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[[alpha].sup.[gamma]]-open sets, [[alpha].sup.[gamma]]-functions and some new separation axioms/ [[alpha].sup.[gamma]]-aberto conjunto, [[alpha].sup.[gamma]]-funcoes e alguns novos axiomas de separacao.

Introduction

In 1965 Njastad (1965) introduced [alpha]-open sets, Kasahara (1979) defined an operation [alpha] on a topological space to introduce [alpha]-closed graphs. Following the same technique, Ogata (1991) defined an operation on a topological space and introduced [gamma]-open sets.

In this paper, we introduce the notion of [[alpha].sup.[gamma]]-open sets, [[alpha].sup.[gamma]]-continuity and [[alpha].sup.([gamma],[beta])]-irresoluteness in topological spaces. By utilizing these notions we introduce some weak separation axioms. Also we show that some basic properties [[alpha].sup.[gamma]]-[T.sub.i] (i = 0, 1/2, 1, 2), [[alpha].sup.[gamma]]-[D.sub.i] (i = 0, 1, 2) spaces and we offer a new notion of the graph of a function called an [[alpha].sup.[gamma]]-closed graph and investigate some of their fundamental properties.

Throughout the paper spaces X and Y mean topological spaces. For a subset A of a space X, cl(A) and int(A) represent the closure of A and the interior of A, respectively.

Preliminaries

A subset A of X is called [alpha]-open if A [subset or equal to] int(cl(int(A))). The complement of [alpha]-open set is called [alpha]-closed set. The family of all [alpha]-open sets of X is denoted by [alpha]O(X). For a subset A of X, the union of all [alpha]-open sets of X contained in A is called the [alpha]-interior (in short [alpha]int(A)) of A, and the intersection of all [alpha]-closed sets of X containing A is called the [alpha]-closure (in short [alpha]cl(A)) of A. An operation [gamma] (KASAHARA, 1979) on a topology [tau] is a mapping from [tau] in to power set P(X) of X such that V [subset or equal to] (V) for each V [member of] [tau], where [gamma](V) denotes the value of at V. A subset A of X with an operation [gamma] on [tau] is called [gamma]-open (OGATA, 1991) 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. Then, [[tau].sub.[gamma]] denotes the set of all [gamma]-open set in X. Clearly [[tau].sub.[gamma]] [subset or equal to] [tau]. Complements of [gamma]-open sets are called [gamma]-closed. The [gamma]-closure (OGATA, 1991) of a subset A of X with an operation [gamma] on [tau] is denoted by [[tau].sub.[gamma]]-cl(A) and is defined to be the intersection of all [gamma]-closed sets containing A, and the [gamma]-interior (OGATA, 1991) of A is denoted by [[tau].sub.[gamma]]-int(A) and defined to be the union of all [gamma]-open sets of X contained in A. A topological (X, [tau]) with an operation [gamma] on [tau] is said to be [gamma]-regular (OGATA, 1991) 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].sup.[gamma]]-open sets

Definition 3.1. Let (X, [tau]) be a topological space, [gamma] an operation on [tau] and A [subset or equal to] X. Then A is called an [[alpha].sup.[gamma]]-open set if A [subset or equal to] int([[tau].sub.[gamma]]-cl(int(A))).

[[alpha].sup.[gamma]]O(X) denotes the collection of all [[alpha].sup.[gamma]]-open sets of (X, [tau]), and [[alpha].sup.[gamma]] O(X, x) is the collection of all [[alpha].sup.[gamma]]-open sets containing the point x of X.

A subset A of X is called [[alpha].sup.[gamma]]-closed if and only if its complement is [[alpha].sup.[gamma]]-open. Moreover, [[alpha].sup.[gamma]]C(X) denotes the collection of all [[alpha].sup.[gamma]]-closed sets of (X, [tau]).

It can be shown that a subset A of X is [[alpha].sup.[gamma]]-closed if and only if cl([[tau].sub.[gamma]]-int(cl(A))) [subset or equal to] A.

Remark 3.2.

(1) Every [alpha]-open set is [[alpha].sup.[gamma]]-open, while in a [gamma]-regular space these concepts are equivalent.

(2) Every [gamma]-open set is [[alpha].sup.[gamma]]-open, but the converse may not be true.

Example 3.3. Let X = {a, b, c} and [tau] = {[phi], {a}, {b}, {a, b}, X}. Define an operation [gamma] on [tau] by [gamma] (A) = {a} if A = {a} and [gamma] (A) = A [union] {c} if A [not equal to] {a}. Clearly, [[tau].sub.[gamma]] = {[phi], {a}, X}.

(1) Then {a, c} is [[alpha].sup.[gamma]]-open but not [alpha]-open.

(2) Also {a, c} is [[alpha].sup.[gamma]]-open but not [gamma]-open.

Theorem 3.4. An arbitrary union of [[alpha].sup.[gamma]]-open sets is [[alpha].sup.[gamma]]-open.

Proof. Let {[A.sub.k]: k [member of] I} be a family of [[alpha].sup.[gamma]]-open sets. Then for each k,

[A.sub.k] [subset or equal to] int([[tau].sub.[gamma]]-cl(int([A.sub.k]))) and so

[U.sub.k] [A.sub.k] [subset or equal to] [[union].sub.k] int([[tau].sub.[gamma]]-cl(int([A.sub.k])))

[subset or equal to] int([[union].sub.k] [[tau].sub.[gamma]]-cl(int([A.sub.k])))

[subset or equal to] int([[tau].sub.[gamma]]-cl(([[union].sub.k] int([A.sub.k])))

[subset or equal to] int([[tau].sub.[gamma]]-cl(int([[union].sub.k] [A.sub.k]))).

Thus, [[union].sub.k] [A.sub.k] is [[alpha].sup.[gamma]]-open.

Remark 3.5.

(1) An arbitrary intersection of [[alpha].sup.[gamma]]-closed sets is [[alpha].sup.[gamma]]-closed.

(2) The intersection of even two [[alpha].sup.[gamma]]-open sets may not be [[alpha].sup.[gamma]]-open.

Example 3.6. Let X = {a, b, c} and [tau] = {9, {a}, {b}, {a, b}, X}. Define an operation [gamma] on [tau] by [gamma] (A) = A if A = {a, b} and [gamma] (A) = X otherwise. Clearly, [[tau].sub.[gamma]] = {[phi], {a, b}, X} and [[alpha].sup.[gamma]] O(X) = {[phi], {a}, {b}, {a, b}, {a, c}, {b, c}, X}, take A = {a, c} and B = {b, c}. Then A 0 B = {c}, which is not an [[alpha].sup.[gamma]]-open set.

Definition 3.7. Let A be a subset of a topological space (X, [tau]) and [gamma] an operation on t.

(1) The union of all [[alpha].sup.[gamma]]-open sets contained in A is called the [[alpha].sup.[gamma]]-interior of A and denoted by [[alpha].sup.[gamma]] int(A).

(2) The intersection of all [[alpha].sup.[gamma]]-closed sets containing A is called the [[alpha].sup.[gamma]]-closure of A and denoted by [[alpha].sup.[gamma]] cl(A).

(3) The set denoted by [[alpha].sup.[gamma]] D(A) and defined by {x: for every [[alpha].sup.[gamma]]-open set U containing x, [union] [intersection] (A\{x}) [not equal to] [phi]} is called the [[alpha].sup.[gamma]]-derived set of A.

