Page 1

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 7, Issue 5 (Jul. - Aug. 2013), PP 38-42 www.iosrjournals.org

On Semi- -Open Sets and Semi- -Continuous Functions 1

2

R. Santhi1, M. Rameshkumar2

Department of Mathematics, NGM College, Pollachi-642 001, Tamil Nadu, India. Department of Mathematics, P. A. College of Engineering and Technology, Pollachi-642 002, Tamil Nadu, India.

Abstract: We study the concepts of semi- -open sets and semi- -continuous functions introduced in [13] and some properties of the functions. Also we introduce notion of semi- - open and semi- -closed functions. Keywords: semi- -open set, semi- -continuous function.

I.

Introduction

Ideal in topological space have been considered since 1930 by Kuratowski[9] and Vaidyanathaswamy[14]. After several decades, in 1990, Jankovic and Hamlett[6] investigated the topological ideals which is the generalization of general topology. Where as in 2010, Khan and Noiri[7] introduced and studied the concept of semi-local functions. The notion of semi-open sets and semi-continuity was first introduced and investigated by Levine [10] in 1963. Finally in 2005, Hatir and Noiri [4] introduced the notion of semi- -open sets and semi- -continuity in ideal topological spaces. Recently we introduced semi- -open sets and semi- -continuity to obtain decomposition of continuity. In this paper, we obtain several characterizations of semi- -open sets and semi- -continuous functions. Also we introduce new functions semi- -open and semi- -closed functions

II.

Preliminaries

Let (X, Ď„) be a topological space with no separation properties assumed. For a subset A of a topological space (X, Ď„), cl(A) and int(A) denote the closure and interior of A in (X, Ď„) respectively. An ideal on a topological space (X, Ď„) is an nonempty collection of subsets of X which satisfies: (1) A ďƒŽ and B ďƒ? A implies B ďƒŽ (2) A ďƒŽ and B ďƒŽ implies A âˆŞ B ďƒŽ . If (X, Ď„) is a topological space and is an ideal on X, then (X, Ď„, ) is called an ideal topological space or an ideal space. Let P(X) be the power set of X. Then the operator ( ) : P(X) ď‚Ž P(X) called a local function [9] of A with respect to ď ´ and , is defined as follows: for A ďƒ? X, A*( , ď ´) = { x ďƒŽ X / U ďƒ‡ A ďƒ? for every open set U containing x}. We simply write A* instead of A*( , ď ´) in case there is no confusion. For every ideal topological space (X, ď ´, ) there exists topology ď ´* finer than ď ´, generated by ď ˘( , ď ´) = { U \ J: U ďƒŽ ď ´ and J ďƒŽ }but in general ď ˘( , ď ´) is not always a topology. Additionally cl *(A) = A ďƒˆ A* defines Kuratowski closure operator for a topology ď ´* finer than ď ´. Throughout this paper X denotes the ideal topological space (X, ď ´, ) and also int*(A) denotes the interior of A with respect to ď ´*. Definition 2.1. Let (X, Ď„) be a topological space. A subset A of X is said to be semi-open [10] if there exists an open set U in X such that U ďƒ? A ďƒ? cl(U). The complement of a semi-open set is said to be semi-closed. The collection of all semi-open (resp. semi-closed) sets in X is denoted by SO(X) (resp. SC(X)). The semi-closure of A in (X, Ď„) is denoted by the intersection of all semi-closed sets containing A and is denoted by scl(A). Definition 2.2. For A ďƒ? X, đ??´âˆ—( , ď ´) = { x ďƒŽ X / U ďƒ‡ A ďƒ? for every U ďƒŽ SO(X)} is called the semi-local function[7] of A with respect to and ď ´, where SO(X, x) = { U ďƒŽ SO(X) / x ďƒŽ U}. We simply write A instead of A ( , ď ´) in this case there is no ambiguity. It is given in [2] that ď ´ď€Şs( ) is a topology on X, generated by the sub basis { U – E : U ďƒŽ SO(X) and E ďƒŽ } or equivalently ď ´ď€Şs( ) = {U ďƒ? X: cls (X – U) = X – U }. The closure operator cls for a topology ď ´ď€Şs( ) is defined as follows: for A ďƒ? X, cls(A) = A ďƒˆ A and ints denotes the interior of the set A in ( X, ď ´ď€Şs, ). It is known that ď ´ ďƒ? ď ´ď€Ş ( ) ďƒ?ď ´ď€Şs( ). A subset A of (X, ď ´, ) is called semi--perfect[8] if A = A. A ďƒ? (X, ď ´, ) is called -semi dense in-itself [8] (resp. Semi--closed [8] ) if A ďƒŒ A (resp. A ďƒ? A). Lemma 2.3. [7] Let (X, ď ´, ) be an ideal topological space and A, B subsets of X. Then for the semi-local function the following properties hold: www.iosrjournals.org

