# On a class of [[alpha].sub.[gamma]]-open sets in a topological space/Uma classe de conjuntos [[alpha].sub.[gamma]]- aberto em um espaco topologico.

IntroductionNjastad (1965) introduced [alpha]-open sets. Kasahara (1979) defined the concept of an operation on topological spaces and introduce the concept of [alpha]-closed graphs of an operation. Ogata (1991) called the operation [alpha] (respectively [alpha]-closed set) as [gamma]-operation (respectively [gamma]-closed set) and introduced the notion of [[tau].sub.[gamma]] which is the collection of all [gamma]-open sets in a topological space. Also he introduced the concept of [gamma]-[T.sub.i] (i = 0, 1/2, 1, 2) and characterized [gamma]-[T.sub.i] using the notion of [gamma]-closed and [gamma]-open sets. In this paper, we introduce the concept of [[alpha].sub.[gamma]]-open sets by using an operation [gamma] on [alpha]O(X, [tau]) and we introduce the concept of [[alpha].sub.[gamma]]-generalized closed sets and [[alpha].sub.[gamma]]-[T.sub.1/2] spaces and characterize [[alpha].sub.[gamma]]-[T.sub.1/2] spaces using the notion of [[alpha].sub.[gamma]]-closed or [[alpha].sub.[gamma]]-open sets. Also, we show that some basic properties of [[alpha].sub.[gamma]][T.sub.i], [[alpha].sub.[gamma]][D.sub.i] for i = 0, 1, 2 spaces and we introduce [[alpha].sub.([gamma],[gamma]')]-continuous mappings and study some of its properties. Let (X, [tau]) be a topological space and A be a subset of X. The closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively. A subset A of a topological space (X, [tau]) is said to be [alpha]-open (NJASTAD, 1965) if A [subset or equal to] Int(Cl(Int(A))). The complement of an [alpha]-open set is said to be [alpha]-closed. The intersection of all [alpha]-closed sets containing A is called the [alpha]-closure of A and is denoted by [alpha]Cl(A).

The family of all [alpha]-open (resp. [alpha]-closed) sets in a topological space (X, [tau]) is denoted by [alpha]O(X, [tau]) (resp. [alpha]C(X, [tau])). An operation [gamma] on a topology [tau] is a mapping from t in to power set P(X) of X such that V [subset or equal to] [gamma](V) for each V [member of] [tau], where [gamma](V) denotes the value of [gamma] at V. A subset A of X with an operation [gamma] on [tau] is called [gamma]-open if for each x [member of] A, there exists an open set U such that x [member of] U and [gamma](U) [subset or equal to] A. Clearly [[tau].sub.[gamma]] [subset or equal to] [tau]. Complements of [gamma]-open sets are called [gamma]-closed. The [gamma]-closure of a subset A of X with an operation [gamma] on [tau] is denoted by [[tau].sub.[gamma]]-C1(A) and is defined to be the intersection of all [gamma]-closed sets containing A. A topological X with an operation [gamma] on [tau] is said to be [gamma]-regular if for each x [member of] X and for each open neighborhood V of x, there exists an open neighborhood U of x such that [gamma](U) contained in V. It is also to be noted that [tau] = [[tau].sub.[gamma]] if and only if X is a [gamma]-regular space (OGATA, 1991).

[[alpha].sub.[gamma]]-open sets

Definition 2.1. Let [gamma]: [alpha]O(X, [tau]) [right arrow] P(X) be a mapping satisfying thefollowing property, V [subset or equal to] [gamma](V) for each V [member of] [alpha]O(X, [tau]). We call the mapping y an operation on [alpha]O(X, [tau]).

Definition 2.2. Let (X, [tau]) be a topological space and [gamma]: [alpha]O(X, [tau]) [right arrow] P(X) an operation on [alpha]O(X, [tau]). A nonempty set A of X is called an [[alpha].sub.[gamma]]-open set of (X, [tau]) if for each point x [member of] A, there exists an [alpha]-open set U containing x such that [gamma](U) [subset or equal to] A. The complement of an [[alpha].sub.[gamma]]-open set is called [[alpha].sub.[gamma]]-closed in (X, [tau]). We suppose that the empty set is [[alpha].sub.[gamma]]-open for any operation [gamma]: [alpha]O(X, [tau]) [right arrow] P(X). We denote the set of all [[alpha].sub.[gamma]]-open (resp. [[alpha].sub.[gamma]]-closed) sets of (X, [tau]) by [alpha]O[(X, [tau]).sub.[gamma]][(resp. [alpha]C(X, [tau]).sub.[gamma]]).

Remark 2.3. A subset A is an [[alpha].sub.id]-open set of (X, [tau]) if and only if A is [alpha]-open in (X, [tau]). The operation id: [alpha]O(X, [tau]) [right arrow] P (X) is defined by id(V) = V for any set V [member of] [alpha]O(X, [tau]), this operation is called the identity operation on [alpha]O(X, [tau]). Therefore, we have that [alpha]O[(X, [tau]).sub.id] = [alpha]O(X, [tau]).

Remark 2.4. The concept of [[alpha].sub.[gamma]]-open and open are independent.

Example 2.5. Consider X = {a, b, c} with the topology [tau] = {[phi], {a}, {a, b}, X} and [alpha]O(X, [tau]) = {[phi], {a}, {a, b}, {a, c}, X}. Define an operation [gamma] on [alpha]O(X, [tau]) by [gamma](A) = A if A = {a, c} or A = [phi] and [gamma](A) = X otherwise. Then [[alpha].sub.[gamma]]-open sets are [phi], {a, c} and X.

Remark 2.6. It is clear from the definition that every [[alpha].sub.[gamma]]-open subset of a space X is [alpha]-open, but the converse need not be true in general as shown in the following example.

Example 2.7. Consider X = {a, b, c} with the topology [tau] = {[phi], {a}, X} and [alpha]O(X, [tau]) = {[phi], {a}, {a, b}, {a, c}, X}. Define an operation [gamma] on [alpha]O(X, [tau]) by [gamma](A) = A if b [member of] A and [gamma](A) = X if b [not member of] A. Then [alpha]O[(X, [tau]).sub.[gamma]] = {[phi], {a, b}, X} and {a} [member of] [alpha]O(X, [tau]), but {a} [not member of] [alpha]O[(X, [tau]).sub.[gamma]].

Theorem 2.8. If A is a [gamma]-open set in (X, [tau]), then A is an [[alpha].sub.[gamma]]-open set.

Proof. Follows from that every open set is [alpha]-open.

The converse of the above theorem need not be true in general as it is shown below.

Example 2.9. Consider X = {a, b, c} with the topology [tau] = {[phi], {a}, X}. Define an operation y on [alpha]O(X, [tau]) by [gamma](A) = A. Then {a, b} is an [[alpha].sub.[gamma]]-open set but not a [gamma]-open set.

The proof of the following result is easy and hence it is omitted.

Proposition 2.10. If (X, [tau]) is [gamma]-regular space, then every open set is [[alpha].sub.[gamma]]-open.