(4) The [[alpha].sup.[gamma]]-frontier of A, denoted by [[alpha].sup.[gamma]] Fr(A) is defined as [[alpha].sup.[gamma]] cl(A) [[alpha].sup.[gamma]] cl(X \ A).

We now state the following theorem without proof.

Theorem 3.8. Let (X, [tau]) be a topological space and [gamma] an operation on [tau]. For any subsets A, B of X we have the following:

(1) A is [[alpha].sup.[gamma]]-open if and only if A = [[alpha].sup.[gamma]] int(A).

(2) A is [[alpha].sup.[gamma]]-closed if and only if A = [[alpha].sup.[gamma]] cl(A).

(3) If A [subset or equal to] B then [[alpha].sup.[gamma]] int(A) [subset or equal to] [[alpha].sup.[gamma]] int(B) and [[alpha].sup.[gamma]] cl(A) [subset or equal to] [[alpha].sup.[gamma]] cl(B).

(4) [[alpha].sup.[gamma]] int (A) [union] [[alpha].sup.[gamma]]int(B) [subset or equal to] [[alpha].sup.[gamma]]int (A [union] B).

(5) [[alpha].sup.[gamma]] int (A [union] B) [subset or equal to] [[alpha].sup.[gamma]]int(A) [intersectiom] [[alpha].sup.[gamma]]int(B).

(6) [[alpha].sup.[gamma]] cl (A) [union] [[alpha].sup.[gamma]]cl(B) [subset or equal to] [[alpha].sup.[gamma]]cl(A [union] B).

(7) [[alpha].sup.[gamma]] cl (A \ B) [subset or equal to] [[alpha].sup.[gamma]]cl(A) [intersection] [[alpha].sup.[gamma]]cl(B).

(8) [[alpha].sup.[gamma]] int (X \ A) = X [[alpha].sup.[gamma]]cl(A).

(9) [[alpha].sup.[gamma]] cl (X \ A) = X [[alpha].sup.[gamma]]int(A).

(10) [[alpha].sup.[gamma]] int(A) = A \ [[alpha].sup.[gamma]]D(X A).

(11) [[alpha].sup.[gamma]] cl(A) = A [union] [[alpha].sup.[gamma]]D(A).

(12) [[tau].sub.[gamma]]-int(A) [subset or equal to] [[alpha].sup.[gamma]] int(A).

(13) [[alpha].sup.[gamma]] cl(A) [subset or equal to] [[tau].sub.[gamma]]-cl(A).

Theorem 3.9. Let A be a subset of a topological space (X, [tau]) and [gamma] be an operation on [tau]. Then x [member of] [[alpha].sup.[gamma]]cl(A) if and only if for every [[alpha].sup.[gamma]]-open set V of

X containing x, A [intersection] V [not equal to] [phi].

Proof. Let x [member of] [[alpha].sup.[gamma]] cl(A) and suppose that V [intersection] A = [phi] for some [[alpha].sup.[gamma]]-open set V which contains x. Then (X \ V) is [[alpha].sup.[gamma]]-closed and A [subset or equal to] (X \ V), thus [[alpha].sup.[gamma]] cl(A) [subset or equal to] (X \ V ). But this implies that x [member of] (X \ V), a contradition. 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].sup.[gamma]]-open set U which contains x, U [intersection] A [not equal to] [phi]. If x [not member of] [[alpha].sup.[gamma]]cl(A), there is an [[alpha].sup.[gamma]]-closed set F such that A [subset or equal to] F and x [not member of] F. Then (X\F) is an [[alpha].sup.[gamma]]-open set with x [member of] (X\F), and thus (X \ F) [intersection] A [not equal to] [phi], which is a contradiction.

Definition 3.10. A subset A of a topological space (X, [tau]) with an operation [gamma] on [tau] is called [tau]-[[alpha].sup.[gamma]]-open (resp. [[alpha].sup.[gamma]]-[gamma]-open) if int(A) = [[alpha].sup.[gamma]]int(A) (resp. [[tau].sub.[gamma]]-int(A) = [[alpha].sup.[gamma]]int(A)).

Definition 3.11. A subset A of a topological space (X, [tau]) with an operation [gamma] on [tau] is called an [[alpha].sup.[gamma]]-generalized closed set ([[alpha].sup.[gamma]]-g closed, for short) if

[[alpha].sup.[gamma]]cl(A) [subset or equal to] U whenever A [subset or equal to] U and U is an [[alpha].sup.[gamma]]- open set in X.

The complement of an [[alpha].sup.[gamma]]-g closed set is called an [[alpha].sup.[gamma]]-g open set. Clearly, A is [[alpha].sup.[gamma]]-g open if and only if F [subset or equal to] [[alpha].sup.[gamma]]int(A) whenever F [subset or equal to] A and F is [[alpha].sup.[gamma]]- closed in X.

Theorem 3.12. Every [[alpha].sup.[gamma]]-closed set is [[alpha].sup.[gamma]]-g closed.

Proof. A set A [subset or equal to] X is [[alpha].sup.[gamma]]-closed if and only if [[alpha].sup.[gamma]]cl(A) = A. Thus [[alpha].sup.[gamma]]cl(A) [subset or equal to] U for every U [member of] [[alpha].sup.[gamma]]O(X) containing A.

Theorem 3.13. A subset A of topological space (X, [tau]) with an operation [gamma] on [tau], is [[alpha].sup.[gamma]]-g closed if and only if [[alpha].sup.[gamma]]cl({x}) [intersection] A [not equal to] [phi], holds for every x [member of] [[alpha].sup.[gamma]]cl(A).

Proof. Let A be an [[alpha].sup.[gamma]]-g closed set in X and suppose if possible there exists an x [member of] [[alpha].sup.[gamma]]cl(A) such that [[alpha].sup.[gamma]]cl({x}) [intersection] A [not equal to] [phi]. Therefore A [subset or equal to] X \ [[alpha].sup.[gamma]]cl({x}),

and so [[alpha].sup.[gamma]]cl(A) [subset or equal to] X\[[alpha].sup.[gamma]]cl({x}). Hence x [member of] [[alpha].sup.[gamma]]cl(A), which is a contradiction.

Conversely, suppose that the condition of the therem holds and let U be any [[alpha].sup.[gamma]]-open set such that A [subset or equal to] U and let x [member of] [[alpha].sup.[gamma]]cl(A). By assumption, there exists a z [member of] [[alpha].sup.[gamma]]cl({x}) and z [member of] A [subset or equal to] U. Thus by the Theorem 3.9,

U [intersection] {x} [not equal to] [phi]. Hence x [member of] U, which implies [[alpha].sup.[gamma]]cl(A) [subset or equal to] U.

Theorem 3.14. Let A be an [[alpha].sup.[gamma]]-g closed set in a topological space (X, [tau]) with operation [gamma] on [tau]. Then [[alpha].sup.[gamma]]cl(A)\A does not contain any nonempty [[alpha].sup.[gamma]]-closed set.

Proof. If possible, let F be an [[alpha].sup.[gamma]]-closed set such that F [subset or equal to] [[alpha].sup.[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].sup.[gamma]]-g closed and X\F is [[alpha].sup.[gamma]]-open, therefore [[alpha].sup.[gamma]]cl(A) [subset or equal to] X\F, that is F [subset or equal to] X\[[alpha].sup.[gamma]]cl(A).