38 | Page


On Semi- -Open Sets and Semi- -Continuous Functions 1. 2. 3. 4. 5. 6.

If A ďƒ? B then A ďƒ? B. If U ďƒŽ ď ´ then U ďƒ‡ A ďƒ? ( U ďƒ‡ A) A = scl(A ) ďƒ? scl(A) and A is semi-closed in X. (A ) ďƒ? A (A ďƒˆ B) = A ďƒˆ B If = {ď Ś}, then A = scl(A).

Definition 2.4. A subset A of a topological space X is said to be 1. ď Ą-open [12] if A ďƒ? int(cl(int(A))), 2. pre-open [11] if A ďƒ? int(cl(A)), 3. ď ˘-open[1] if A ďƒ? cl(int(cl(A))). Definition 2.5. A subset A of an ideal topological space (X, ď ´, ) is said to be 1. ď Ą- -open[4] if A ďƒ? int(cl (int(A))), 2. semi- -open [4] if A ďƒ? cl (intA)), 3. pre- -open [3] if A ďƒ? int(cl(A)). Definition2.6. A subset A of an ideal topological space (X, ď ´, ) is said to be 1. ď Ą- -open[13] if A ďƒ? int(cls(int(A))), 2. semi- -open [13] if A ďƒ? cls(intA)), 3. pre- -open [13] if A ďƒ? int(cls(A)). Remark 2.7. In [13], the authors obtained the following diagram:

By SISO(X,ď ´), we denote the family of all semi- -open sets of a space (X, ď ´, ).

III.

Semi- -open sets

Theorem3.1. A subset A of a space (X, ď ´, ) is semi- -open if and only if cls(A) = cls(int(A)). Proof. Let A be semi- -open, we have A ďƒ? cls(int(A)). Then cls(A) ďƒ? cls(int(A)). Obviously cls(int(A)) ďƒ? cls(A). Hence cls(A) = cls(int(A)). The converse is obvious. Theorem3.2. A subset A of a space (X, ď ´, ) is semi- -open if and only if there exists U ďƒŽ ď ´ such that U ďƒ? A ďƒ? cls(U). Proof. Let A be semi- -open, we have A ďƒ? cls(int(A)). Take int(A) = U. Then we have U ďƒ? A ďƒ? cls(U). Conversely, let U ďƒ? A ďƒ? cls(U) for some U ďƒŽ ď ´. Since U ďƒ? A we have U ďƒ? int(A) and hence cls(U) ďƒ? cls(int(A)). Thus we obtain A ďƒ? cls(int(A)). Theorem3.3. If A is semi- -open set in a space (X, ď ´, ) and A ďƒ? B ďƒ? cls(A), then B is semi- -open in (X, ď ´, ). Proof. Since A is semi- -open, there exists an open set U such that U ďƒ? A ďƒ? cls(U). Then we have U ďƒ? A ďƒ? B ďƒ? cls(A) ďƒ? cls(cls(U)) = cls(U) and hence U ďƒ? B ďƒ? cls(A). By Theorem 3.2, we obtain B is semi- -open. Theorem3.4. Let (X, ď ´, ) be an ideal topological space and A, B subsets of X. 1. If Uď Ą ďƒŽ SISO(X, ď ´) for each ď Ą ďƒŽ ď „, then ďƒˆ{Uď Ą : ď Ą ďƒŽ ď „} ďƒŽ SISO(X, ď ´), 2. If A ďƒŽ SISO(X, ď ´) and B ďƒŽ ď ´, then A ďƒ‡ B ďƒŽ SISO(X, ď ´). Proof. 1. Since Uď Ą ďƒŽ SISO(X, ď ´), we have Uď Ą ďƒ? cls(int(Uď Ą)) for each ď Ą ďƒŽ ď „. Thus by using Lemma 2.3, we obtain cl∗s (int(UÎą )) ďƒ?