Theorem 2.11. Let [{A[alpha]}.sub.[alpha][member of]j] be a collection of [[alpha].sub.[gamma]]-open sets in a topological space (X, [tau]), then [U.sub.[alpha][member of]J] [A.sub.[alpha]] is [[alpha].sub.[gamma]]-open.

Proof. Let x [member of] [U.sub.[alpha][member of]J] [A.sub.[alpha]], then x [member of] [A.sub.[alpha]] for some [alpha] [member of] J. Since [A.sub.[alpha]] is an [[alpha].sub.[gamma]]-open set, implies that there exists an [alpha]-open set U containing x such that [gamma](U) [subset or equal to] A[alpha] [subset or equal to] [U.sub.[alpha][member of]J][A.sub.[alpha]]. Therefore [U.sub.[alpha][member of]J][A.sub.[alpha]] is an [[alpha].sub.[gamma]]-open set of (X, [tau]).

If A and B are two [[alpha].sub.[gamma]]-open sets in (X, [tau]), then the following example shows that A [intersection] B need not be [[alpha].sub.[gamma]]-open.

Example 2.12. Consider X = {a, b, c} with the discrete topology on X. Define an operation y on [alpha]O(X, [tau]) by [gamma](A) = {a, b} if A = {a} or {b} and [gamma](A) = A otherwise. Then A = {a, b} and B = {a, c} are [[alpha].sub.[gamma]]-open sets but A [intersection] B = {a} is not an [[alpha].sub.[gamma]]open set.

From the above example we notice that the family of all [[alpha].sub.[gamma]]-open subsets of a space X is a supratopology and need not be a topology in general.

Proposition 2.13. The set A is [[alpha].sub.[gamma]]-open in the space (X, [tau]) if and only if for each x [member of] A, there exists an [[alpha].sub.[gamma]]-open set B such that x [member of] B [subset or equal to] A.

Proof. Suppose that A is [[alpha].sub.[gamma]]-open set in the space (X, [tau]). Then for each x [member of] A, put B = A is an [[alpha].sub.[gamma]]-open set such that x [member of] B [subset or equal to] A.

Conversely, suppose that for each x [member of] A, there exists an [[alpha].sub.[gamma]]-open set B such that x [member of] B [subset or equal to] A, thus A = [union][B.sub.x] where [B.sub.x] [member of] [alpha]O(X, [tau]) [gamma] for each x. Therefore, A is an [[alpha].sub.[gamma]]-open set.

Definition 2.14. An operation [gamma] on [alpha]O(X, [tau]) is said to be [alpha]-regular if for every [alpha]-open sets U and V of each x [member of] X, there exists an [alpha]-open set W of x such that [gamma](W) [subset or equal to] [gamma](U) [intersection] [gamma](V).

Definition 2.15. An operation [gamma] on [alpha]O(X, [tau]) is said to be [alpha]-open if for every [alpha]-open set U of each x [member of] X, there exists an [[alpha].sub.[gamma]]-open set V such that x [member of] V and V [subset or equal to] y(U).

In the following two examples, we show that [alpha]-regular operation is incomparable with the [alpha]-open operation.

Example 2.16. Consider X = {a, b, c} with the topology [tau] = {[phi], {a}, {a, b}, {a, c}, X}. Define an operation [gamma] on [alpha]O(X, [tau]) by [gamma](A) = {a, b} if A = {a} and [gamma](A) = X if A [not equal to] {a}. Then [gamma] is [alpha]-regular but not [alpha]-open.

Example 2.17. Consider X = {a, b, c} with the topology [tau] = {[phi], {a}, {a, b}, {a, c}, X}. Define an operation [gamma] on [alpha]O(X, [tau]) by [gamma](A) = A if A = {a, b} or {a, c} and [gamma](A) = X otherwise. Then [gamma] is not [alpha]-regular but [gamma] is [alpha]-open.

In the following proposition the intersection of two [[alpha].sub.[gamma]]-open sets is also an [[alpha].sub.[gamma]]-open set.

Proposition 2.18. Let [gamma] be an [alpha]-regular operation on [alpha]O(X, [tau]). If A and B are [[alpha].sub.[gamma]]-open sets in X, then A [intersection] B is also an [[alpha].sub.[gamma]]-open set.

Proof. Let x [member of] A [intersection] B, then x [member of] A and x [member of] B. Since A and B are [[alpha].sub.[gamma]]-open sets, there exist [alpha]-open sets U and V such that x [member of] U and [gamma](U) [subset or equal to] A, x [member of] V and [gamma](V) [subset or equal to] B. Since [gamma] is an [alpha]-regular operation, then there exists an [alpha]-open set W of x such that [gamma](W) [subset or equal to] [gamma](U) [intersection] [gamma](V) [subset or equal to] A [intersection] B. This implies that A [intersection] B is [[alpha].sub.[gamma]] open set.

Remark 2.19. By the above proposition, if [gamma] is an [alpha]-regular operation on [alpha]O(X, [tau]). Then [alpha]O[(X, [tau]).sub.[gamma]] form a topology on X.

Definition 2.20. A point x [member of] X is in [alpha][Cl.sub.[gamma]]-closure of a set A [subset or equal to] X, if [gamma](U) [intersection] A [not equal to] [phi] for each [alpha]-open set U containing x. The [alpha][Cl.sub.[gamma]]-closure of A is denoted by [alpha][Cl.sub.[gamma]] (A).

Definition 2.21. Let A be a subset of (X, [tau]), and [gamma]: [alpha]O(X, [tau]) [right arrow] P (X) be an operation on [alpha]O(X, [tau]). Then the [[alpha].sub.[gamma]]-closure of A is denoted by [[alpha].sub.[gamma]]Cl(A) and defined as follows, [[alpha].sub.[gamma]]C1(A) = [intersection] {F: F is [[alpha].sub.[gamma]]-closed and A [subset or equal to] F}.

The proof of the following theorem is obvious and hence omitted.

Theorem 2.22. Let (X, [tau]) be a topological space and [gamma] be an operation on [alpha]O(X, [tau]). For any subsets A, B of X, we have the following properties:

(1) A [subset or equal to] [[alpha].sub.[gamma]]Cl(A).

(2) [[alpha].sub.[gamma]]C1(A) is [[alpha].sub.[gamma]]-closed set in X.

(3) A is [[alpha].sub.[gamma]]-closed set if and only if A = [[alpha].sub.[gamma]]C1(A).

(4) [[alpha].sub.[gamma]]C1([phi]) = [phi] and [[alpha].sub.[gamma]]C1(X) = X.

(5) If A [subset or equal to] B, then [[alpha].sub.[gamma]]Cl(A) [subset or equal to] [[alpha].sub.[gamma]]Cl(B).

(6) [[alpha].sub.[gamma]]Cl(A [union] B) [contains or equal to] [[alpha].sub.[gamma]]C1(A) [union] [[alpha].sub.[gamma]]Cl(B).