Hence F [subset or equal to] [[alpha].sup.[gamma]]cl(A) [intersection] (X\[[alpha].sup.[gamma]]cl(A)) = [phi]. This shows that, F = [phi] which is a contradiction.

Theorem 3.15. In a topological space (X, [tau]) with an operation [gamma] on [tau], either {x} is [[alpha].sup.[gamma]]-closed or X\{x} is [[alpha].sup.[gamma]]-g closed.

Proof. Suppose that {x} is not [[alpha].sup.[gamma]]-closed, then X\{x} is not [[alpha].sup.[gamma]]-open. Then X is the only [[alpha].sup.[gamma]]-open set such that X\{x} [subset or equal to] X. Hence X\{x} is [[alpha].sup.[gamma]]-g closed set.

[[alpha].sup.[gamma]]-Functions

Definition 4.1. Let (X, [tau]) and (Y, [sigma]) be two topological spaces and [gamma] an operation on [tau]. Then a function f: (X, [tau]) [right arrow] (Y, [sigma]) is said to be [[alpha].sup.[gamma]]-continuous at a point x [member of] X if for each open set V of Y containing f(x), there exists an [[alpha].sup.[gamma]]-open set U of X containing x such that f(U) [subset or equal to] V.

If f is [[alpha].sup.[gamma]]-continuous at each point x of X, then f is called [[alpha].sup.[gamma]]-continuous on X.

Theorem 4.2. Let (X, [tau]) be a topological space with an operation [gamma] on [tau]. For a function f: (X, t) [right arrow] (Y, [sigma]), the following statements are equivalent:

(1) f is [[alpha].sup.[gamma]]-continuous.

(2) [f.sup.-1](V) is [[alpha].sup.[gamma]]-open set in X, for each open set V in Y.

(3) [f.sup.-1](V) is [[alpha].sup.[gamma]]-closed set in X, for each closed set V in Y.

(4) f([[alpha].sup.[gamma]]cl(U)) [subset or equal to] cl(f(U)), for each subset U of X.

(5) [[alpha].sup.[gamma]]cl([f.sup.-1](V)) [subset or equal to] [f.sup.-1](cl(V)), for each subset V of Y.

(6) [f.sup.-1](int(V)) [subset or equal to] [[alpha].sup.[gamma]]int([f.sup.-1](V)), for each subset V of Y.

(7) int(f(U)) [subset or equal to] f([[alpha].sup.[gamma]]int(U)), for each subset U of X.

Proof. (1) [??] (2) [??] (3). Obvious.

(3) [??] (4). Let U be any subset of X. Then f(U) [subset or equal to] cl(f(U)) and cl(f(U)) is closed set in Y. Hence U [subset or equal to] [f.sup.-1](cl(f(U))). By (3), we have [f.sup.-1](cl(f(U))) is [[alpha].sup.[gamma]]-closed set in X. Therefore, [[alpha].sup.[gamma]]cl(U) [subset or equal to] [f.sup.-1](cl(f(U))).

Hence f([[alpha].sup.[gamma]]cl(U)) [subset or equal to] cl(f(U)).

(4) [??] (5). Let V be any subset of Y. Then [f.sup.-1](V) is a subset of X. By (4), we have f([[alpha].sup.[gamma]]cl([f.sup.-1](V))) [subset or equal to] cl(f([f.sup.-1](V))) = cl(V). Hence [[alpha].sup.[gamma]]cl([f.sup.-1](V)) [subset or equal to] [f.sup.-1](cl(V)).

(5) [??] (6). Let V be any subset of Y. Then apply (5) to Y \ V we obtain

[[alpha].sup.[gamma]]cl([f.sup.-1](Y\V )) [subset or equal to] [f.sup.-1](cl(Y\V )) [??][[alpha].sup.[gamma]]cl(X\ [f.sup.-1](V)) [subset or equal to] [f.sup.-1](Y\int(V)) [??] X\[[alpha].sup.[gamma]] int([f.sup.-1](V)) [subset or equal to] X\ [f.sup.-1](int(V)) [??] [f.sup.-1] (int(V)) [subset or equal to] [[alpha].sup.[gamma]]int([f.sup.-1](V)). Therefore, [f.sup.-1](int(V)) [subset or equal to] [[alpha].sup.[gamma]]int([f.sup.-1](V)).

(6) [??] (7). Let U be any subset of X. Then f(U) is a subset of Y. By (6), we have [f.sup.-1](int(f(U))) [subset or equal to] [[alpha].sup.[gamma]]int ([f.sup.-1](f(U))) = [[alpha].sup.[gamma]]int(U). Therefore, int(f(U)) [subset or equal to] f([[alpha].sup.[gamma]]int(U)).

(7) [??] (1). Let x [member of] X and let V be any open set of Y containing f(x). Then x [member of] [f.sup.-1](V) and [f.sup.-1](V) is a subset of X. By (7), we have int(f([f.sup.-1](V))) [subset or equal to] f([[alpha].sup.[gamma]]int([f.sup.-1](V))). Then int(V) [subset or equal to] f([[alpha].sup.[gamma]]int([f.sup.-1](V))). Since V is an open set. Then V [subset or equal to] f([[alpha].sup.[gamma]]int([f.sup.-1](V))) implies that [f.sup.-1](V) [subset or equal to] [[alpha].sup.[gamma]]int([f.sup.-1](V)). Therefore, [f.sup.-1](V) is [[alpha].sup.[gamma]]-open set in X which contains x and clearly f([f.sup.-1](V)) [subset or equal to] V. Hence f is [[alpha].sup.[gamma]]-continuous.

Theorem 4.3. For a function f: (X, [tau]) [right arrow] (Y, [sigma]) with an operation [gamma] on [tau], the following statements are equivalent:

(1) [f.sup.-1](V) is [[alpha].sup.[gamma]]-open set in X, for each open set V in Y.

(2) [[alpha].sup.[gamma]]Fr([f.sup.-1](V)) [subset or equal to] [f.sup.-1] Fr(V)), for each subset V in Y.

Proof. (1) [??] (2). Let V be any subset of Y. Then, we have [f.sup.-1](Fr(V))=[f.sup.-1](cl(V)\int(V)) = [f.sup.-1] (cl(V))\[f.sup.1](int(V)) = [[alpha].sup.[gamma]]cl([f.sup.-1](V))\[f.sup.-1](int(V)) [contains or equal to] [[alpha].sup.[gamma]]cl([f.sup.-1] (V))[[alpha].sup.[gamma]]int[f.sup.-1] int(V)) [contains or equal to] [[alpha].sup.[gamma]]cl([f.sup.-1](V))[[alpha].sup.[gamma]]int([f.sup.-1](V)) = [[alpha].sup.[gamma]]Fr [f.sup.-1] V), and hence [f.sup.-1](Fr(V)) [contains or equal to] [[alpha].sup.[gamma]]Fr ([f.sup.-1] (V)).