UÎą ďƒ? đ?›źâˆˆâˆ†

ď ĄďƒŽď „

(int(UÎą ))∗ ďƒˆ(int(UÎą )) ďƒ? đ?›ź ∈∆

www.iosrjournals.org

ďƒˆ int

(int(UÎą )) đ?›źâˆˆâˆ†

∗

UÎą Îąâˆˆâˆ†

39 | Page


On Semi- -Open Sets and Semi- -Continuous Functions ďƒ?

int Îąâˆˆâˆ†

UÎą = cl∗s int

ďƒˆ int

UÎą

Îąâˆˆâˆ†

∗

UÎą

.

Îąâˆˆâˆ†

This shows that Îąâˆˆâˆ† UÎą ďƒŽ SISO(X, ď ´). 2. Let A ďƒŽ SISO(X, ď ´) and B ďƒŽ ď ´. Then A ďƒ? cls(int(A)) and by using Lemma 2.3, we have A ďƒ‡ B ďƒ? cls(int(A)) ďƒ‡ B = ((int(A)) ďƒˆ (int(A))) ďƒ‡ B = ((int(A)) ďƒ‡ B) ďƒˆ (int(A) ďƒ‡ B) ďƒ? (int(A) ďƒ‡ B) ďƒˆ int(A ďƒ‡ B) = (int(A ďƒ‡ B)) ďƒˆ int(A ďƒ‡ B) = cls(int(A ďƒ‡ B)). This shows that A ďƒ‡ B ďƒŽ SISO(X, ď ´). Definition3.5. A subset F of a space (X, ď ´, ) is said to be semi- -closed if its complement is semi- -open. Remark3.6. For a subset A of a space (X, ď ´, ), we have X− int(cls(A)) ď‚š cls(int(X − A)) as shown from the following example. Example3.7. Let X = {a, b, c, d}, ď ´ = {ď Ś, X, {a}, {b}, {a, b}} and = {ď Ś, {b}, {c}, {b, c}}. Then we put A = {b}, we have cls(int(X − A)) = cls({a}) = {a} and X − int(cls(A)) = X − int({b}) = {a, c, d}. Theorem 3.8. If a subset A of a space (X, ď ´, ) is semi- -closed, then int(cls(A)) ďƒ? A. Proof. Since A is semi- -closed, X − A ďƒŽ SISO(X, ď ´). Since ď ´ď€Şs(I) is finer than ď ´, we have X − A ďƒ? cls(int(X − A)) ďƒ? cl(int(X − A)) = X − int(cl(A)) ďƒ? X − int(cls(A)). Therefore we obtain int(cls(A)) ďƒ? A. Corollary3.9. Let A be a subset of a space (X, ď ´, ) such that X − int(cls(A)) = cls(int(X − A)). Then A is semi- -closed if and only if int(cls(A)) ďƒ? A. Proof. This is an immediate consequence of Theorem 3.8. Theorem3.10. [8] Let (X, ď ´, A( , ď ´|Y) = A( , ď ´) ďƒ‡ Y.

) be an ideal space and A ďƒ? Y ďƒ? X, where Y is ď Ą-open in X. Then

Theorem3.11. Let (X, ď ´, ) be an ideal topological space. If Y ďƒŽ ď ´ and W ďƒŽ SISO(X), then Y ďƒ‡ W ďƒŽ SISO(Y, ď ´|Y, ). Proof. Since Y is open, we have intY(A) = int(A) for any subset A of Y. Now Y ďƒ‡ W ďƒ? Y ďƒ‡ cls(int(W)) = Y ďƒ‡ ((int(W)) ďƒˆ int(W)) = [(Y ďƒ‡ (int(W)) ) ďƒˆ (Y ďƒ‡ int(W))] ďƒ‡ Y = [Y ďƒ‡ (Y ďƒ‡ (int(W)))] ďƒˆ [(Y ďƒ‡ int(W)) ďƒ‡ Y] = Y ďƒ‡ [intY (Y ďƒ‡W)] ďƒˆ (Y ďƒ‡ [intY (Y ďƒ‡ W)]) = [intY (Y ďƒ‡W)] ( , ď ´|Y) ďƒˆ [intY (Y ďƒ‡ W)] = cl∗s Y (intY(Y ďƒ‡ W)). This shows that Y ďƒ‡ W ďƒŽ SISO(Y, ď ´|Y, ).