(7) [[alpha].sub.[gamma]]Cl(A [intersection] B) [subset or equal to] [[alpha].sub.[gamma]]Cl(A) [intersection] [[alpha].sub.[gamma]]C1(B).

Theorem 2.23. For a point x [member of] X, x [member of] [[alpha].sub.[gamma]]Cl(A) if and only if for every [[alpha].sub.[gamma]]-open set V of X containing x such that A [intersection] V [not equal to] [phi].

Proof. Let x [member of] [[alpha].sub.[gamma]]Cl(A) and suppose that V [intersection] A = [phi] for some [[alpha].sub.[gamma]]-open set V which contains x. Then (X\V) is [[alpha].sub.[gamma]]-closed and A [subset or equal to] (X\V), thus [[alpha].sub.[gamma]]Cl(A) [subset or equal to] (X\V). But this implies that x [member of] (X\V), a contradiction. Therefore V [intersection] A [not equal to] [phi].

Conversely, Let A [subset or equal to] X and x [member of] X such that for each [[alpha].sub.[gamma]]-open set U which contains x, U [intersection] A [not equal to] [phi]. If x [not equal to] [[alpha].sub.[gamma]] C1(A), there is an [[alpha].sub.[gamma]]-closed set F such that A [subset or equal to] F and x [not equal to] F. Then (X\F) is an [[alpha].sub.[gamma]]-open set with x [member of] (X\F), and thus (X\F) [intersection] A [not equal to] [phi], which is a contradiction.

The proof of the following theorems are obvious and hence omitted.

Theorem 2.24. Let A be any subset of a topological space (X, [tau]) and [gamma] be an operation on [alpha]O(X, [tau]). Then the following relation holds.

A [subset or equal to] [alpha]Cl(A) [subset or equal to] [alpha][Cl.sub.[gamma]](A) [subset or equal to] [[alpha].sub.[gamma]]Cl(A) [subset or equal to] [[tau].sub.[gamma]]-Cl(A).

Theorem 2.25. Let A be a subset of a topological space (X, [tau]) and [gamma] be an operation on [alpha]O(X, [tau]). Then, the following conditions are equivalent:

(1) A is [[alpha].sub.[gamma]]-open.

(2) [alpha][Cl.sub.[gamma]] (X\A) = X\A.

(3) [[alpha].sub.[gamma]]Cl(X\A) = X\A.

(4) X\A is [[alpha].sub.[gamma]]-closed.

Theorem 2.26. Let [gamma]: [alpha]O(X, [tau]) [right arrow] P (X) be an operation on [alpha]O(X, [tau]) and A be a subset of X, then:

(1) A subset [alpha][Cl.sub.[gamma]](A) is an [alpha]-closed set in (X, [tau]).

(2) If [gamma] is [alpha]-open, then [alpha][Cl.sub.[gamma]](A) = [[alpha].sub.[gamma]]C1(A), and [alpha][Cl.sub.[gamma]] ([alpha][Cl.sub.[gamma]](A)) = [alpha][Cl.sub.[gamma]](A), and [alpha][Cl.sub.[gamma]](A) is [[alpha].sub.[gamma]]-closed.

Proof. To prove that [alpha][Cl.sub.[gamma]](A) is [alpha]-closed. Let x [member of] [alpha]Cl([alpha][Cl.sub.[gamma]](A)). Then U [intersection] [alpha][Cl.sub.[gamma]](A) [not equal to] [phi] for every [alpha]-open set U of x. Let y [member of] U [intersection] [alpha][Cl.sub.[gamma]](A), [gamma] [member of] U and y [member of] [alpha][Cl.sub.[gamma]](A). Since U is [alpha]-open set containing y, implies [gamma](U) [intersection] A [not equal to] [phi]. Therefore x [member of] [alpha][Cl.sub.[gamma]](A). Hence [alpha]Cl([alpha][Cl.sub.[gamma]](A)) [subset or equal to] [alpha][Cl.sub.[gamma]](A). This implies [alpha][Cl.sub.[gamma]](A) is an [alpha]-closed set.

(2) By Theorem 2.24, we have [alpha][Cl.sub.[gamma]](A) [subset or equal to] a[Cl.sub.[gamma]]C1(A). Now to prove that a[Cl.sub.[gamma]]C1(A) [subset or equal to] [alpha][Cl.sub.[gamma]](A). Let x [not member of] [alpha][Cl.sub.[gamma]](A), then there exists an [alpha]-open set U such that [gamma](U) [intersection] A = [phi]. Since [gamma] is [alpha]-open, there exists an [[alpha].sub.[gamma]]-open set V such that x [member of] V [subset or equal to] [gamma] (U). Therefore V [intersection] A = [phi]. This implies x [not member of] [[alpha].sub.[gamma]]Cl(A). Hence [[alpha.sub.][gamma]]C1(A) [subset or equal to] [alpha][Cl.sub.[gamma]](A). Therefore [alpha][Cl.sub.[gamma]](A) = [[alpha].sub.[gamma]]C1(A). Now, [alpha][Cl.sub.[gamma]]([alpha][Cl.sub.[gamma]](A)) = [[alpha].sub.[gamma]]Cl([[alpha].sub.[gamma]]Cl(A)) = [[alpha].sub.[gamma]]Cl(A) = [alpha][Cl.sub.[gamma]](A).

Definition 2.27. A subset A of the space (X, [tau]) is said to be [[alpha].sub.[gamma]]-generalized closed (Briefly. [[alpha].sub.[gamma]]-g.closed) if [[alpha].sub.[gamma]]C1(A) [subset or equal to] U whenever A [subset or equal to] U and U is an [[alpha].sub.[gamma]]-open set in (X, [tau]). The complement of an [[alpha].sub.[gamma]]g.closed set is called an [[alpha].sub.[gamma]]-g.open set.

It is clear that every [[alpha].sub.[gamma]]-closed subset of X is also an [[alpha].sub.[gamma]]-g.closed set. The following example shows that an [[alpha].sub.[gamma]]-g.closed set need not be [[alpha].sub.[gamma]]-closed.

Example 2.28. Consider X = {a, b, c} with the topology [tau] = {[phi], {a}, {b}, {a, b}, {a, c}, X}. Define an operation [gamma] on [alpha]O(X, [tau]) by [gamma](A) = A if A = {b} or {a, c} or [phi] and [gamma](A)=X otherwise. Now, if we let A = {a}, since the only [[alpha].sub.[gamma]]-open supersets of A are {a, c} and X, then A is [[alpha].sub.[gamma]]-g.closed. But it is easy to see that A is not [[alpha].sub.[gamma]]-closed.

Theorem 2.29. A subset A of (X, [tau]) is [[alpha].sub.[gamma]]-g.closed if and only if [[alpha].sub.[gamma]]Cl({x}) [intersection] A [not equal to] [phi], holds for every x [member of] [[alpha].sub.[gamma]]Cl(A).

