# On quasi sg-open and quasi sg-closed functions.

1. Introduction and Preliminaries

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 considers openness. Its importance is significant in various areas of mathematics and related sciences.

As a generalization of closed sets, the notion of sg-closed sets were introduced and studied by Bhattacharyya and Lahiri . In this paper, we will continue the study of related functions by involving sg-open sets. We introduce and characterize the concept of quasi sg-open functions.

Throughout this paper, spaces mean topological spaces on which no separation axioms are assumed unless otherwise mentioned and f: (X, [tau]) [right arrow] (Y, [sigma]) (or simply f : X [right arrow] Y) denotes a function f of a space (X, [tau]) into a space (Y, [sigma]). Let A be a subset of a space X. The closure and the interior of A are denoted by Cl(A) and Int(A), respectively.

Definition 1.1. A subset A of a space (X, [tau]) is called semi-open  if A [subset] Cl(Int(A)).

The complement of a semi-open set is called semi-closed.

The semi-closure  of a subset A of X, denoted by sCl(A), is defined to be the intersection of all semi-closed sets containing A in X.

Definition 1.2. A subset A of a space X is called:

sg-closed  if sCl(A) [subset] U whenever A [subset] U and U is semi-open in X.

The complement of sg-closed set is called sg-open.

The union of all sg-open sets, each contained in a set A in a space X is called the sg-interior of A and is denoted by sg-Int(A) .

The intersection of all sg-closed sets containing a set A in a space X is called the sg-closure of A and is denoted by sg-Cl(A) .

Definition 1.3. A function f: (X, [tau]) [right arrow] (Y, [sigma]) is called:

(i) sg-irresolute  (resp. sg-continuous ) if [f.sup.-1](V) is sg-closed in X for every sg-closed (resp. closed) subset V of Y ;

(ii) sg-open (resp. sg-closed ) if f (V) is sg-open (resp. sg-closed) in Y for every open (resp. closed) subset of X.

2. Quasi sg-open Functions

We introduce a new definition as follows:

Definition 2.1. A function f: X [right arrow] Y is said to be quasi sg-open if the image of every sg-open set in X is open in Y.

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

Theorem 2.1. A function f: X [right arrow] Y is quasi sg-open if and only if for every subset U of X, f (sg-Int(U)) [subset] Int(f (U)).

Proof. Let f be a quasi sg-open function. Now, we have Int(U) [subset] U and sg- Int(U) is a sg-open set. Hence, we obtain that f(sg-Int(U)) [subset] f (U). As f (sg- Int(U)) is open, f (sg-Int(U)) [subset] Int(f (U)). Conversely, assume that U is a sg-open set in X. Then, f (u) = f (sg-Int(U)) [subset] Int(f (U)) but Int(f (U)) [subset] f(U). Consequently, f (U) = Int(f (U)) and hence f is quasi sg-open.

Lemma 2.1. If a function f: X [right arrow] Y is quasi sg-open, then sg- Int([f.sup.-1](G)) [subset] [f.sup.-1](Int(G)) for every subset G of Y .

Proof. Let G be any arbitrary subset of Y. Then, sg-Int([f.sup.-1](G)) is a sg- open set in X and f is quasi sg-open, then f (sg-Int([f.sup.-1](G))) [subset] Int(f ([f.sup.- 1](G))) [subset] Int(G). Thus, sg-Int([f.sup.-1](G)) [subset] [f.sup.-1](Int(G)).

Recall that a subset S is called a sg-neighbourhood  of a point x of X if there exists a sg-open set U such that x [member of] U [subset] S.

Theorem 2.2. For a function f: X [right arrow] Y, the following are equivalent:

(i) f is quasi sg-open;

(ii) For each subset U of X, f (sg-Int(U)) [subset] Int(f (U));

(iii) For each x [member of] X and each sg-neighbourhood U of x in X, there exists a neighbourhood f(U) of f (x) in Y such that V [subset] f (U).

Proof. (i) [??] (ii): It follows from Theorem 2.1.

(ii) [??] (iii): Let x [member of] X and U be an arbitrary sg-neighbourhood of x in X. Then there exists a sg-open set V in X such that x [member of] V [subset] U. Then by (ii), we have f (V) = f (sg-Int(V)) [subset] Int(f (V)) and hence f (V) = Int(f(V)). Therefore, it follows that f (V) is open in Y such that f (x) e f (V) [subset] f (u).