(2) [??] (1). Let V be open in Y and F = Y\V. Then by (2), we obtain [[alpha].sup.[gamma]]Fr([f.sup.-1](F)) [subset or equal to] [f.sup.-1](Fr(F)) [subset or equal to] [f.sup.-1](cl(F)) = [f.sup.-1](F) and hence [[alpha].sup.[gamma]]cl([f.sup.-1](F)) = [[alpha].sup.[gamma]]int([f.sup.-1](F)) U [[alpha].sup.[gamma]]Fr([f.sup.-1](F)) [subset or equal to] [f.sup.-1](F). Thus [f.sup.-1](F) is [[alpha].sup.[gamma]]- closed and hence [f.sup.-1](V) is [[alpha].sup.[gamma]]-open in X.

Theorem 4.4. let (X, [tau]) be a topological space with an operation [gamma] on [tau] and let f: (X, [tau]) [right arrow] (Y, [sigma]) be a function. Then

X\[[alpha].sup.[gamma]]C(f) = [union]{ [[alpha].sup.[gamma]]Fr([f.sup.-1](V)): V [member of] [sigma], f(x) [member of] V, x [member of] X},

where [[alpha].sup.[gamma]]C(f) denotes the set of points at which f is [[alpha].sup.[gamma]]-continuous.

Proof. Let x [member of]X\[[alpha].sup.[gamma]]C(f). Then there exists V [member of] [sigma] containing f(x) such that f(U) [not subset or equal to] V, for every [[alpha].sup.[gamma]]-open set U containing x. Hence U [intersection] [X \ [f.sup.-1](V)] [not equal to] [phi] for every [[alpha].sup.[gamma]]-open set U containing x. Therefore, by

Theorem 3.9, x [member of] [[alpha].sup.[gamma]]cl(X[f.sup.-1](V)). Then x [member of] [f.sup.-1] (V) [intersection] [[alpha].sup.[gamma]]cl(X\[f.sup.-1](V)) [subset or equal to] [[alpha].sup.[gamma]]Fr([f.sup.-1](V). So, X\[[alpha].sup.[gamma]]C(f) [subset or equal to] [union]{ [[alpha].sup.[gamma]]Fr([f.sup.-1](V)):V [member of] [sigma], f(x) [member of] V, x [member of] X}

Conversely, let x [not member of] X\[[alpha].sup.[gamma]]C(f). Then for each V [member of] [sigma] containing f(x), [f.sup.-1](V) is an [[alpha].sup.[gamma]]-open set containing x. Thus x [member of] [[alpha].sup.[gamma]]int([f.sup.-1](V) and hence x [not member of] [[alpha].sup.[gamma]]Fr ([f.sup.-1](V)), for every V [member of] [sigma] containing f(x). Therefore,

X\[[alpha].sup.[gamma]]C(f) [contains or equal to] U{[[alpha].sup.[gamma]]Fr([f.sup.-1](V)):V [member of] [sigma], f(x) [member of] V, x [member of] X}.

Remark 4.5. Every [gamma]-continuous function is [[alpha].sup.[gamma]]-continuous, but the converse is not true.

Example 4.6. Let X = {a, b, c}, [tau] = {[phi], {a}, X} and [sigma] = {[phi], {a}, {b}, {a, b}, X}. Define an operation [gamma] on [tau] by [gamma](A) = A if A={a} and [gamma](A) = A [union] {b} if A [not equal to] {a}. Define a function f: (X, [tau]) [right arrow] (X, [sigma]) as follows: f(x) = a if x = a, f(x) = a if x = b and f(x) = c if x = c.

Then f is [[alpha].sup.[gamma]]-continuous but not [gamma]-continuous at b, because {a, b} is an open set in (X, [sigma]) containing f(b) = a, there exist no [[alpha].sup.[gamma]]-open set U in (X, [tau]) containing b such that f(U) [subset or equal to] {a, b}.

Remark 4.7. Let [gamma] and [beta] be operations on the topological spaces (X, [tau]) and (Y, [sigma]), respectively. If the functions f: (X, [tau]) [right arrow] (Y, [sigma]) and g: (Y, [sigma]) [right arrow] (Z, u) are [[alpha].sup.[gamma]]-continuous and continuous, respectively, then gof is [[alpha].sup.[gamma]]-continuous.

Definition 4.8. Let (X, [tau]) be a topological space with an operation [gamma] on [tau]. A function f: (X, t) [right arrow] (Y, [sigma]) is called [tau]- [[alpha].sup.[gamma]]-continuous (resp. [[alpha].sup.[gamma]]-[gamma]-continuous) if for each open set V in Y, [f.sup.-1](V) is [tau]-[[alpha].sup.[gamma]]-open (resp. [[alpha].sup.[gamma]]-[gamma]-open) in X.

Theorem 4.9. Let f: (X, [tau]) [right arrow] (Y, [sigma]) be a mapping and [gamma] an operation on [tau]. Then the following are equivalent:

(1) f is [gamma]-continuous.

(2) f is [[alpha].sup.[gamma]]-continuous and [[alpha].sup.[gamma]]-[gamma]-continuous.

Proof. (1) [??] (2). Let f be [gamma]-continuous. Then f is [[alpha].sup.[gamma]]-continuous. Now, let G be any open set in Y, then [f.sup.-1](G) is [gamma]-open in X. Then [[tau].sub.[gamma]]-int([f.sup.-1](G)) = [f.sup.-1](G) = [[alpha].sup.[gamma]]int([f.sup.-1](G)). Thus, [f.sup.-1](G) is [[alpha].sup.[gamma]]-[gamma]-open in X. Therefore f is [[alpha].sup.[gamma]]-[gamma]-continuous.

(2) [??] (1). Let f be [[alpha].sup.[gamma]]-continuous and [[alpha].sup.[gamma]]-[gamma]-continuous. Then for any open set G in Y, [f.sup.-1](G) is both [[alpha].sup.[gamma]]-open and [[alpha].sup.[gamma]]-[gamma]-open in X. So [f.sup.-1](G) = [[alpha].sup.[gamma]]int([f.sup.-1](G)) = [[tau].sub.[gamma]]-int([f.sup.-1](G)).

Thus [f.sup.-1](G) is [gamma]-open and hence f is [gamma]-continuous.

Theorem 4.10. Let f: (X, [tau]) [right arrow] (Y, [sigma]) be [tau]-[[alpha].sup.[gamma]]-continuous, where [gamma] is an operation on [tau]. Then f is continuous if and only if f is [[alpha].sup.[gamma]]-continuous.

Proof. Let V [member of] [sigma]. Since f is continuous as well as [tau]- [[alpha].sup.[gamma]]-continuous, [f.sup.-1](V) is open as well as [tau]-[[alpha].sup.[gamma]]-open in X and hence [f.sup.-1](V) = int([f.sup.-1](V)) = [[alpha].sup.[gamma]] int([f.sup.-1](V}) [member of] [[alpha].sup.[gamma]] O(X). Therefore, f is [[alpha].sup.[gamma]]-continuous.

Conversely, let V [member of] [sigma]. Then [f.sup.-1](V) is [[alpha].sup.[gamma]]-open and [tau]-[[alpha].sup.[gamma]]-open. So [f.sup.-1](V) = [[alpha].sup.[gamma]] int([f.sup.-1](V}) = int([f.sup.-1](V)). Hence [f.sup.-1](V) is open in X. Therefore f is continuous.

Definition 4.11. A function f: (X, [tau]) [right arrow] (Y, c), where [gamma] and [beta] are operations on [tau] and [sigma], respectively, is called [[alpha].sup.[beta]]-g-closed if for every [[alpha].sup.[gamma]]-closed set F in X, f(F) is [[alpha].sup.[beta]]-g closed in Y.