Proof. Let U be an [[alpha].sub.[gamma]]-open set such that A [subset or equal to] U and let x [member of] [[alpha].sub.[gamma]]Cl(A). By assumption, there exists a z [member of] [[alpha].sub.[gamma]]Cl({x}) and z [member of] A [subset or equal to] U. It follows from Theorem 2.23, that U [intersection] {x} [not equal to] [phi], hence x [member of] U, this implies [[alpha].sub.[gamma]]Cl(A) [subset or equal to] U. Therefore A is [[alpha].sub.[gamma]]-g.closed.

Conversely, suppose that x [member of] [[alpha].sub.[gamma]]Cl(A) such that [[alpha].sub.[gamma]]Cl({x}) [intersection] A = [phi]. Since, [[alpha].sub.[gamma]]Cl({x}) is [[alpha].sub.[gamma]]- closed, therefore X\[[alpha].sub.[gamma]]Cl({x}) is an [[alpha].sub.[gamma]]-open set in X. Since A [subset or equal to] X\([[alpha].sub.[gamma]]Cl({x})) and A is [[alpha].sub.[gamma]]-g.closed implies that [[alpha].sub.[gamma]]Cl(A) [subset or equal to] X\[[alpha].sub.[gamma]]Cl({x}) holds, and hence x [not member of] [[alpha].sub.[gamma]]Cl(A). This is a contradiction. Therefore [[alpha].sub.[gamma]]Cl({x}) [intersection] A [not equal to] [phi].

Theorem 2.30. A set A of a space X is [[alpha].sub.[gamma]]-g.closed if and only if [[alpha].sub.[gamma]]Cl(A)\A does not contain any nonempty [[alpha].sub.[gamma]]-closed set.

Proof. Necessity. Suppose that A is [[alpha].sub.[gamma]]-g.closed set in X. We prove the result by contradiction. Let F be an [[alpha].sub.[gamma]]-closed set such that F [subset or equal to] [[alpha].sub.[gamma]]Cl(A)\A and F [not equal to] [phi]. Then F [subset or equal to] X\A which implies A [subset or equal to] X\F. Since A is [[alpha].sub.[gamma]]-g.closed and X\F is [[alpha].sub.[gamma]]-open, therefore [[alpha].sub.[gamma]]Cl(A) [subset or equal to] X\F, that is F [subset or equal to] X\[[alpha].sub.[gamma]]Cl(A). Hence F [subset or equal to] [[alpha].sub.[gamma]]Cl(A) 0 (X\[[alpha].sub.[gamma]]Cl(A)) = [phi]. This shows that, F = [phi] which is a contradiction. Hence [[alpha].sub.[gamma]]Cl(A)\A does not contains any non-empty [[alpha].sub.[gamma]]-closed set in

Sufficiency. Let A [subset or equal to] U, where U is [[alpha].sub.[gamma]]-open in (X, [tau]). If [[alpha].sub.[gamma]]Cl(A) is not contained in U, then [[alpha].sub.[gamma]]Cl(A) [intersection] X\U [not equal to] [phi]. Now, since [[alpha].sub.[gamma]]Cl(A) [intersection] X\U [subset or equal to] [[alpha].sub.[gamma]]Cl(A)\A and [[alpha].sub.[gamma]]Cl(A) [intersection] X\U is a non-empty [[alpha].sub.[gamma]]closed set, then we obtain a contradication and therefore A is [[alpha].sub.[gamma]]-g.closed.

Corollary 2.31. If a subset A of X is [[alpha].sub.[gamma]]-g.closed set in X, then [[alpha].sub.[gamma]]Cl(A)\A dose not contain any non-empty [gamma]-closed set in X.

Proof. Proof follows from the Theorem 2.8.

The converse of the above corollary is not true in general as it is shown in the following example.

Example 2.32. Consider X = {a, b, c} with the topology [tau] = {[phi], {c}, X}. Define an operation y on [alpha]O(X, [tau]) by [gamma](A) = A. If we let A = {a, c} then A is not [[alpha].sub.[gamma]]-g.closed, since A [subset or equal to] {a, c} [member of] [alpha]O[(X, [tau]).sub.[gamma]] and Cl(A) = X [??] {a, c}, where [[alpha].sub.[gamma]]Cl(A)\A = {b} dose not contain any non-empty [gamma]-closed set in X.

Theorem 2.33. If A is an [[alpha].sub.[gamma]]-g.closed set of a space X, then the following are equivalent:

(1) A is [[alpha].sub.[gamma]]-closed.

(2) [[alpha].sub.[gamma]]Cl(A)\A is [[alpha].sub.[gamma]]-closed.

Proof. (1) [??] (2). If A is an [[alpha].sub.[gamma]]-g.closed set which is also [[alpha].sub.[gamma]]-closed, then by Theorem 2.30, [[alpha].sub.[gamma]]Cl(A)\A = [phi] which is [[alpha].sub.[gamma]]-closed.

(2) [??] (1). Let [[alpha].sub.[gamma]]Cl(A)\A be [[alpha].sub.[gamma]]-closed set and A be [[alpha].sub.[gamma]]-g.closed. Then by Theorem 2.30, [[alpha].sub.[gamma]]Cl(A)\A does not contain any non-empty [[alpha].sub.[gamma]]-closed subset. Since [[alpha].sub.[gamma]]Cl(A)\A is [[alpha].sub.[gamma]]-closed and [[alpha].sub.[gamma]]Cl(A)\A = [phi], this shows that A is [[alpha].sub.[gamma]]-closed.

Theorem 2.34. For a space (X, [tau]), the following are equivalent:

(1) Every subset of X is [[alpha].sub.[gamma]]-g.closed.

(2) [alpha]O[(X, [tau]).sub.[gamma]] = [alpha]C(X, [tau])[gamma].

Proof. (1) [??] (2). Let U [member of] [alpha]O(X, [tau])[gamma]. Then by hypothesis, U is [[alpha].sub.[gamma]]-g.closed which implies that [[alpha].sub.[gamma]]Cl(U) [subset or equal to] U, so, [[alpha].sub.[gamma]]Cl(U) = U, therefore U [member of] [[alpha].sub.[gamma]][(X, [tau]).sub.[gamma]]. Also let V [member of] [[alpha].sub.[gamma]] (X, [tau])y. Then X\V [member of] [[alpha].sub.[gamma]] [(X, [tau]).sub.[gamma]], hence by hypothesis X\V is [[alpha].sub.[gamma]]-g.closed and then X\V [member of] [[alpha].sub.[gamma]] [(X, [tau]).sub.[gamma]], thus V [member of] [alpha]O[(X, [tau]).sub.[gamma]] according above we have [alpha]O[(X, [tau]).sub.[gamma]] = [alpha]C[(X, [tau]).sub.[gamma]].

(2) [??] (1). If A is a subset of a space X such that A [subset or equal to] U where U [member of] [alpha]O[(X, [tau]).sub.[gamma]], then U [member of] [alpha]C[(X, [tau]).sub.[gamma]] and therefore [[alpha].sub.[gamma]]Cl(U) [subset or equal to] U which shows that A is [[alpha].sub.[gamma]]-g.closed.

Proposition 2.35. If A is y-open and [[alpha].sub.[gamma]]-g.closed then A is [[alpha].sub.[gamma]]-closed.

