem need not be true which follows from the following example:

Let

Theorem 3.15. Let

Proof. Here

In this section the concept of IF α-generalized* minimal open set is introduced and some theorems related to this newly constructed set are studied and also related properties are discussed.

Definition 4.1. An IF set B is said to be an IF α-generalized* minimal closed set, if there exist at least one IF Minimal Open Set A containing B such that

Example 4.2. Let

Theorem 4.3.

(i) Let

(ii) If

Proof. (i) Let

(ii) Since B is an IF α-generalized* minimal closed set i.e.

Remark 4.4. There does not exist any IF Minimal Open Set between A and B such that

Theorem 4.5 If A is an IF α-generalized* minimal open set then

Proof. As

Remark 4.6. The converse of the above theorem may not be true and it can be shown with the help of an example:

Let

Theorem 4.7. Every IF Minimal Open Set is an IF α-generalized* minimal open set in itself.

Proof. Let A is an IF Minimal Open Set. We know that

Remark 4.8. The converse of the above theorem need not be true, as IF Set C in example 4.2 is IF α-generalized* minimal open set but it is not an IF Minimal Open Set. According to the theorem 4.5 the converse is true if the set is not a rare set. i.e. for a set which is not rare, IF minimal open set and IF α-generalized* minimal open set are similar concepts.

Theorem 4.9. If

Proof. Let B is an IF Open Set which is IF α-generalized* minimal open set. From definition there exist a IF minimal open set A containing B such that

Conversely, let B is an IF Minimal Open Set, then as proved in theorem 4.7, B is an IF α-generalized* minimal open set.

Theorem 4.10. Every IF-dense set is an IF α-generalized* minimal open set if it is a subset of some IF Minimal Open Set but the converse is not true.

Proof. Let A is an IFτ dense set

The converse is not true as shown in example 4.6. C is an IF α-generalized* minimal open set, but C is not an IFτ dense set as

Theorem 4.11. An IF α-generalized* minimal open set A is IF α-generalized closed set if and only if

Proof. Since A is an IF α-generalized* minimal open set,

But A is IF α-generalized closed set which implies

So from (1) and (2)

Conversely let

Theorem 4.12. If the IF minimal open set containing a α-generalized* minimal closed set is IF closed set then the α-generalized* minimal closed set is a IF Pre-open set.

Proof. Let U be a IF minimal open set containing A. Since A is a α-generalized* minimal closed set

Theorem 4.13. Let A be an closed set and an IF α-generalized* minimal closed set then A is the minimal open set.

Proof. Let U be a IF minimal open set containing A. Since A is an IF α-generalized* minimal closed set,

Theorem 4.14. Arbitrary union of IF α-generalized* minimal open set is an IF α-generalized* minimal open set if it is contained in an If minimal open set.

Proof. Let _{i} is an IF α-generalized* minimal open set}

Remark 4.15. The collection of all IF α-generalized* minimal open set forms an IF supra topological space if

Theorem 4.16. An IF set A of X is both IF α-generalized minimal closed set and IF α-generalized* minimal closed set iff

Theorem 4.17. The union of an IF α-generalized minimal closed set and an IF α-generalized* minimal closed set is an IF α-generalized* minimal closed set.

Proof. Let A be an IF α-generalized minimal closed set and B be an IF α-generalized* minimal closed set in the same IF topological space. Let

Mani Parimala,Sivaraman Murali, (2016) Intuitionistic Fuzzy α-Generalized Closed Sets in Terms of Minimal Structure Spaces. Circuits and Systems,07,1486-1491. doi: 10.4236/cs.2016.78130