Definition 4.12. Let (X, [tau]) and (Y, [sigma]) be two topological spaces and [gamma], [beta] operations on [tau], [sigma], respectively. A mapping f: (X, [tau]) [right arrow] (Y, [sigma]) is called [[alpha].sup.([gamma],[beta])]-irresolute at x if and only if for each [[alpha].sup.[beta]]-open set V in Y containing f(x), there exists an [[alpha].sup.[gamma]]-open set U in X containing x such that f(U) [subset or equal to] V.

If f is [[alpha].sup.([gamma],[beta])]-irresolute at each point x [member of] X, then f is called [[alpha].sup.([gamma],[beta])]-irresolute on X.

Theorem 4.13. let (X, [tau]), (Y, [sigma]) be topological spaces and [gamma], [beta] operations on [tau], [sigma], respectively. If f: (X, [tau]) [right arrow] (Y, [sigma]) is [[alpha].sup.([gamma],[beta])]-irresolute and [[alpha].sup.[beta]]-g-closed, and A is [[alpha].sup.[gamma]]-g closed in X, then f(A) is [[alpha].sup.[beta]]-g closed.

Proof. Suppose A is an [[alpha].sup.[gamma]]-g closed set in X and that U is an [[alpha].sup.[beta]]-open set in Y such that f(A) [subset or equal to] U. Then A [subset or equal to] [f.sup.-1](U). Since f is [[alpha].sup.([gamma],[beta])]-irresolute, [f.sup.-1](U) is [[alpha].sup.[gamma]]-open set in X. Again A is an [[alpha].sup.[gamma]]-g closed set, therefore [[alpha].sup.[gamma]]cl(A) [subset or equal to] [f.sup.-1](U) and hence f([[alpha].sup.[gamma]]cl(A)) [subset or equal to] U. Since f is an [[alpha].sup.[beta]]-g-closed map, f([[alpha].sup.[gamma]]cl(A)) is an [[alpha].sup.[beta]]-g closed set in Y. Therefore, [[alpha].sup.[beta]]cl(f([[alpha].sup.[gamma]]cl(A)) [subset or equal to] U, which implies [[alpha].sup.[beta]]cl(f(A)) [subset or equal to] U.

We now state the following theorem without proof.

Theorem 4.14. Let f: (X, [tau]) [right arrow] (Y, [sigma]) be a mapping and [gamma], [beta] operations on [tau], [sigma], respectively. Then the following are equivalent:

(1) f is [[alpha].sup.([gamma],[beta])]-irresolute.

(2) The inverse image of each [[alpha].sup.[beta]]-open set in Y is an [[alpha].sup.[gamma]]-open set in X.

(3) The inverse image of each [[alpha].sup.[beta]]-closed set in Y is an [[alpha].sup.[gamma]]-closed set in X.

(4) [[alpha].sup.[gamma]]cl([f.sup.-1](V}) [subset or equal to] [f.sup.-1] ([[alpha].sup.[beta]]cl(V)), for all V [subset or equal to] Y.

(5) f([[alpha].sup.[gamma]]cl(U)) [subset or equal to] [[alpha].sup.[beta]]cl(f(U)), for all U [subset or equal to] X.

(6) [[alpha].sup.[gamma]]Fr([f.sup.-1](V)) [subset or equal to] [f.sup.-1] ([[alpha].sup.[beta]]Fr(V)), for all V [subset or equal to] Y.

(7) f([[alpha].sup.[gamma]]D(U)} [subset or equal to] [[alpha].sup.[beta]]cl(f(U)), for all U [subset or equal to] X.

(8) [f.sup.-1]([[alpha].sup.[beta]]int(V)) [subset or equal to] [[alpha].sup.[gamma]]int([f.sup.-1](V}), for all V [subset or equal to] Y.

[[alpha].sup.[gamma]]-Separation Axioms

Definition 5.1. A topological space (X, [tau]) with an operation [gamma] on [tau] is said to be

(1) [[alpha].sup.[gamma]]-[T.sub.0] if for each pair of distinct points x, y in X, there exists an [[alpha].sup.[gamma]]-open set U such that either x [member of] U and y [not member of] U or x [not member of] U and y [member of] U.

(2) [[alpha].sup.[gamma]]-[T.sub.1] if for each pair of distinct points x, y in X, there exist two [[alpha].sup.[gamma]]-open sets U and V such that x [member of] U but y [not member of] U and y [member of] V but x [not member of] V.

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

(4) [[alpha].sup.[gamma]]-[T.sub.1/2] if every [[alpha].sup.[gamma]]-g closed set is [[alpha].sup.[gamma]]-closed.

Theorem 5.2. A topological space (X, t) with an operation [gamma] on [tau] is [[alpha].sup.[gamma]]-[T.sub.0] if and only if for each pair of distinct points x, y of X, [[alpha].sup.[gamma]]cl({x}) [not equal to] [[alpha].sup.[gamma]]cl({y}).

Theorem 5.3. The following statements are equivalent for a topological space (X, [tau]) with an operation [gamma] on [tau]:

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

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

Theorem 5.4. A topological space (X, [tau]) with an operation [gamma] on [tau] is [[alpha].sup.[gamma]]-[T.sub.1] if and only if the singletons are [[alpha].sup.[gamma]]-closed sets.

Theorem 5.5. The following statements are equivalent for a topological space (X, [tau]) with an operation [gamma] on [tau]:

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

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

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

Corollary 5.6. If (X, [tau]) is a topological space and y be an operation on [tau], then the following statements are hold:

(1) Every [[alpha].sup.[gamma]]-[T.sub.1] space is [[alpha].sup.[gamma]]-[T.sub.1/2].

(2) Every [[alpha].sup.[gamma]]-[T.sub.1/2] space is [[alpha].sup.[gamma]]-[T.sub.0].

Proof. (1) By deinition and Theorem 5.4 we prove it.

(2) Let x and y be any two distinct points of X. By Theorem 5.3, the singleton set {x} is [[alpha].sup.[gamma]]-closed or [[alpha].sup.[gamma]]-open.

(a) If {x} is [[alpha].sup.[gamma]]-closed, then X\{x} is [[alpha].sup.[gamma]]-open. So y [member of] X\{x} and x [not member of] X\{x}. Therefore, we have X is [[alpha].sup.[gamma]]-[T.sub.0].

(b) If {x} is [[alpha].sup.[gamma]]-open. Then x [member of]{x} and y [not member of] {x}. Therefore, we have X is [[alpha].sup.[gamma]]-[T.sub.0].

Definition 5.7. A subset A of a topological space X is called an [[alpha].sup.[gamma]]Difference set (in short [[alpha].sup.[gamma]]D-set) if there are U, V [member of] [[alpha].sup.[gamma]]O(X) such that U [not equal to] X and A = U\V.

It is true that every [[alpha].sup.[gamma]]-open set U different from X is an [[alpha].sup.[gamma]]D-set if A = U and V = [phi]. So, we can observe the following.

Remark 5.8. Every proper [[alpha].sup.[gamma]]-open set is a [[alpha].sup.[gamma]]D-set.

Now we define another set of separation axioms called [[alpha].sup.[gamma]]-[D.sub.i], i = 0, 1, 2 by using the [[alpha].sup.[gamma]]D-sets.