Proof. Suppose that A is y-open and [[alpha].sub.[gamma]]-g.closed. As every y-open is [[alpha].sub.[gamma]]-open and A [subset or equal to] A, we have [[alpha].sub.[gamma]]Cl(A) [subset or equal to] A, also A [subset or equal to] [[alpha].sub.[gamma]]Cl(A), therefore [[alpha].sub.[gamma]]Cl(A) = A. That is A is [[alpha].sub.[gamma]]-closed.

Theorem 2.36. If a subset A of X is [[alpha].sub.[gamma]]-g.closed and A [subset or equal to] B [subset or equal to] [[alpha].sub.[gamma]]Cl(A), then B is an [[alpha].sub.[gamma]]- g.closed set in X.

Proof. Let A be [[alpha].sub.[gamma]]-g.closed set such that A [subset or equal to] B [subset or equal to] [[alpha].sub.[gamma]]Cl(A). Let U be an [[alpha].sub.[gamma]]-open set of X such that B [subset or equal to] U. Since A is [[alpha].sub.[gamma]]-g.closed, we have [[alpha].sub.[gamma]]Cl(A) [subset or equal to] U. Now [[alpha].sub.[gamma]]Cl(A) [subset or equal to] [[alpha].sub.[gamma]]Cl(B) [subset or equal to] [[alpha].sub.[gamma]]Cl[[[alpha].sub.[gamma]]Cl(A)] = [[alpha].sub.[gamma]]Cl(A) [subset or equal to] U. That is [[alpha].sub.[gamma]]Cl(B) [subset or equal to] U, where U is [[alpha].sub.[gamma]]- open. Therefore B is an [[alpha].sub.[gamma]]-g.closed set in X.

The converse of the above theorem need not be true as seen from the following example.

Example 2.37. Consider X = {a, b, c} with the topology [tau] = {[phi], {a}, {c}, {a, c}, {b, c}, X}. Define an operation [gamma] on [alpha]O(X, [tau]) by [gamma](A) = A. Let A = {b} and B = {b, c}. Then A and B are [[alpha].sub.[gamma]]-g.closed sets in (X, [tau]). But A [subset or equal to] B [[alpha].sub.[gamma]]Cl(A).

Proposition 2.38. Let [gamma] be an operation on [alpha]O(X, [tau]). Then for each x [member of] X, {x} is [[alpha].sub.[gamma]]-closed or X\{x} is [[alpha].sub.[gamma]]-g.closed in (X, [tau]).

Proof. Suppose that {x} is not [[alpha].sub.[gamma]]-closed, then X\{x} is not [[alpha].sub.[gamma]]-open. Let U be any [[alpha].sub.[gamma]]-open set such that X\{x} [subset or equal to] U, implies U = X. Therefore [[alpha].sub.[gamma]]Cl(X\{x}) [subset or equal to] U. Hence X\{x} is [[alpha].sub.[gamma]]-g.closed.

[[alpha].sub.[gamma]]-Separation axioms

Definition 3.1. A space (X, [tau]) is said to be [[alpha].sub.[gamma]]-[T.sub.1/2] if every [[alpha].sub.[gamma]]-g.closed set is [[alpha].sub.[gamma]]-closed.

Theorem 3.2. The following statements are equivalent for a topological space (X, [tau]) with an operation [gamma] on [alpha]O(X, [tau]):

(1) (X, [tau]) is [[alpha].sub.[gamma]]-T1/2.

(2) Each singleton {x} of X is either [[alpha].sub.[gamma]]-closed or [[alpha].sub.[gamma]]-open.

Proof. (1) ^ (2). Suppose {x} is not [[alpha].sub.[gamma]]-closed. Then by Proposition 2.38, X\{x} is [[alpha].sub.[gamma]]-g.closed. Now since (X, [tau]) is [[alpha].sub.[gamma]]-T1/2, X\{x} is [[alpha].sub.[gamma]]-closed, that is {x} is [[alpha].sub.[gamma]]-open.

(2) [??] (1). Let A be any [[alpha].sub.[gamma]]-g.closed set in (X, [tau]) and x [member of] [[alpha].sub.[gamma]]Cl(A). By (2) we have {x} is [[alpha].sub.[gamma]]-closed or [[alpha].sub.[gamma]]-open. If {x} is [[alpha].sub.[gamma]]-closed then x [not member of] A will imply x [member of] [[alpha].sub.[gamma]]Cl(A)\A, which is not possible by Theorem 2.30. Hence x [member of] A. Therefore, [[alpha].sub.[gamma]]Cl(A) = A, that is A is [[alpha].sub.[gamma]]-closed. So, (X, [tau]) is [[alpha].sub.[gamma]]-T1/2. On the other hand, if {x} is [[alpha].sub.[gamma]]-open then as x [member of] [[alpha].sub.[gamma]]Cl(A), {x} [intersection] A [not equal to] [phi]. Hence x [member of] A. So A is [[alpha].sub.[gamma]]-closed.

Definition 3.3. A subset A of a topological space (X, [tau]) is called an [[alpha].sub.[gamma]]Dset if there are two U, V [member of] [alpha]O[(X, [tau]).sub.[gamma]] such that U [not equal to] X and A = U\V. It is true that every [[alpha].sub.[gamma]]-open set U different from X is an [[alpha].sub.[gamma]]D-set if A = U and V = [phi]. So, we can observe the following.

Remark 3.4. Every proper [[alpha].sub.[gamma]]-open set is an [[alpha].sub.[gamma]]D-set.

Definition 3.5. A topological space (X, [tau]) with an operation [gamma] on [alpha]O(X, [tau]) is said to be

(1) [[alpha].sub.[gamma]]D0 if for any pair of distinct points x and y of X there exists an [[alpha].sub.[gamma]]D-set of X containing x but not y or an [[alpha].sub.[gamma]]D-set of X containing y but not x.

(2) [[alpha].sub.[gamma]][D.sub.1] if for any pair of distinct points x and y of X there exists an [[alpha].sub.[gamma]]D-set of X containing x but not y and an [[alpha].sub.[gamma]]D-set of X containing y but not x.

(3) [[alpha].sub.[gamma]]D2 if for any pair of distinct points x and y of X there exist disjoint [[alpha].sub.[gamma]]D-sets G and E of X containing x and y, respectively.

Definition 3.6. A topological space (X, [tau]) with an operation [gamma] on [alpha]O(X, [tau]) is said to be:

(1) [[alpha].sub.[gamma]][T.sub.1] if for any pair of distinct points x and y of X there exists an [[alpha].sub.[gamma]]-open set U in X containing x but not y or an [[alpha].sub.[gamma]]-open set V in X containing y but not x.

(2) [[alpha].sub.[gamma]][T.sub.1] if for any pair of distinct points x and y of X there exists an [[alpha].sub.[gamma]]-open set U in X containing x but not y and an [[alpha].sub.[gamma]]-open set V in X containing y but not x.

