1. Introduction

Functions and of course open functions stand among the most important notions in the whole of mathematical science. Many different forms of open functions have been introduced over the years. Various interesting problems arise when one consider openness. Its important is significant in various areas of mathematics and related sciences.

Introduced semi-generalized continuous maps in general topology as a generalization of continuous maps M.Caldas[1]. This concept was found to be useful and many results in general topology were improved. Many researchers like Balachan-dran, Sundaram and Maki[2], Bhattacharyya and Lahiri[4], Palaniappan and Rao[8], Park[9]and N.Levine[5], Benchalli and Wali[3], Vadivel, A. and Vairamanickam have worked on general-ized closed sets, quasi sg-open and quasi sg-closed functions Ravi, O[11], Dontchev,J.and Noiri,T.,[12] Quasi-Normal Spaces and πg-Closed Sets ,their generalizations and related concepts in general topology.

As a generalization of closed sets, the notion of rw-closed sets were introduced and studied in[3]. In this paper, we will continue the study of related functions by involving αm-open sets. We introduce and characterize the concept of quasi αm-open functions.Throughout this paper, by spaces we mean topological spaces on which no separation axioms are assumed unless otherwise mentioned and f : (X, τ ) → (Y, σ) (or simply f : X → Y ) denotes a function f of a space (X, τ ) into a space (Y, σ).Let A be a subset of a space X. The closure (A) and interior(A) are denote the Cl(A) and Int(A), respectively.

Definition 1.1 A function f : (X, τ ) → (Y, σ) is called a αm-open map[3] if the image f (A) is αm-open in (Y, σ) for each open set A in (X, τ ).

Definition 1.2 A map f : (X, τ ) → (Y, σ) is called a αm-irresolute[3] if the inverse image of every αm-closed in (Y, σ) is αm-closed in (X, τ ).

2. Quasi αm - open Functions

We have introduce a new definition as follows:

Definition 2.1. A function f : X → Y is said to be quasi αm -open if the image of every αm -open set in X is open in Y .

It is evident that, the concepts of quasi αm -openness and αm -continuity coincide if the function is a bijection.

Theorem 2.2 A function f : X → Y is said to be quasi αm-open iff for every subset U of X, f (αm-int(U )) ⊂ int(f (U )).

Proof:

Let f be a quasi αm-open function. Now, we have int(U ) ⊂ U and αm-int(U ) is a αm -open set. Hence we obtain that f (αm -int(U )) ⊂ f (U ). As f (αm -int(U )) is open, f (αm -int(U )) ⊂ int (f (U )).

Conversely, assume that U is a αm -open set in X. Then, f (U ) = f (αm -int(U )) ⊂ int (f (U )) but int (f (U )) ⊂ f (U ). Consequently f (U ) = int (f (U )) And hence f is quasi αm –open.

Lemma 2.3 If a function f : X→ Y is quasi αm -open, then αm -int (f ⁻¹ (G)) f ⁻¹ (int (G)) for every subset G of Y .

Proof:

Let G be any arbitrary subset of Y . Then, αm -int(f ⁻¹(G)) is a αm -open set in X and f is quasi αm -open, then f (αm -int(f ⁻¹(G))) ⊂ int(f (f ⁻¹(G))) ⊂ int(G). Thus αm -int(f ⁻¹(G)) ⊂ f ⁻¹(int(G))).

Recall that a subset S is called a αm-neighborhood[4] of a point x of X, if here exists a αm -open set U such that x ∈ U ⊂ S.

Theorem 2.4 For a function f : X → Y , the following are equivalent.

(a) f is quasi αm -open.

(b)For each subset U of X, f (αm - int(U )) ⊂int (f (U )).

(c) For each x ∈ X and each αm-neighbourhood U of x in X, there exists a neighborhood V of f (x) in Y such that V ⊂ f (U ).

Proof:

(a) ⇒ (b) : It follows from Theorem 2.2

(b) ⇒ (c) : Let x ∈ X and U be an arbitrary αm-neighbourhood of x in X. Then there exist a αm -open set V in X such that x ∈ V ⊂ U . Then by (b), we have f (V ) = f (αm-int(V )) ⊂ int (f (V )) and hence f (V ) = int (f (V )). Therefore, it follows that f (V ) is open in Y such that f (x) ∈ f (V ) ⊂ f (U ).