Definition 5.9. A topological space (X, [tau]) with an operation [gamma] on [tau] is said to be

(1) [[alpha].sup.[gamma]]-[D.sub.0] if for any pair of distinct points x and y of X there exists an [[alpha].sup.[gamma]]D-set of X containing x but not y or an [[alpha].sup.[gamma]]D-set of X containing y but not x.

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

(3) [[alpha].sup.[gamma]]-[D.sub.2] if for any pair of distinct points x and y of X there exist disjoint [[alpha].sup.[gamma]]D-set G and E of X containing x and y, respectively.

Remark 5.10. For a topological space (X, [tau]) with an operation [gamma] on [tau], the following properties hold:

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

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

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

Theorem 5.11. A space X is [[alpha].sup.[gamma]]-[D.sub.1] if and only if it is [[alpha].sup.[gamma]]-[D.sub.2].

Proof. Necessity. Let x; y [member of] X, x [not equal to] y. Then there exist [[alpha].sup.[gamma]]D-sets [G.sub.1], [G.sub.2] in X such that x [member of] [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].sup.[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] [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].sup.[gamma]]- [D.sub.2].

Sufficiency. Follows from Remark 5.10 (3).

Theorem 5.12. A space is [[alpha].sup.[gamma]]-[D.sub.0] if and only if it is [[alpha].sup.[gamma]]-[T.sub.0].

Proof. Suppose that X is [[alpha].sup.[gamma]]-[D.sub.0]. Then for each distinct pair x, y [member of] X, at least one of x, y, say x, belongs to an [[alpha].sup.[gamma]]D-set G but y [not member of] G. Let G = [U.sub.1]\[U.sub.2] where [U.sub.1] [not equal to] X and [U.sub.1], [U.sub.2] [member of] [[alpha].sup.[gamma]]O(X). Then x [member of] [U.sub.1], and for y [not member of] G we have two cases: (a) y [not member of] [U.sub.1], (b) y [member of] [U.sub.1] and y [member of] [U.sub.2].

In case (a), x [member of] [U.sub.1] but y [not member of] [U.sub.1].

In case (b), y [member of] [U.sub.2] but x [not member of] [U.sub.2].

Thus in both the cases, we obtain that X is [[alpha].sup.[gamma]]-[T.sub.0].

Conversely, if X is [[alpha].sup.[gamma]]-[T.sub.0], by Remark 5.10 (2), X is [[alpha].sup.[gamma]]-[D.sub.0].

Corollary 5.13. If (X, [tau]) is [[alpha].sup.[gamma]]-[D.sub.1], then it is [[alpha].sup.[gamma]]-[T.sub.0].

Proof. Follows from Remark 5.10 (3) and Theorem 5.12.

Definition 5.14. A point x [member of]X which has only X as the [[alpha].sup.[gamma]]-neighborhood is called an [[alpha].sup.[gamma]]-neat point.

Theorem 5.15. For an [[alpha].sup.[gamma]]-[T.sub.0] topological space (X, [tau]) the following are equivalent:

(1) (X, [tau]) is [[alpha].sup.[gamma]]-[D.sub.1].

(2) (X, [tau]) has no [[alpha].sup.[gamma]]-neat point.

Proof. (1) [??] (2). Since (X, t) is [[alpha].sup.[gamma]]-[D.sub.1], then each point x of X is contained in an [[alpha].sup.[gamma]]D-set A = U\V and thus in U. By definition U [not equal to] X. This implies that x is not an [[alpha].sup.[gamma]]-neat point.

(2) [??] (1). If X is [[alpha].sup.[gamma]]-[T.sub.0], then for each distinct pair of points x, y [member of] X, at least one of them, x (say) has an [[alpha].sup.[gamma]]-neighborhood U containing x and not y. Thus U which is different from X is a [[alpha].sup.[gamma]]D-set. If X has no [[alpha].sup.[gamma]]-neat point, then y is not an [[alpha].sup.[gamma]]-neat point. This means that there exists an [[alpha].sup.[gamma]]-neighbourhood V of y such that V [not equal to] X. Thus y [member of] V\U but not x and V\U is an [[alpha].sup.[gamma]] D-set. Hence X is [[alpha].sup.[gamma]]-[D.sub.1].

Corollary 5.16. An [[alpha].sup.[gamma]]-[T.sub.0] space X is not [[alpha].sup.[gamma]]-[D.sub.1] if and only if there is a unique [[alpha].sup.[gamma]]-neat point in X.

Proof. We only prove the uniqueness of the [[alpha].sup.[gamma]]-neat point. If x and y are two [[alpha].sup.[gamma]]-neat points in X, then since X is [[alpha].sup.[gamma]]-[T.sub.0], at least one of x and y, say x, has an [[alpha].sup.[gamma]]-neighborhood U containing x but not y. Hence U [not equal to] X. Therefore x is not an [[alpha].sup.[gamma]]-neat point which is a contradiction.

Definition 5.17. A topological space (X, [tau]) with an operation y on t, is said to be [[alpha].sup.[gamma]]-symmetric if for x and y in X, x [member of] [[alpha].sup.[gamma]]cl({y}) implies y [member of] [[alpha].sup.[gamma]]cl({x}).

Theorem 5.18. If (X, [tau]) is a topological space with an operation [gamma] on [tau], then the following are equivalent:

(1) (X, [tau]) is [[alpha].sup.[gamma]]-symmetric space.

(2) Every singleton is [[alpha].sup.[gamma]]-g closed, for each x [member of] X.

Proof. (1) [??] (2). Assume that {x} [subset or equal to] U [member of] [[alpha].sup.[gamma]]O(X), but [[alpha].sup.[gamma]]cl({x}) [not subset or equal to] U. Then [[alpha].sup.[gamma]]cl({x}) [intersection] X\U [not equal to] [phi]. Now, we take y [member of] [[alpha].sup.[gamma]]cl({x}) [intersection] X\U, then by hypothesis x [member of] [[alpha].sup.[gamma]]cl({y}) [subset or equal to] X \ U and x [not member of] U, which is a contradiction. Therefore {x} is [[alpha].sup.[gamma]]-g closed, for each x [member of] X

(2) [??] (1). Assume that x [member of] [[alpha].sup.[gamma]]cl({y}), but y [not member of] [[alpha].sup.[gamma]]cl({x}). Then {y} [subset or equal to] X\[[alpha].sup.[gamma]]cl({x}) and hence [[alpha].sup.[gamma]]cl({y}) [member of] X\[[alpha].sup.[gamma]]cl({x}). Therefore x [member of] X\[[alpha].sup.[gamma]]cl({x}), which is a contradiction and hence y [member of] [[alpha].sup.[gamma]]cl({x}).

Corollary 5.19. If a topological space (X, t) with an operation y on t is an [[alpha].sup.[gamma]]-[T.sub.1] space, then it is [[alpha].sup.[gamma]]-symmetric.

Proof. In an [[alpha].sup.[gamma]]-[T.sub.1] space, every singleton is [[alpha].sup.[gamma]]-closed (Theorem 5.4) and therefore is [[alpha].sup.[gamma]]-g closed (Theorem 3.12). Then by Theorem 5.18, (X, t) is [[alpha].sup.[gamma]]-symmetric.