(3) [[alpha].sub.[gamma]][T.sub.2] if for any pair of distinct points x and y of X there exist disjoint [[alpha].sub.[gamma]]-open sets U and V in X containing x and y, respectively.

Remark 3.7. For a topological space (X, [tau]) with an operation [gamma] on [alpha]O(X, [tau]), the following properties hold:

(1) If (X, [tau]) is [[alpha].sub.[gamma]]T;, then it is [[alpha].sub.[gamma]] [T.sub.i-1], for i = 1, 2.

(2) If (X, [tau]) is [alpha][T.sub.i], then it is [[alpha].sub.[gamma]][D.sub.i], for i = 0, 1, 2.

(3) If (X, [tau]) is [[alpha].sub.[gamma]][D.sub.i], then it is [[alpha].sub.[gamma]][D.sub.i-j1, for i = 1, 2.

Theorem 3.8. A topological space (X, [tau]) is [[alpha].sub.[gamma]][D.sub.1] if and only if it is [[alpha].sub.[gamma]][D.sub.2].

Proof. Sufficiency. Follows from Remark 3.7.

Necessity. Let x, y [member of] X, x [not equal to] y. Then there exist [[alpha].sub.[gamma]]D-sets [G.sub.1], [G.sub.2] in X such that x [member to] [G.sub.1], y [not member of] [G.sub.1] and y [member of] [G.sub.2], x [not member of] [G.sub.2]. Let [G.sub.1] = [U.sub.1]\[U.sub.2] and [G.sub.2] = [U.sub.3]\[U.sub.4], where [U.sub.1], [U.sub.2], [U.sub.3] and [U.sub.4] are [[alpha].sub.[gamma]]-open sets in X. From x [not member of] [G.sub.2], it follows that either x [not member of] [U.sub.3] or x [member of] [U.sub.3] and x [member of] [U.sub.4]. We discuss the two cases separately.

(i) x [not member of] [U.sub.3]. By y [not member of] [G.sub.1] we have two subcases:

(a) y [not member of] [U.sub.1]. From x [member of] [U.sub.1]\[U.sub.2], it follows that x [member of] [U.sub.1]\ ([U.sub.2] U [U.sub.3]), and by y [member of] [U.sub.3]\[U.sub.4] we have y [member of] [U.sub.3]\([U.sub.1] U [U.sub.4]). Therefore ([U.sub.1]\([U.sub.2] U [U.sub.3])) [intersection] ([U.sub.3]\([U.sub.1] U [U.sub.4])) = [phi].

(b) y [member of] [U.sub.1] and y [member of] [U.sub.2]. We have x [member of] [U.sub.1]\[U.sub.2], and y [member of] [U.sub.2]. Therefore ([U.sub.1]\[U.sub.2]) [intersection] [U.sub.2] = [phi].

(ii) x [member of] [U.sub.3] and x [member of] [U.sub.4]. We have y [member of] [U.sub.3]\[U.sub.4] and x [member of] [U.sub.4]. Hence ([U.sub.3]\[U.sub.4]) [intersection] [U.sub.4] = [phi]. Therefore, X is [[alpha].sub.[gamma]][D.sub.2].

Theorem 3.9. A topological space (X, [tau]) with an operation [gamma] on [alpha]O(X, [tau]) is [[alpha].sub.[gamma]][T.sub.0] if and only if for each pair of distinct points x, y of X, [[alpha].sub.[gamma]]Cl({x}) [not equal to] [[alpha].sub.[gamma]]Cl({y}).

Proof. Clear.

Theorem 3.10. A topological space (X, [tau]) with an operation [gamma] on [alpha]O(X, [tau]) is [[alpha].sub.[gamma]][T.sub.1] if and only if the singletons are [[alpha].sub.[gamma]]-closed sets.

Proof. Let (X, [tau]) be [[alpha].sub.[gamma]][T.sub.1] and x any point of X. Suppose y [member of] X\{x}, then

x [not equal to] y and so there exists an [[alpha].sub.[gamma]]-open set U such that y [member of] U but x [not member of] U. Consequently y [member of] U [subset or equal to] X\{x} that is X\{x} = U{U : y [member of] X\{x}} which is [[alpha].sub.[gamma]]-open.

Conversely, suppose {p} is [[alpha].sub.[gamma]]-closed for every p [member of] X. Let x, y [member of] X with x [not equal to] y. Now x [not equal to] y implies y [member of] X\{x}. Hence X\{x} is an [[alpha].sub.[gamma]]-open set contains y but not x. Similarly X\{y} is an [[alpha].sub.[gamma]]-open set contains x but not y. Accordingly X is an [[alpha].sub.[gamma]][T.sub.1] space.

Proposition 3.11. The following statements are equivalent for a topological space (X, [tau]) with an operation [gamma] on [alpha]O(X, [tau]):

(1) X is [[alpha].sub.[gamma]][T.sub.2].

(2) Let x [member of] X. For each y [not equal to] x, there exists an [[alpha].sub.[gamma]]open set U containing x such that y g [[alpha].sub.[gamma]]Cl(U).

(3) For each x [member of] X, [intersection]{[[alpha].sub.[gamma]]Cl(U) : U [member of] [alpha]O[(X, [tau]).sub.[gamma]] and x [member of] U} = {x}.

Proof. (1) [??] (2). Since X is [[alpha].sub.[gamma]][T.sub.2], there exist disjoint [[alpha].sub.[gamma]]-open sets U and V containing x and y respectively. So, U [member of] X\V. Therefore, [[alpha].sub.[gamma]]Cl(U) [subset or equal to] X\V. So y [not member of] [[alpha].sub.[gamma]]Cl(U).

(2) [??] (3). If possible for some y [not equal to] x, we have y [member of] [[alpha].sub.[gamma]]Cl(U) for every [[alpha].sub.[gamma]]-open set U containing x, which then contradicts (2).

(3) [??] (1). Let x, y [member of] X and x [not equal to] y. Then there exists an [[alpha].sub.[gamma]]-open set U containing x such that y [not member of] [[alpha].sub.[gamma]]Cl(U). Let V = X\[[alpha].sub.[gamma]]Cl(U), then y [member of] V and x [member of] U and also U [intersection] V = [phi].

[[alpha].sub.([gamma],[gamma]')]-Continuous maps

Throughout this section, let (X, [tau]) and (Y, [sigma]) be two topological spaces and let y : [alpha]O(X, [tau]) [right arrow] P (X) and [gamma]': [alpha]O(Y, [sigma]) [right arrow] P(Y) be the operations on [alpha]O(X, t) and [alpha]O(Y, [sigma]), respectively.

Definition 4.1. A mapping f : (X, [tau]) [right arrow] (Y, [sigma]) is said to be [[alpha].sub.([gamma],[gamma])]-continuous if for each x of X and each [[alpha].sub.[gamma]]-open set V containing f(x), there exists an [[alpha].sub.[gamma]]-open set U such that x [member of] U and f(U) [subset or equal to] V.

