On Some Maps Concerning gα-Open Sets

In this paper, we consider a new generalization of α-open maps via the concept of gα-closed sets which we call approximately α-open maps. We study some of its fundamental properties. It turns out that we can use this notion to obtain a new characterization of α-T1/2 spaces. 2000 Mathematics Subject Classification: 54B05, 54C08


Introduction and preliminaries
The notion of α-open set in topological spaces was introduced in 1965 by Njåstad [16].Since then, many mathematicians turned their attention to the generalizations of various concepts in topology by considering α-open sets (see for example [1], [4], [5], [6], [12], [13], [14], [15], [17]).In 1994 Maki, Devi and Balachandran [15] generalized the notion of closed sets to generalized α-closed sets (briefly gα-closed sets) with the help of α-open sets.In this direction, we introduce the notion of ap-α-open maps by using gα-closed sets and study some of their basic properties.Finally we characterize the class of α-T 1/2 spaces in terms of ap-α-open maps.
Throughout this paper (X, τ ), (Y, σ) and (Z, γ) represent non-empty topological spaces on which no separation axioms are assumed, unless otherwise mentioned.For a subset A of a space (X, τ ), Cl(A) and Int(A) denote the closure and the interior of A, respectively.
In order to make the contents of the paper as self contained as possible, we briefly describe certain definitions, notations and some properties.For those not described, we refer the reader to [16].A subset A of a space (X, τ ) is said to be α-open [16] The α-interior of A, denoted by α Int(A), is the union of all α-open sets of (X, τ ) contained in A. A is α-open [1] if and only if α Int(A) = A. Also, we have α Int(A) = A ∩ Int(Cl(Int(A))) [1].A subset B of (X, τ ) is said to be α-closed if its complement B c is α-open in (X, τ ).By αO(X, τ ) (resp.αC(X, τ )) we mean the collection of all α-open sets (resp.α-closed sets) in (X, τ ).The α-closure of a set B of (X, τ ) denoted by α Cl X (B) (briefly α Cl(B)), is defined to be the intersection of all α-closed sets of (X, τ ) containing B. B is α-closed [1] if and only if α Cl(B) = B. Also, we have α Cl(B) = B ∪ Cl(Int(Cl(B))) [1].
A subset F of (X, τ ) is said to be generalized α-closed

Ap-α-open maps
We introduce the following notion: Clearly pre-α-open maps are ap-α-open, but not conversely.
Example 2.1.Let X = {a, b} be the Sierpinski space with the topology, τ = {∅, {a}, X}.Let f : X → X be defined by In the following theorem, we show that under certain conditions the converse of Theorem 2.1 is true.
Proof.(i) Suppose that A is an arbitrary α-open subset in (X, τ ) and B a gαclosed subset of (Z, γ) for which Now we state the following theorem whose proof is straightforward and hence omitted.
It is well-known that the graph G(f ) of f is a closed set of X × Y , whenever f is continuous and Y is Hausdorff.The following theorem is a modification of this fact, i.e., we give a condition under which a contra pre α-open map has α-closed graph.
Regarding the restriction f A of a map f : (X, τ ) → (Y, σ) to a subset A of X, we have the following: Proof.Let O be an arbitrary α-open subset of (A, τ A ) and B a gα-closed subset of (Y, σ) for which Observe that restrictions of ap-α-open maps can fail to be ap-α-open.Let X be an indiscrete space.Then X and ∅ are the only α-closed subsets of X. Hence the α-open subsets of X are X and ∅.Let A be a nonempty proper subset of X.The identity map f : In recent years, the class of α-T 1/2 spaces has been of some interest (see for example [15,9,7]).In the following theorem, we give a new characterization of α-T 1/2 spaces by using the notion of ap-α-open maps.We recall that a topological space (X, τ ) is said to be α-T 1/2 space [15, Definition 5 and Theorem 2.3], if every αg-closed set is α-closed.Theorem 2.9.Let (Y, σ) be a topological space.Then the following statements are equivalent.

Remark 2 . 1 .
Let (X, τ ) be the topological space defined in Example 2.1.Then the identity map on (X, τ ) is ap-α-open.It is clear that the converse of Theorem 2.1 does not hold (see Example 2.2).

Remark 2 . 2 .
In fact contra pre-α-open maps and pre-α-open are independent notions.Example 2.1 shows that contra-pre-α-openness does not imply pre-α-openness while the converse is shown in the following example.Example 2.2.The identity map on the same topological space (X, τ ) where τ = {∅, {a}, X} is an example of a pre-α-open map which is not contra pre-α-open.Remark 2.3.By Theorem 2.1 and Remark 2.1 we have that every contra pre-αopen map is ap-α-open but the converse does not hold.For the definitions of ap-semi-open, contra pre-semi-open, pre-semi-open and semi-open maps see Caldas and Baker [8], Sundaram, Maki and Balachandran [18] and Biswas [3].The following diagram holds: The next theorem establishes conditions under which the inverse map of every gα-open set from codomain is a gα-open set.Theorem 2.3.If a map f : (X, τ ) → (Y, σ) is surjective α-irresolute and ap-α-open, then f −1 (A) is gα-open whenever A is gα-open subset of (Y, σ).