Corollary 5.20. For a topological space (X, [tau]) with an operation [gamma] on [tau], the following statements are equivalent:

(1) (X, [tau]) is [[alpha].sup.[gamma]]-symmetric and [[alpha].sup.[gamma]]-[T.sub.0].

(2) (X, [tau]) is [[alpha].sup.[gamma]]-[T.sub.1].

Proof. By Remark 5.10 and Corollary 5.19, it suffices to prove only (1) [??] (2).

Let x [not equal to] y and by [[alpha].sup.[gamma]]-[T.sub.0], we may assume that x [member of] U [subset or equal to] X\{y}for some U [member of] [[alpha].sup.[gamma]]O(X). Then x [not member of] [[alpha].sup.[gamma]]cl({y}) and hence y [not member of] [[alpha].sup.[gamma]]cl({x}). There exists an [[alpha].sup.[gamma]]-open set V such that y [member of] V [subset or equal to] X\{x} and thus (X, [tau]) is an [[alpha].sup.[gamma]]-[T.sub.1] space.

Corollary 5.21. For an [[alpha].sup.[gamma]]-symmetric topological space (X, [tau]) the following are equivalent:

(1) (X, [tau]) is [[alpha].sup.[gamma]]-[T.sub.0].

(2) (X, [tau]) is [[alpha].sup.[gamma]]-[D.sub.1].

(3) (X, [tau]) is [[alpha].sup.[gamma]]-[T.sub.1].

Proof. (1) [??] (3). Corollary 5.20.

(3) [??] (2) [??] (1). Remark 5.10 (2) and Corollary 5.13.

Remark 5.22. If (X, [tau]) is an [[alpha].sup.[gamma]]-symmetric space with an operation [gamma] on [tau], then the following statements are equivalent:

(1) (X, [tau]) is an [[alpha].sup.[gamma]]-[T.sub.0] space.

(2) (X, [tau]) is an [[alpha].sup.[gamma]]-[T.sub.1/2] space.

(3) (X, [tau]) is an [[alpha].sup.[gamma]]-[T.sub.1] space.

Definition 5.23. Let A be a subset of a topological space (X, [tau]) and [gamma] an operation on [tau]. The [[alpha].sup.[gamma]]-kernel of A, denoted by [[alpha].sup.[gamma]]ker(A) is defined to be the set [[alpha].sup.[gamma]]ker(A) = [intersection]{U [member of] [[alpha].sup.[gamma]]O(X): A [subset or equal to] U}.

Theorem 5.24. Let (X, [tau]) be a topological space with an operation [gamma] on [tau] and x [member of] X. Then y [member of] [[alpha].sup.[gamma]]ker({x}) if and only if x [member of] [[alpha].sup.[gamma]]cl({y}).

Proof. Suppose that y [not member of] [[alpha].sup.[gamma]]ker({x}). Then there exists an [[alpha].sup.[gamma]]-open set V containing x such that y [not member of] V. Therefore, we have x [not member of] [[alpha].sup.[gamma]]cl({y}). The proof of the converse case can be done similarly.

Theorem 5.25. Let (X, [tau]) be a topological space with an operation [gamma] on [tau] and A be a subset of X. Then, [[alpha].sup.[gamma]]ker(A) = {x [member of] X: [[alpha].sup.[gamma]]cl({x}) [intersection] A [not equal to] [phi]}.

Proof. Let x [member of] [[alpha].sup.[gamma]]ker(A) and suppose [[alpha].sup.[gamma]]cl({x}) [intersection] A [not equal to] [phi]. Hence x [not member of] X\[[alpha].sup.[gamma]]cl({x}) which is an [[alpha].sup.[gamma]]-open set containing A. This is impossible, since x [member of] [[alpha].sup.[gamma]]ker(A). Consequently, [[alpha].sup.[gamma]]cl({x}) [intersection] A [not equal to] [phi]. Next, let x [member of] X such that [[alpha].sup.[gamma]]cl({x}) [intersection] A [intersection] [phi] and suppose that x [not member of] [[alpha].sup.[gamma]]ker(A). Then, there exists an [[alpha].sup.[gamma]]-open set V containing A and x [not member of] V. Let y [member of] [[alpha].sup.[gamma]]cl({x}) [intersection] A. Hence, V is an [[alpha].sup.[gamma]]-neighborhood of y which does not contain x. By this contradiction x [member of] [[alpha].sup.[gamma]]ker(A) and the claim.

Theorem 5.26. If a singleton {x} is an [[alpha].sup.[gamma]]D-set of (X, [tau]), then [[alpha].sup.[gamma]]ker({x}) [not equal to] X.

Proof. Since {x} is an [[alpha].sup.[gamma]]D-set of (X, t), then there exist two subsets [U.sub.1] [member of] [[alpha].sup.[gamma]]O(X) and [U.sub.2] [member of] [[alpha].sup.[gamma]]O(X) such that {x} = [U.sub.1]\[U.sub.2], {x} [subset or equal to] [U.sub.1] and [U.sub.1] [not equal to] X. Thus, we have that [[alpha].sup.[gamma]]ker({x}) [subset or equal to] [U.sub.1] [not equal to] X and so [[alpha].sup.[gamma]]ker({x}) [not equal to] X.

Theorem 5.27. If f:(X, [tau]) [right arrow] (Y, [sigma]) is an [[alpha].sup.[beta]] [right arrow] (Y, [sigma]) is an [[alpha].sup.([gamma],[beta])]-irresolute surjective function and A is an [[alpha].sup.[beta]]D-set in Y, then the inverse image of A is an [[alpha].sup.[gamma]]D-set in X.

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

Theorem 5.28. If (Y, [sigma]) is [[alpha].sup.[beta]]-[D.sub.1] and f: (X, [tau]) [right arrow] (Y, [sigma]) is [[alpha].sup.([gamma],[beta])]-irresolute bijective, then (X, [tau]) is [[alpha].sup.[gamma]]- [D.sub.1].

Proof. Suppose that Y is an [[alpha].sup.[beta]]-[D.sub.1] space. Let x and y be any pair of distinct points in X. Since f is injective and Y is [[alpha].sup.[beta]]-[D.sub.1], there exist [[alpha].sup.[beta]]D-set [O.sub.x] and [O.sub.y] of Y containing f(x) and f(y) respectively, such that f(x) [not member of] [O.sub.y] and f(y) [not member of] [O.sub.x]. By Theorem 5.27, [f.sup.-1]([O.sub.x]) and [f.sup.-1]([O.sub.y]) are [[alpha].sup.[gamma]]D-set in X containing x and y, respectively, such that x [not equal to] [f.sup.-1]([O.sub.y]) and y [not member of] [f.sup.-1]([O.sub.x]). This implies that X is an [[alpha].sup.[gamma]]-[D.sub.1] space.