Theorem 4.2. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an [[alpha].sub.([gamma],[gamma])]-continuous mapping. Then:

(1) f([[alpha].sub.[gamma]]Cl(A)) [subset or equal to] [[alpha].sub.[gamma]]Cl(f(A)) holds for every subset A of (X, [tau]).

(2) For every [[alpha].sub.[gamma]]-closed set B of (Y, [sigma]), [f.sup.-1](B) is [[alpha].sub.[gamma]]-closed in (X, [tau]).

Proof. (1) Let y [member of] f([[alpha].sub.[gamma]]Cl(A)) and V be the [[alpha].sub.[gamma]]open set containing y, then there exists a point x [member of] X and an [[alpha].sub.[gamma]]-open set U such that f(x) = y, x [member of] U and f(U) [member of] V. Since x [member of] [[alpha].sub.[gamma]]Cl(A), we have U [intersection] A [not equal to] [phi], and hence [phi] [not equal to] f(U [intersection] A) [member of] f(U) [subset or equal to] f(A) [subset or equal to] V [intersection] f(A). This implies y [member of] [[alpha].sub.[gamma]]Cl(f(A)).

(2) It is sufficient to prove that (1) implies (2). Let B be the [[alpha].sub.[gamma]]-closed set in (Y, [sigma]). That is [[alpha].sub.[gamma]]Cl(B) = B. By using (1) we have f([[alpha].sub.[gamma]]Cl([f.sup.-1](B))) [subset or equal to] [[alpha].sub.[gamma]]Cl(f([f.sup.-1](B))) [subset or equal to] [[alpha].sub.[gamma]']Cl(B) = B holds. Therefore [[alpha].sub.[gamma]]Cl([f.sup.-1](B)) [subset or equal to] [f.sup.-1](B), and hence [f.sup.-1](B) = [[alpha].sub.[gamma]]Cl([f.sup.-1](B)). Hence [f.sup.-1](B) is [[alpha].sub.[gamma]]-closed set in (X, [tau]).

Definition 4.3. A mapping f : (X, [tau]) [right arrow] (Y, [sigma]) is said to be [[alpha].sub.([gamma],[gamma])]-closed if for any [[alpha].sub.[gamma]]-closed set A of (X, [tau]), f(A) is [[alpha].sub.[gamma]]-closed (Y, [sigma]).

Definition 4.4. If f is [[alpha].sub.(id, [gamma]')]-closed, then f(F) is [[alpha].sub.[gamma]']-closed for any O-closed set F of (X, [tau]).

Remark 4.5. If f is bijective mapping and [f.sup.-1]: (Y, [sigma]) [right arrow] (X, [tau]) is [[alpha].sub.([gamma],id)]-continuous, then f is [[alpha].sub.(id, [gamma]')]-closed.

Proof. Proof follows from the Definitions 4.3 and 4.4.

Theorem 4.6. Suppose f : (X, [tau]) [right arrow] (Y, [sigma]) is [[alpha].sub.([gamma],[gamma])]-continuous and f is [[alpha].sub.([gamma],[gamma])]-closed, then

(1) For every [[alpha].sub.[gamma]]-g.closed set A of (X, [sigma]) the image f(A) is [[alpha].sub.[gamma]]-g.closed.

(2) For every [[alpha].sub.[gamma]]-g.closed set B of (Y, [sigma]) the inverse set [f.sup.-1](B) is [[alpha].sub.[gamma]]-g.closed.

Proof. (1) Let V be any [[alpha].sub.[gamma]]-open set in (Y, [sigma]) such that f(A) [subset or equal to] V, then by Therem 4.2 (2), [f.sup.-1](V) is [[alpha].sub.[gamma]]open. Since A is [[alpha].sub.[gamma]]-g.closed and A [member of] [f.sup.-1](V), we have [[alpha].sub.[gamma]]Cl(A) [subset or equal to] [f.sup.-1](V), and hence f([[alpha].sub.[gamma]]Cl(A)) [member of] V. By assumption f([[alpha].sub.[gamma]]Cl(A)) is an [[alpha].sub.[gamma]']-closed set, therefore [[alpha].sub.[gamma]']Cl(f(A)) [subset or equal to] [[alpha].sub.[gamma]']Cl(f([[alpha].sub.[gamma]]Cl(A))) = f([[alpha].sub.[gamma]]Cl(A)) [subset or equal to] V. This implies f(A) is [[alpha].sub.[gamma]']-g.closed.

(2) Let U be any [[alpha].sub.[gamma]]-open set such that [f.sup.-1](B) [subset or equal to] U. Let F = [[alpha].sub.[gamma]]Cl([f.sup.-1](B)) [intersection] (X\U), then F is [[alpha].sub.[gamma]]-closed in (X, [tau]). This implies f(F) is [[alpha].sub.[gamma]']-closed set in (Y, [sigma]). Since f(F) = f([[alpha].sub.[gamma]]Cl([f.sup.-1](B)) [intersection] (X\U))) [subset or equal to] [[alpha].sub.[gamma]']Cl(B) [intersection] f(X\U) [member of] [[alpha].sub.[gamma]']Cl(B) [intersection] (Y\B). This implies f(F) = [phi] and hence F = [phi]. Therefore [[alpha].sub.[gamma]]Cl([f.sup.-1](B)) [subset or equal to] U. This implies [f.sup.-1](B) is [[alpha].sub.[gamma]]-g.closed.

Theorem 4.7. Suppose f : (X, [tau]) [right arrow] (Y, [sigma]) is [[alpha].sub.([gamma],[gamma]')]-continuous and [[alpha].sub.([gamma],[gamma]')]-closed, then:

(1) If f is injective and (Y, [sigma]) is [[alpha].sub.[gamma]]-[T.sub.1/2], then (X, [tau]) is [[alpha].sub.[gamma]]-[T.sub.1/2].

(2) If f is surjective and (X, [tau]) is [[alpha].sub.[gamma]]-[T.sub.1/2], then (Y, g) is [[alpha].sub.[gamma]]-[T.sub.1/2].

Proof. (1) Let A be an [[alpha].sub.[gamma]]-g.closed set of (X, [tau]). Now to prove that A is [[alpha].sub.[gamma]]-closed. By Theorem 4.6 (1), f(A) is [[alpha].sub.[gamma]]-g.closed. Since (Y, [sigma]) is [[alpha].sub.[gamma]]-[T.sub.1/2], this implies that f(A) is [[alpha].sub.[gamma]]-closed. Since f is ar,y)continuous, then by Theorem 4.2, we have A = [f.sup.-1](f(A)) is [[alpha].sub.[gamma]]-closed. Hence (X, [tau]) is [[alpha].sub.[gamma]]-[T.sub.1/2].

(2) Let B be an [[alpha].sub.[gamma]]-g.closed set in (Y, [sigma]). Then [f.sup.-1](B) is [[alpha].sub.[gamma]]-closed, since (X, [tau]) is [[alpha].sub.[gamma]]-[T.sub.1/2] space. It follows from the assumption that B is [[alpha].sub.[gamma]]-closed.