(iii) [??] (i): Let U be an arbitrary sg-open set in X. Then for each y [member of] f (U), by (iii) there exists a neighbourhood Vy of y in Y such that Vy [subset] f (U). As Vy is a neighbourhood of y, there exists an open set Wy in Y such that y [member of] Wy [subset] Vy . Thus f (U) = [union]{Wy: y [member of] f (U)} which is an open set in Y. This implies that f is quasi sg-open function.

Theorem 2.3. A function f: X [right arrow] Y is quasi sg-open if and only if for any subset B of Y and for any sg-closed set F of X containing [f.sup.-1](B), there exists a closed set G of Y containing B such that [f.sup.-1] (G) [subset] F.

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

Conversely, let U be a sg-open set of X and put B = Y\f (U). Then X\U is a sg- closed set in X containing [f.sup.-1](B). By hypothesis, there exists a closed set F of Y such that B [subset] F and [f.sup.-1](F) [subset] X\U. Hence, we obtain f (U) [subset] Y\F. On the other hand, it follows that B [subset] F, Y\F [subset] Y\B = f (U). Thus, we obtain f (U) = Y\F which is open and hence f is a quasi sg-open function.

Theorem 2.4. A function f: X [right arrow] Y is quasi sg-open if and only if [f.sup.-1](Cl(B)) [subset] sgCl([f.sup.-1](B)) for every subset B of Y .

Proof. Suppose that f is quasi sg-open. For any subset B of Y, [f.sup.-1](B) [subset] sg-Cl([f.sup.-1](B)). Therefore by Theorem 2.3, there exists a closed set F in Y such that B [subset] F and [f.sup.-1](F) [subset] sg-Cl([f.sup.-1](B)). Therefore, we obtain [f.sup.-1](Cl(B)) [subset] [f.sup.-1](F) [subset] sg-Cl([f.sup.-1](B)).

Conversely, let B [subset] Y and F be a sg-closed of X containing [f.sup.-1](B). Put W = [CI.sub.Y] (B), then we have B [subset] W and W is closed and f1(W) [subset] sg-Cl ([f.sup.-1]B)) [subset] F. Then by Theorem 2.3, f is quasi sg-open.

Lemma 2.2. Let f: X [right arrow] Y and g: Y [right arrow] Z be two functions and g o f: X [right arrow] Z is quasi sg-open. If g is continuous injective, then f is quasi sg-open.

Proof. Let U be a sg-open set in X. Then (g o f)(U) is open in Z since g o f is quasi sg-open. Again g is an injective continuous function, f (U) = [g.sup.-1] (g o f (U)) is open in Y. This shows that f is quasi sg-open.

3. Quasi sg-closed Functions

Definition 3.1. A function f: X [right arrow] Y is said to be quasi sg-closed if the image of each sg-closed set in X is closed in Y.

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

Remark 3.1. Every sg-closed (resp. closed) function need not be quasi sg-closed as shown by the following example.

Example 3.1. Let X = Y = {a, b, c}, [tau] = {[phi], {a, b}, X} and [sigma] = {9, {a}, {b, c}, Y}. Define a function f: (X, [tau]) -- (Y, [sigma]) by f (a) = b, f (b) = [subset] and f (c) = a. Then clearly f is sg-closed as well as closed but not quasi sg-closed.

Lemma 3.1. If a function f: X [right arrow] Y is quasi sg-closed, then ([f.sup.- 1](Int(B)) [subset] sg-Int ([f.sup.-1](B)) for every subset B of Y .

Proof. This proof is similar to the proof of Lemma 2.1.

Theorem 3.1. A function f: X [right arrow] Y is quasi sg-closed if and only if for any subset B of Y and for any sg-open set G of X containing [f.sup.-1](B), there exists an open set U of Y containing B such that [f.sup.-1](U) [subset] G.

Proof. This proof is similar to that of Theorem 2.3.

Definition 3.2. A function f: X [right arrow] Y is called sg*-closed if the image of every sg-closed subset of X is sg-closed in Y .

Theorem 3.2. If f: X [right arrow] Y and g: Y [right arrow] Z are two quasi sg- closed function, then g o f: X [right arrow] Z is a quasi sg-closed function.

Proof. Obvious.