IV.

Semi-

-continuous functions

Definition4.1. A function f : (X, ď ´, ) ď‚Ž (Y, ď ł) is said to be semi- -continuous [13] ( resp. semi- -continuous [4], semi-continuous [10]) if đ?‘“ −1 (V) is semi- -open (resp. semi- -open, semi-open) in (X, ď ´, ) for each open set V of (Y, ď ł). Definition4.2. A function f : (X, ď ´, ) ď‚Ž (Y, ď ł, ) is said to be -irresolute ( resp. -irresolute [5]) if đ?‘“ −1 (V) is semi- -open (resp. semi- -open) in (X, ď ´, ) for each semi- -open set(resp. semi- -open set) V of (Y, ď ł, ). Remark4.3. It is obvious that continuity implies semi- -continuity, semi- -continuity implies semi- continuity and semi- -continuity implies semi-continuity. Theorem 4.4. For a function f : (X, ď ´, ) ď‚Ž (Y, ď ł), the following are equivalent: 1. f is semi- -continuous, 2. for each x ďƒŽ X and each V ďƒŽ ď ł containing f (x), there exists W ďƒŽ SISO(X, ď ´) containing x such that f (W) ďƒ? V, 3. the inverse image of each closed set in Y is semi- -closed. Proof. (1) ďƒž (2). Let x ďƒŽ X and V be any open set of Y containing f (x). Set W = đ?‘“ −1 (V), then by Definition 4.1, W is a semi- -open set containing x and f (W) ďƒ? V. (2) ďƒž (3). Let F be a closed set of Y. Set V = Y − F, then V is open in Y. Let x ďƒŽ đ?‘“ −1 (V), by (2), there exists a semi- -open set W of X containing x such that f (W) ďƒ? V. Thus, we obtain x ďƒŽ W ďƒ? cls(int(W)) ďƒ? www.iosrjournals.org

40 | Page


On Semi- -Open Sets and Semi- -Continuous Functions cls(int(đ?‘“ −1 (V))) and hence đ?‘“ −1 (V) ďƒ? cls(int(đ?‘“ −1 (V))). This shows that đ?‘“ −1 (V) is semi- -open in X. Hence đ?‘“ −1 (F) = X − đ?‘“ −1 (Y − F) = X − đ?‘“ −1 (V) is semi- -closed in X. (3) ďƒž (1). Let V be a open set of Y. Set F = Y − V, then F is closed in Y. đ?‘“ −1 (V) = X − đ?‘“ −1 (Y − V) = X − đ?‘“ −1 (F). By (3) đ?‘“ −1 (V) is semi- -open in X. Theorem 4.5. Let f : (X, ď ´, ) ď‚Ž (Y, ď ł), be semi- -continuous and U ďƒŽ ď ´. Then the restriction f |U : (U, ď ´|U, ) ď‚Ž (Y, ď ł) is semi- -continuous. Proof. Let V be any open set of (Y, ď ł). Since f is semi- -continuous, đ?‘“ −1 (V) ďƒŽ SISO(X, ď ´) and by Theorem 3.11, (f |U)−1(V) = đ?‘“ −1 (V) ďƒ‡ U ďƒŽ SISO(U, ď ´|U). This shows that f |U: (U, ď ´|U, ) ď‚Ž (Y, ď ł) is semi- -continuous. Theorem 4.6. For function f : (X, ď ´, ) ď‚Ž (Y, ď ł, ) and g : (Y, ď ł, ) ď‚Ž (Z, ď ¨), the following hold. 1. g f is semi- -continuous if f is semi- -continuous and g is continuous. 2. g f is semi- -continuous if f is -irresolute and g is semi- -continuous. Proof. It is Obvious. Theorem 4.7. A function f : (X, ď ´, ) ď‚Ž (Y, ď ł) is semi- -continuous if and only if the graph function g : X ď‚Ž X Ă— Y, defined by g(x) = (x, f (X)) for each x ďƒŽ X, is semi- -continuous. Proof. Necessity. Suppose that f is semi- -continuous. Let x ďƒŽ X and W be any open set of X Ă— Y containing g(x). Then there exists a basic open set U Ă— V such that g(x) = (x, f (x)) ďƒŽ U Ă— V ďƒ? W. Since f is semi- continuous, then there exists a semi- -open set U of X containing x such that f (U ) ďƒ? V. By Theorem 3.4 U ďƒ‡ U ďƒŽ SISO(X, ď ´) and g(U ďƒ‡ U) ďƒ? U Ă— V ďƒ? W. This shows that g is semi- -continuous. Sufficiency: Suppose that g is semi- -continuous. Let x ďƒŽ X and V be any open set of Y containing f (x). Then X Ă— V is open in X Ă— Y and by semi- -continuity of g, there exists U ďƒŽ SISO(X, ď ´) containing x such that g(U) ďƒ? X Ă— V. Therefore we obtain f (U) ďƒ? V. This shows that f is semi- -continuous. Theorem 4.8. Let f : (X, ď ´, ) ď‚Ž (Y, ď ł, ) be semi- -continuous and đ?‘“ −1 (V*) ďƒ? (đ?‘“ −1 (V))* for each V ďƒŽ ď ł. Then f is -irresolute. Proof. Let B be any semi- -open set of (Y, ď ł, ). By Theorem 3.2, there exists V ďƒŽ ď ł such that V ďƒ? B ďƒ? cls(V). Therefore, we have đ?‘“ −1 (V) ďƒ? đ?‘“ −1 (B) ďƒ? đ?‘“ −1 (cls(V)) ďƒ? cls(đ?‘“ −1 (V)). Since f is semi- continuous and VďƒŽ ď ł, đ?‘“ −1 (V) ďƒŽ SISO(X, ď ´) and hence by Theorem 3.3, đ?‘“ −1 (B) is semi- -open in (X, ď ´, ). This shows that f is -irresolute.