(c) ⇒ (a) : Let U be an arbitrary αm -open set in X. Then for each y ∈ f (U ), by (c) there exist a neighbourhood V_y of y in Y such that V_y ⊂ f (U ). As V_y is a neighbourhood of y, there exist an open set W_y in Y such that y ∈ W_y ⊂ V_y. Thus f (U ) = {W_y : y ∈ f (U )} which is a open set in Y . This implies that f is quasi αm -open function.

Theorem 2.5. A function f : X → Y is quasi αm-open iff for any subset B of Y and for any αm-closed set F of X, containing f ⁻¹ (B), there exist a closed set G of Y containing B such that f ⁻¹ (G) ⊂ F .

Proof:

Suppose f is quasi αm -open. Let B ⊂ Y and F be a αm -closed set of X containing f ⁻¹ (B). Now, put G = Y −f (X − F ). It is clear that f ⁻¹ (B) ⊂ F implies B ⊂ G. Since f is quasi αm-open, we obtain G as a closed set of Y . Moreover, we have f ⁻¹ (G) ⊂ F .

Conversely, Let U be a αm -open set of X and put B = Y \f (U ). ThenX\U is a αm-closed set in X containing f ⁻¹ (B). By hypothesis, there exists a closed set F of Y such that B ⊂ F and f ⁻¹ (F ) ⊂ X\U . Hence, we obtain f (U ) ⊂ Y \F . On the other hand, it follows that B ⊂ F , Y \F ⊂ Y \B = f (U ). Thus, we obtain f (U ) = Y \F which is open and hence f is a quasi αm -open function.

Theorem 2.6 A function f : X→ Y is quasi αm -open iff f ⁻¹ (cl (B)) ⊂αm - cl (f ⁻¹ (B)) for every subset B of Y .

Proof:

Suppose thst f is quasi αm -open. For any subset B of Y , f ⁻¹ (B) ⊂αm -cl (f ⁻¹ (B)). Therefore by Theorem 2.4, there existsa closed set F in Y such that B⊂F and f ⁻¹ (F ) ⊂αm -cl (f ⁻¹ (B)). Therefore, we obtain f ⁻¹ (cl (B)) ⊂ f ⁻¹ (F ) ⊂ αm -cl (f ⁻¹ (B)).

Conversely, let B⊂Y and F be a αm -closed set of X containing f ⁻¹ (B).Put W = cl_Y (B), then we have B ⊂ W and W is closed set and f ⁻¹ (W ) ⊂ αm -cl (f ⁻¹ (B)) ⊂ F . Then by Theorem 2.4, f is quasi αm -open.

Lemma 2.7 Let f : X → Y and g : Y → Z be two functions and g ◦f : X → Z is quasi αm -open. If g is continuous injective, then f is quasi αm -open.

Proof. Let U be a αm -open set in X, then (g ◦ f )(U ) is open in Z. Since g ◦ f is quasi αm -open. Again g is an injective continuous function, f (U ) = g⁻¹(g ◦ f (U ))) is open in Y . This shows that f is quasi αm -open.

3. Quasi αm -closed Functions

Definition3.1 A function f : X → Y is said to be quasi αm -closed if the image of each αm -closed set in X is closed in Y .

Clearly, every quasi αm -closed function is closed as well as αm -closed.

Remark 3.2 Every αm -closed (resp. closed) function need not be quasi αm-closed as shown by the following example.

Example 3.3 Let X = Y = {a, b, c }, τ = {X, φ, {a, b}} and σ = {Y, φ, {a} , {b, c}}. Define a function f : (X, τ ) → (Y, σ) by f (a) = b, f (b) = c, and f (c) = a. Then clearly f is αm -closed as well as closed but not quasi αm -closed.

Lemma 3.4 If a function f : X → Y is quasi αm-closed, then f ⁻¹ (αm-int (f ⁻¹ (B)) for every subset B of Y .

Proof:

This proof is smilar to the proof of Lemma 2.3.

Theorem 3.5 A function f : X → Y is quasi αm-closed iff for any subset B of Y and for any αm-open set G of X containing f ⁻¹ (B), there exists an open set U of Y containing B such that f ⁻¹ (U ) ⊂ G.

Proof. This proof is similar to that Theorem 2.4

Definition 3.6 A function f : X → Y is called αm -closed if the image of αm-closed subset of X is αm -closed set in Y .