Furthermore, we have the following theorem:

Theorem 3.3. Let f: X [right arrow] Y and g: Y [right arrow] Z be any two functions. Then:

(i) if f is sg-closed and g is quasi sg-closed, then g o f is closed;

(ii) if f is quasi sg-closed and g is sg-closed, then g o f is sg*-closed;

(iii) if f is sg*-closed and g is quasi sg-closed, then g o f is quasi sg- closed.

Proof. Obvious.

Theorem 3.4. Let f: X [right arrow] Y and g: Y [right arrow] Z be two functions such that g o f: X -- Z is quasi sg-closed. Then:

(i) if f is sg-irresolute surjective, then g is closed.

(ii) if g is sg-continuous injective, then f is sg*-closed.

Proof. (i) Suppose that F is an arbitrary closed set in Y. As f is sg- irresolute, [f.sup.-1](F) is sg-closed in X. Since g o f is quasi sg-closed and f is surjective, (g o f ([f.sup.-1](F))) = g(F), which is closed in Z. This implies that g is a closed function.

(ii) Suppose F is any sg-closed set in X. Since g o f is quasi sg-closed, (g o f)(F) is closed in Z. Again g is a sg-continuous injective function, [g.sup.-1](g o f (F)) = f (F), which is sg-closed in Y. This shows that f is sg*-closed.

Theorem 3.5. Let X and Y be topological spaces. Then the function g: X [right arrow] Y is a quasi sg-closed if and only if g(X) is closed in Y and g(V)\g(X\V) is open in g(X) whenever V is sg-open in X.

Proof. Necessity: Suppose g: X [right arrow] Y is a quasi sg-closed function. Since X is sg-closed, g(X) is closed in Y and g(V)\g(X\V) = g(V) n g(X)\g(X\V) is open in g(X) when V is sg-open in X.

Sufficiency: Suppose g(X) is closed in Y, g(V)\g(X\V) is open in g(X) when V is sg-open in X, and let C be closed in X. Then g(C) = g(X)\(g(X\C)\g(C)) is closed in g(X) and hence, closed in Y .

Corollary 3.1. Let X and Y be topological spaces. Then a surjective function g: X [right arrow] Y is quasi sg-closed if and only if g(V)\g(X \V) is open in Y whenever V is sg-open in X.

Proof. Obvious.

Corollary 3.2. Let X and Y be topological spaces and let g: X [right arrow] Y be a sg-continuous quasi sg-closed surjective function. Then the topology on Y is {g(V)\ g(X\V): V is sg-open in X}.

Proof. Let W be open in Y. Then [g.sup.-1] (W) is sg-open in X, and g([g.sup.- 1](W))\g(X\ [g.sup.-1](W)) = W. Hence, all open sets in Y are of the form g(V)\g(X\V), V is sg-open in X. On the other hand, all sets of the form g(V)\g(X\V), V is sg-open in X, are open in Y from Corollary 3.1.

Definition 3.3. A topological space (X, [tau]) is said to be sg*-normal if for any pair of disjoint sg-closed subsets F1 and F2 of X, there exist disjoint open sets U and V such that F1 [subset] U and F2 [subset] V .

Theorem 3.6. Let X and Y be topological spaces with X is sg*-normal. If g: X [right arrow] Y is a sg-continuous quasi sg-closed surjective function, then Y is normal.

Proof. Let K and M be disjoint closed subsets of Y. Then [g.sup.-1](K), [g.sup.- 1](M) are disjoint sg-closed subsets of X. Since X is sg*-normal, there exist disjoint open sets V and W such that [g.sup.-1](K) [subset] V and [g.sup.-1](M) [subset] W. Then K [subset] g(V)\g (X\V) and M c g(W)\g(X\W). Further by Corollary 3.1, g(V)\g(X\V) and g(W)\g(X\W) are open sets in Y and clearly (g(V) \g(X\V)) n (g (w)\g(X\W)) = 9. This shows that Y is normal.

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(1) O. Ravi, (2) G. Ramkumar and (3) S. Chandrasekar

(1) Department of Mathematics, P.M.Thevar College, Usilampatti, Madurai, Tamil Nadu India. E-mail: siingam@yahoo.com

(2) Department of Mathematics, Rajapalayam Rajus' College, Rajapalayam, Virudhunagar, Tamil Nadu, India

E-mail: ramanujam 1729@yahoo.com.

(3) Department of Mathematics, Muthayammal Engineering College, Rasipuram, Namakkal, Tamil Nadu, India

E-mail:chandrumat@gmail.com