V.

Semi-

-open and semi-

-closed functions

Definition 5.1. A function f : (X, ď ´) ď‚Ž (Y, ď ł, ) is called semi- -open (resp. semi- -closed) if for each U ďƒŽ ď ´ (resp. U is closed) f (U) ďƒŽ SISO(Y, ď ł, ) (resp. f (U) is semi- -closed set). Definition 5.2. [5] A function f : (X, ď ´) ď‚Ž (Y, ď ł, ) is called semi- -open (resp. semi- -closed) if for each U ďƒŽ ď ´ (resp. U is closed) f (U) is semi- -open (resp. f (U) is semi- -closed) set in (Y, ď ł, ). Remark 5.3. 1. Every semi- -open (resp. semi- -closed) function is semi-open (resp. semi-closed) and the converses are false in general. 2. Every semi- -open (resp. semi- -closed) function is semi- -open (resp. semi- -closed) and the converses are false in general. 3. Every open function is semi- -open but the converse is not true in general. Example 5.4. Let X = {a, b, c, d}, ď ´ = {ď Ś, X, {a, b}}, ď ł = {ď Ś, X, {a}, {b}, {a, b}} and = {ď Ś, {b}, {c}, {b, c}}. Define a function f : (X, ď ´) ď‚Ž (Y, ď ł, ) as follows f (a) = b, f (b) = c, f (c) = f (d) = a. Then f is semi-open, but it is not semi- -open. Example 5.5. Let X = {a, b, c, d}, ď ´ = {ď Ś, X, {a, b}}, ď ł = {ď Ś, X, {a}, {b}, {a, b}} and = {ď Ś, {b}, {c}, {b, c}}. Define a function f : (X, ď ´) ď‚Ž (X, ď ł, ) as follows f (a) = a, f (b) = c, f (c) = f (d) = d. Then f is semi- open, but it is not semi- -open.