Definition 4.8. A mapping f : (X, [tau]) [right arrow] (Y, [sigma]) is said to be [[alpha].sub.[gamma]]-homeomorphic, if f is bijective, [[alpha].sub.([gamma],[gamma]')]-continuous and [f.sup.-1] is [[alpha].sub.([gamma],[gamma]')]-continuous.

Remark 4.9. If f : (X, [tau]) [right arrow] (Y, [sigma]) is bijective and [f.sup.-1]: (Y, [sigma]) [right arrow] (X, [tau]) is [[alpha].sub.([gamma],[gamma]')]-continuous, then f is [[alpha].sub.([gamma],[gamma]')]-closed.

Theorem 4.10. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be [[alpha].sub.([gamma],[gamma]')]-homeomorphic. The space (X, [tau]) is [[alpha].sub.[gamma]]-[T.sub.1/2] if and only if (Y, [sigma]) is [[alpha].sub.[gamma]]-[T.sub.1/2].

Proof. Necessity. Let B be an [[alpha].sub.[gamma]]-g.closed set of (Y, [sigma]). By Theorem 4.6, [f.sup.-1](B) is [[alpha].sub.[gamma]]-g.closed and hence [[alpha].sub.[gamma]]-closed. Since f is [[alpha].sub.([gamma],[gamma]')]-closed, we have B = f([f.sup.-1](B)) is [[alpha].sub.[gamma]]-closed.

Sufficiency. Let A be an [[alpha].sub.[gamma]]-g.closed set of (X, [tau]). By Theorem 4.6, f(A) is [[alpha].sub.[gamma]]-g.closed and hence [[alpha].sub.[gamma]]closed. Since f is [[alpha].sub.[gamma]]-continuous, then by Theorem 4.2, we have A = [f.sup.-1](f(A)) is [[alpha].sub.[gamma]]-closed.

Theorem 4.11. If f : (X, [tau]) [right arrow] (Y, [sigma]) is an [[alpha].sub.([gamma],[gamma]')]-continuous surjective mapping and E is an [[alpha].sub.[gamma]]D-set in Y, then the inverse image of E is an [[alpha].sub.[gamma]]D-set in X.

Proof. Let E be an [[alpha].sub.[gamma]]D-set in Y. Then there are [[alpha].sub.[gamma]]-open sets [U.sub.1] and [U.sub.2] in Y such that E = [U.sub.1]\[U.sub.2] and [U.sub.1] [not equal to] Y. By the [[alpha].sub.([gamma],[gamma]')]-continuous of f, [f.sup.-1]([U.sub.1]) and [f.sup.-1]([U.sub.2]) are [[alpha].sub.[gamma]]-open in X. Since [U.sub.1] [not equal to] Y and f is surjective, we have [f.sup.-1]([U.sub.1]) [not equal to] X. Hence, [f.sup.-1](E) = [f.sup.-1]([U.sub.1])\[f.sup.-1]([U.sub.2]) is an [[alpha].sub.[gamma]]D-set.

Theorem 4.12. If (Y, [sigma]) is [[alpha].sub.[gamma]][D.sub.1] and f : (X, [tau]) [right arrow] (Y, g) is [[alpha].sub.([gamma],[gamma]')]-continuous bijective, then (X, [tau]) is [[alpha].sub.[gamma]][D.sub.1].

Proof. Suppose that Y is an [[alpha].sub.[gamma]]D space. Let x and y be any pair of distinct points in X. Since f is injective and Y is [[alpha].sub.[gamma]]D1, there exist [[alpha].sub.[gamma]]D-sets [G.sub.x] and [G.sub.y] of Y containing f(x) and f(y) respectively, such that f (x) [not member of] [G.sub.y] and f(y) [right arrow] [G.sub.x]. By Theorem 4.11, [f.sup.-1]([G.sub.x]) and [f.sup.-1]([G.sub.y]) are [[alpha].sub.[gamma]]D-sets in X containing x and y, respectively, such that x [not member of] [f.sup.-1]([G.sub.y]) and y [not member of] [f.sup.-1]([G.sub.x]). This implies that X is an [[alpha].sub.[gamma]][D.sub.1] space.

Theorem 4.13. A topological space (X, [tau]) is [[alpha].sub.[gamma]][D.sub.1] if for each pair of distinct points x, y [member of] X, there exists an [[alpha].sub.([gamma],[gamma]')]-continuous surjective mapping f : (X, [tau]) [right arrow] (Y, [sigma]), where Y is an [[alpha].sub.[gamma]],D1space such that f(x) and f(y) are distinct.

Proof. Let x and y be any pair of distinct points in X. By hypothesis, there exists an [[alpha].sub.([gamma],[gamma]')]-continuous, surjective mapping f of a space X onto an [[alpha].sub.[gamma]'][D.sub.1]space Y such that f(x) [not equal to] f(y). By Theorem 3.8, there exist disjoint [[alpha].sub.[gamma]]D-sets [G.sub.x] and [G.sub.y] in Y such that f(x) [member of] [G.sub.x] and f(y) [member of] [G.sub.y]. Since f is [[alpha].sub.([gamma],[gamma]')]-continuous and surjective, by Theorem 4.11, [f.sup.-1]([G.sub.x]) and [f.sup.-1]([G.sub.y]) are disjoint [[alpha].sub.[gamma]]D-sets in X containing x and y, respectively. Hence by Theorem 3.8, X is [[alpha].sub.[gamma]][D.sub.1] space.

Conclusion

In this paper, we introduce the concept of an operation [gamma] on a family of [alpha]-open sets in a topological space (X, [tau]). Using this operation [gamma], we introduce the concept of [[alpha].sub.[gamma]]-open sets as a generalization of [gamma]-open sets in a topological space (X, [tau]). Using this set, we introduce [[alpha].sub.[gamma]][T.sub.0], [[alpha].sub.[gamma]]-[T.sub.1/2], [[alpha].sub.[gamma]][T.sub.2], [[alpha].sub.[gamma]][D.sub.0], [[alpha].sub.[gamma]]D and [[alpha].sub.[gamma]][D.sub.2] spaces and study some of their properties. Finally, we introduce [[alpha].sub.([gamma],[gamma]')]-continuous mappings and give some properties of such mappings.

Doi: 10.4025/actascitechnol.v35i3.15788

References

KASAHARA, S. Operation-compact spaces. Mathematica Japonica, v. 24, n. 1, p. 97-105, 1979.

NJASTAD, O. On some classes of nearly open sets. Pacific Jounal of Mathematics, v. 15, n. 3, p. 961-970, 1965.

OGATA, H. Operation on topological spaces and associated topology. Mathematica Japonica, v. 36, n. 1, p. 175-184, 1991.

Received on January 21, 2012.

Accepted on August 7, 2012.

Hariwan Zikri Ibrahim

Department of Mathematics, Faculty of Science, University of Zakh, Duhok, Kurdistan-Region, Iraq. E-mail: hariwan_math@yahoo.com

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Author: | Ibrahim, Hariwan Zikri |
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Publication: | Acta Scientiarum. Technology (UEM) |

Date: | Jul 1, 2013 |

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