Theorem 3.7 If f : X → Y and g : Y → Z are two quasi αm-closed function, then g ◦ f is quasi αm-closed function.

Proof. Obvious.

Furthermore, we have the following Theorem:2.5

Theorem 3.8 Let f : X → Y and g : Y → Z be any two functions. Then

(i) If f is αm-closed and g is quasi αm-closed, then g ◦ f is closed.

(ii) If f is quasi αm-closed and g is αm-closed, then g◦ f is αm -closed.

(iii) If f is αm-closed and g is quasi αm -closed, then g◦ f is quasi αm-closed.

Proof. Obvious.

Theorem 3.9 Let f : X → Y and g : Y → Z be two functions such that g ◦ f : X → Z is quasi αm-closed.

(i) If f is αm -irresolute surjective, then is α closed.

(ii) If g is αm-continuous injective, then f is αm-closed.

Proof:

(i) Suppose that F is an arbitrary closed set in Y . As f is αm -irresolute, f ⁻¹ (F ) is αm-closed in X. Since g ◦ f is quasi αm -closed and f is surjective, g ◦f (f ⁻¹ (F )) = g (F ) which is closed in Z. This implies g is a closed function.

(ii) Suppose F is any αm-closed set in X. Since g ◦ f is quasi αm -closed, (g ◦ f ) (F ) is closed in Z. Again g is a αm -continuous injective function, g⁻¹ (g ◦ f (F )) = f (F ), which is αm -closed in Y . This shows that f is αm -closed.

Theorem 3.10 Let X and Y be topological spaces. Then the function f : X → Y is a quasiαm-closed if and only if f (X) is closed in Y and f (V ) − f (X − V ) is open in f (X) whenever V is αm-open in X.

Proof. Necessity: Suppose f : X → Y is a quasi αm-closed function. Since

X is αm-closed, f (X) is closed in Y and f (V ) − f (X − V ) = f (V ) ∩ f (X) − f (X − V ) is open in f (X) when V is αm-open in X.

Sufficiency: Suppose f (X) is closed in Y , f (V ) − f (X − V ) is open in f (X) when V is αm-open in X and let C be closed in X. Then f (C) = f (X) − (f (C − X) − f (C)) is closed in f (X) and hence closed in Y .

Corollary 3.11 Let X and Y be topological spaces. Then a surjective function f : X → Y is quasi αm-closed if and only if f (V ) − f (X − V ) is open in Y whenever U is αm-open in X.

Proof. It is obvious.

Theorem 3.12 Let X and Y be topological spaces and let f : X → Y be αm-continuous and quasi αm-closed surjective function. Then the topology on Y is {f (V ) − f (X − V ) : V is αm-open in X}.

Proof. Let W be open in Y . Let f ^{− 1}(W ) is αm-open in X, and f (f ⁻¹(W )) − f (X −f ⁻¹(W )) = W . Hence all open sets an Y are of the form f (V )−f (X −V ),

V is αm-open in X. On the other hand, all sets of the form f (V ) − f (X − V ). V is αm-open in X, are open in Y from corollary 3.11.

Definition 3.13 A topological space (X, τ ) is said to be αm-normal if for any pair of disjoint αm-closed subsets F₁ and F₂ of X, there exists disjoint open sets U and V such that F₁ ⊂ U and F₂ ⊂ V .

Theorem 3.14 Let X and Y be topological spaces with X is αm-normal. If f : X → Y is αm-continuous and quasi αm-closed surjective function. Then

Y is normal.

Proof. Let K and M be disjoint closed subsets of Y . Then f ⁻¹(K), f ⁻¹(M ) are disjoint αm-closed subsets of X. Since X is αm-normal, there exists disjoint open sets V and W such that f ⁻¹(K) ⊂ V , f ⁻¹(M ) ⊂ W . Then K ⊂ f (V ) − f (X − V ) and M ⊂ f (W ) − f (X − W ), further by corollary 3.11, f (V ) − f (X − V ) and f (W ) − f (X − W ) are open sets in Y and clearly (f (V ) − f (X − V )) ∩ (f (W ) − f (X − W )) = φ. This shows that Y is normal.

4. Conclusions

The author in this chapter, we define what a topological space is, and we study a number of ways of constructing a topology on a set so as to make it into a topological space. Open and closed sets, limit points, and continuous functions and corresponding ideas for the real line and topological spaces.