www.iosrjournals.org

41 | Page


On Semi- -Open Sets and Semi- -Continuous Functions Example 5.6. Let X = {a, b, c, d}, ď ´ = {ď Ś, X, {c}, {a, b, d}}, ď ł = {ď Ś, X, {c}, {a, b}, {a, b, c}} and = {ď Ś, {a}}. The identity function f : (X, ď ´) ď‚Ž (Y, ď ł, ) is semi- -open, but it is not open. Example 5.7. Let X = {a, b, c, d}, ď ´ = {ď Ś, X, {a, b}}, ď ł = {ď Ś, X, {a}, {b}, {a, b}} and = {ď Ś, {b}, {c}, {b, c}}. The identity function f : (X, ď ´) ď‚Ž (Y, ď ł, ) is semi-closed, but it is not semi- -closed. Theorem 5.8. A function f : (X, ď ´) ď‚Ž (Y, ď ł, ) is semi- -open if and only if for each x ďƒŽ X and each neighbourhood U of x, there exists V ďƒŽ SISO(Y, ď ł) containing f(x) such that V ďƒ? f (U). Proof. Suppose that f is a semi- -open function. For each x ďƒŽ X and each neighbourhood U of x, there exists U ďƒŽ ď ´ such that x ďƒŽU ďƒ? U. Since f is semi- -open, V = f (U ) ďƒŽ SISO(Y, ď ł) and f (x) ďƒŽ V ďƒ? f (U). Conversely, let U be an open set of (X, ď ´) . For each x ďƒŽ U, there exists Vx ďƒŽ SISO(Y, ď ł) such that f (x) ďƒŽ Vx ďƒ? f (U). Therefore we obtain f (U) = ďƒˆ{Vx : x ďƒŽ U} and hence by Theorem 3.4, f (U) ďƒŽ SISO(Y, ď ł). This shows that f is semi- -open. Theorem 5.9. Let f : (X, ď ´) ď‚Ž (Y, ď ł, ) be a semi- -open (resp. semi- -closed) function. If W is any subset of Y and F is a closed (resp. open) set of X containing đ?‘“ −1 (W), then there exists a semi- -closed (resp. semi- open) subset H of Y containing W such that đ?‘“ −1 (H) ďƒ? F. Proof. Suppose that f is a semi- -open function. Let W be any subset of Y and F a closed subset of X containing đ?‘“ −1 (W). Then X − F is open and since f is semi- -open, f (X  F) is semi- -open. Hence H = Y – f (X − F) is semi- -closed. It follows from đ?‘“ −1 (W) ďƒ? F that W ďƒ? H. Moreover we obtain đ?‘“ −1 (H) ďƒ? F. For a semi- -closed function can be proved similarly. Theorem 5.10. For any bijective function f : (X, ď ´) ď‚Ž (Y, ď ł, ), the following are equivalent: 1. đ?‘“ −1 : (X, ď ł, ) ď‚Ž (X, ď ´ ) is semi- -continuous, 2. f is semi- -open, 3. f is semi- -closed, Proof. Obvious.

References

[1]. [2]. [3]. [4]. [5]. [6]. [7]. [8]. [9]. [10]. [11]. [12]. [13]. [14].

M. E. Abd El-Monsef, S. N. El Deep and R. A. Mahmoud, ď ˘-open sets and ď ˘-continuous mappings, Bull. Fac. Sci. Assiut Univ., 12 (1983), 77-90. M.E. Abd El-Monsef, E.F. Lashien and A.A. Nasef, Some topological operators via ideals, Kyungpook Math. J., 32, No. 2 (1992), 273-284. J. Dontchev, Idealization of Ganster-Reilly decomposition theorems, http://arxiv.org/abs/ Math. GN/9901017, 5 Jan. 1999(Internet). E. Hatir and T.Noiri, On decompositions of continuity via idealization, Acta. Math. Hungar. 96(4)(2002), 341-349. E. Hatir and T.Noiri, On semi- -open sets and semi- -continuous functions, Acta. Math. Hungar. 107(4)(2005), 345-353. D. Jankovic and T. R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97(4) (1990), 295-310. M. Khan and T. Noiri, Semi-local functions in ideal topological spaces, J. Adv. Res. Pure Math., 2(1) (2010), 36-42. M. Khan and T. Noiri, On gI -closed sets in ideal topological space, J. Adv. Stud. in Top., 1(2010),29-33. K. Kuratowski. Topology, Vol. I, Academic press, New York, 1966. N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36-41. A. S. Mashour, M. E. Abd. El-Monsef and S. N. El-deeb, On pre-continuous and weak pre-continuous mapping, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47-53. O. Njastad, On some classes of nearly open sets, Pacific J. Math., 15 (1965), 961-970. R. Santhi and M. Rameshkumar, A decomposition of continuity in ideal by using semi-local functions, (Submitted). R. Vaidyanathaswamy, Set Topology, Chelsea Publishing Company, 1960.

www.iosrjournals.org

42 | Page

F0753842  
Read more
Read more
Similar to
Popular now
Just for you