Theorem 5.29. A topological space (X, [tau]) is [[alpha].sup.[gamma]]-[D.sub.1] if for each pair of distinct points x, y [member of] X, there exists an [[alpha].sup.([gamma],[beta])]-irresolute surjective function f: (X, [tau]) [right arrow] (Y, [sigma]), where Y is an [[alpha].sup.[beta]]-[D.sub.1] space 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].sup.([gamma],[beta])]-irresolute, surjective function f of a space X onto an [[alpha].sup.[beta]]-[D.sub.1] space Y such that f(x) [not equal to] f(y). It follows from Theorem 5.11 that [[alpha].sup.[beta]]-[D.sub.1] = [[alpha].sup.[beta]]-[D.sub.2]. Hence, there exist disjoint [[alpha].sup.[beta]]D-set [O.sub.x] and [O.sub.y] in Y such that f(x) [member of] [O.sub.x] and f(y) [member of] [O.sub.y]. Since f is [[alpha].sup.([gamma],[beta])]-irresolute and surjective, by Theorem 5.27, [f.sup.-1]([O.sub.x]) and [f.sup.-1]([O.sub.y]) are disjoint [[alpha].sup.[gamma]]D-sets in X containing x and y, respectively. So, the space (X, [tau]) is [[alpha].sup.[gamma]]-[D.sub.1].

Functions With [[alpha].sup.[gamma]]-Closed Graphs

In this section, functions with [[alpha].sup.[gamma]]-closed graphs are introduced and studied, and some properties and characterizations of [[alpha].sup.[gamma]]-closed graphs are explained.

Definition 6.1. Let f: (X, [tau]) [right arrow] (Y, [sigma]) be any function, the graph of the function f is denoted by G(f) and is said to be [[alpha].sup.[gamma]]-closed if for each (x, y) [not member of] G(f), there exist U [member of] [[alpha].sup.[gamma]]O(X, x) and an open set V of Y containing y such that (U x V ) [intersection] G(f) = [phi].

A useful characterisation of functions with [[alpha].sup.[gamma]]-closed graph is given below.

Lemma 6.2. The function f: (X, [tau]) [right arrow] (Y, [sigma]) has an [[alpha].sup.[gamma]]-closed graph if and only if for each x [member of] X and y [member of] Y such that y [not equal to] f(x), there exist an [[alpha].sup.[gamma]]-open set U and an open set V containing x and y respectively, such that f(U) [intersetion] V = [phi].

Proof. It follows readily from the above definition.

Theorem 6.3. Suppose that a function f: (X, [tau]) [right arrow] (Y, [sigma]) has an [[alpha].sup.[gamma]]-closed graph, then the following are true:

(1) If f is surjective, then Y is [T.sub.1].

(2) If f is injective, then X is [[alpha].sup.[gamma]]-[T.sub.1].

(3) If a function f is [[alpha].sup.[gamma]]-continuous and injective, then X is [[alpha].sup.[gamma]]-[T.sub.2].

(4) For each x [member of] X, {f(x)} = [intersection]{f(cl(f(U)): U [member of] [[alpha].sup.[gamma]]O(X, x)}.

Proof. (1) Let [y.sub.1] and [y.sub.2] be two distinct points of Y. Since f is surjective, there exists x in X such that f(x) = [y.sub.2], then (x, [y.sub.1]) [not member of] G(f). By Lemma 6.2, there exist [[alpha].sup.[gamma]]-open set U and open set V containing x and [y.sub.1] respectively, such that f(U) [intersection] V = [phi]. We obtain an open set V containing [y.sub.1] which does not contain [y.sub.2]. Similarly we can obtain an open set containing [y.sub.2] but not [y.sub.1]. Hence, Y is [T.sub.1].

(2) Let [x.sub.1] and [x.sub.2] be two distinct points of X. The injectivity of f implies f([x.sub.1]) [not equal to] f([x.sub.2]) whence one obtains that ([x.sub.1], f([x.sub.2])) [member of] (X x Y)\G(f). The [[alpha].sup.[gamma]]-closedness of G(f), by Lemma 6.2, ensures the existence of U [member of] [[alpha].sup.[gamma]]O(X, [x.sub.1]), V [member of] O(Y, f([x.sub.2])) such that f(U) [intersection] V = [phi]. Therefore, f([x.sub.2]) [not member of] f(U) and a fortiori [x.sub.2] [not member of] U. Again

([x.sub.2], f([x.sub.1])) [member of] (X x Y)\G(f) and [[alpha].sup.[gamma]]-closedness of G(f), as before gives A [member of] [[alpha].sup.[gamma]]O(X, [x.sub.2]), B [member of] O(Y, f([x.sub.1])) with f(A) [intersection] B = [phi], which guarantees that f([x.sub.1]) [not member of] f(A) and so [x.sub.1] [not member of] A. Therefore, we obtain sets U and A [member of] [[alpha].sup.[gamma]]O(X) such that [x.sub.1] [member of] U but [x.sub.2] [member of] U while [x.sub.2] [member of] A but [x.sub.1] [not member of] A. Thus X is [[alpha].sup.[gamma]]-[T.sub.1].

(3) Let [x.sub.1] and [x.sub.2] be any distinct points of X. Then f([x.sub.1]) [not member of] f([x.sub.2]), so ([x.sub.1], f([x.sub.2])) [member of] (X x Y)\G(f). Since the graph G(f) is [[alpha].sup.[gamma]]-closed, there exist an [[alpha].sup.[gamma]]-open set U containing [x.sub.1] and open set V containing f([x.sub.2]) such that f(U) [intersection] V = [phi]. Since f is [[alpha].sup.[gamma]]-continuous, [f.sup.-1](V) is an [[alpha].sup.[gamma]]-open set containing [x.sub.2] such that U [intersection] [f.sup.-1](V) = [phi]. Hence X is [[alpha].sup.[gamma]]-[T.sub.2].

(4) Suppose that y [not equal to] f(x) and y [member of] {cl(f(U)): U [member of] [[alpha].sup.[gamma]]O(X, x)}. Then y [member of] cl(f(U)) for each U [member of] [[alpha].sup.[gamma]]O(X, x). This implies that for each open set V containing y, V [intersection] f(U) [not equal to] [phi]. Since (x, [tau]) [right arrow] G(f) and G(f) is an [[alpha].sup.[gamma]]-closed graph, this is a contradiction.

Theorem 6.4. If f: (X, [tau]) [right arrow] (Y, [sigma]) is [[alpha].sup.[gamma]]-continuous and Y is [T.sub.2] space, then G(f) is [[alpha].sup.[gamma]]-closed graph.

Proof. Suppose that (x, y) [not member of] G(f), then f(x) [not equal to] y. By the fact that Y is [T.sub.2], there exist open sets W and V such that f(x) [member of] W, y [member of] V and V [intersection] W = [phi]. Since f is [[alpha].sup.[gamma]]-continuous, there exists U [member of] [[alpha].sup.[gamma]]O(X, x) such that f(U) [subset or equal to] W. Hence, we have f(U) [intersection] V = [phi]. This means that G(f) is [[alpha].sup.[gamma]]-closed.

Conclusion

In this paper, we introduce the notion of [[alpha].sup.[gamma]]-open sets, [[alpha].sup.[gamma]]-continuity and [[alpha].sup.([gamma],[beta])]-irresoluteness in topological spaces. By utilizing these notions we introduce some weak separation axioms. Also we show that some basic properties [[alpha].sup.[gamma]]-[T.sub.1] (i = 0, 1/2, 1, 2), [[alpha].sup.[gamma]]-[D.sub.i] (i = 0, 1, 2) spaces and we offer a new notion of the graph of a function called an [[alpha].sup.[gamma]]-closed graph and investigate some of their fundamental properties.

Doi: 10.4025/actascitechnol.v35i4.15728

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 Journal 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 11, 2012.

Accepted on January 28, 2013.

Hariwan Zikri Ibrahim

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