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&#x201E;) be a topological space with no separation properties assumed. For a subset A of a topological space (X, Ď&#x201E;), cl(A) and int(A) denote the closure and interior of A in (X, Ď&#x201E;) respectively. An ideal on a topological space (X, Ď&#x201E;) is an nonempty collection of subsets of X which satisfies: (1) A ď&#x192;&#x17D; and B ď&#x192;? A implies B ď&#x192;&#x17D; (2) A ď&#x192;&#x17D; and B ď&#x192;&#x17D; implies A â&#x2C6;Ş B ď&#x192;&#x17D; . If (X, Ď&#x201E;) is a topological space and is an ideal on X, then (X, Ď&#x201E;, ) is called an ideal topological space or an ideal space. Let P(X) be the power set of X. Then the operator ( )ď&#x20AC;Ş : P(X) ď&#x201A;Ž P(X) called a local function [9] of A with respect to ď ´ and , is defined as follows: for A ď&#x192;? X, A*( , ď ´) = { x ď&#x192;&#x17D; X / U ď&#x192;&#x2021; A ď&#x192;? 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 ď&#x192;&#x17D; ď ´ and J ď&#x192;&#x17D; }but in general ď ˘( , ď ´) is not always a topology. Additionally cl *(A) = A ď&#x192;&#x2C6; 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&#x201E;) 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 ď&#x192;? A ď&#x192;? 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&#x201E;) is denoted by the intersection of all semi-closed sets containing A and is denoted by scl(A). Definition 2.2. For A ď&#x192;? X, đ??´â&#x2C6;&#x2014;( , ď ´) = { x ď&#x192;&#x17D; X / U ď&#x192;&#x2021; A ď&#x192;? for every U ď&#x192;&#x17D; SO(X)} is called the semi-local function[7] of A with respect to and ď ´, where SO(X, x) = { U ď&#x192;&#x17D; SO(X) / x ď&#x192;&#x17D; U}. We simply write Aď&#x20AC;Ş instead of Aď&#x20AC;Ş ( , ď ´) in this case there is no ambiguity. It is given in [2] that ď ´ď&#x20AC;Şs( ) is a topology on X, generated by the sub basis { U â&#x20AC;&#x201C; E : U ď&#x192;&#x17D; SO(X) and E ď&#x192;&#x17D; } or equivalently ď ´ď&#x20AC;Şs( ) = {U ď&#x192;? X: clď&#x20AC;Şs (X â&#x20AC;&#x201C; U) = X â&#x20AC;&#x201C; U }. The closure operator clď&#x20AC;Şs for a topology ď ´ď&#x20AC;Şs( ) is defined as follows: for A ď&#x192;? X, clď&#x20AC;Şs(A) = A ď&#x192;&#x2C6; Aď&#x20AC;Ş and intď&#x20AC;Şs denotes the interior of the set A in ( X, ď ´ď&#x20AC;Şs, ). It is known that ď ´ ď&#x192;? ď ´ď&#x20AC;Ş ( ) ď&#x192;?ď ´ď&#x20AC;Şs( ). A subset A of (X, ď ´, ) is called semi-ď&#x20AC;Ş-perfect[8] if A = Aď&#x20AC;Ş. A ď&#x192;? (X, ď ´, ) is called ď&#x20AC;Ş-semi dense in-itself [8] (resp. Semi-ď&#x20AC;Ş-closed [8] ) if A ď&#x192;&#x152; Aď&#x20AC;Ş (resp. Aď&#x20AC;Ş ď&#x192;? 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

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

If A ď&#x192;? B then Aď&#x20AC;Ş ď&#x192;? Bď&#x20AC;Ş. If U ď&#x192;&#x17D; ď ´ then U ď&#x192;&#x2021; Aď&#x20AC;Ş ď&#x192;? ( U ď&#x192;&#x2021; A)ď&#x20AC;Ş Aď&#x20AC;Ş = scl(Aď&#x20AC;Ş ) ď&#x192;? scl(A) and Aď&#x20AC;Ş is semi-closed in X. (Aď&#x20AC;Ş )ď&#x20AC;Ş ď&#x192;? Aď&#x20AC;Ş (A ď&#x192;&#x2C6; B)ď&#x20AC;Ş = Aď&#x20AC;Ş ď&#x192;&#x2C6; Bď&#x20AC;Ş If = {ď Ś}, then Aď&#x20AC;Ş = scl(A).

Definition 2.4. A subset A of a topological space X is said to be 1. ď Ą-open [12] if A ď&#x192;? int(cl(int(A))), 2. pre-open [11] if A ď&#x192;? int(cl(A)), 3. ď ˘-open[1] if A ď&#x192;? 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 ď&#x192;? int(clď&#x20AC;Ş (int(A))), 2. semi- -open [4] if A ď&#x192;? clď&#x20AC;Ş (intA)), 3. pre- -open [3] if A ď&#x192;? int(clď&#x20AC;Ş(A)). Definition2.6. A subset A of an ideal topological space (X, ď ´, ) is said to be 1. ď Ą- -open[13] if A ď&#x192;? int(clď&#x20AC;Şs(int(A))), 2. semi- -open [13] if A ď&#x192;? clď&#x20AC;Şs(intA)), 3. pre- -open [13] if A ď&#x192;? int(clď&#x20AC;Şs(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 clď&#x20AC;Şs(A) = clď&#x20AC;Şs(int(A)). Proof. Let A be semi- -open, we have A ď&#x192;? clď&#x20AC;Şs(int(A)). Then clď&#x20AC;Şs(A) ď&#x192;? clď&#x20AC;Şs(int(A)). Obviously clď&#x20AC;Şs(int(A)) ď&#x192;? clď&#x20AC;Şs(A). Hence clď&#x20AC;Şs(A) = clď&#x20AC;Şs(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 ď&#x192;&#x17D; ď ´ such that U ď&#x192;? A ď&#x192;? clď&#x20AC;Şs(U). Proof. Let A be semi- -open, we have A ď&#x192;? clď&#x20AC;Şs(int(A)). Take int(A) = U. Then we have U ď&#x192;? A ď&#x192;? clď&#x20AC;Şs(U). Conversely, let U ď&#x192;? A ď&#x192;? clď&#x20AC;Şs(U) for some U ď&#x192;&#x17D; ď ´. Since U ď&#x192;? A we have U ď&#x192;? int(A) and hence clď&#x20AC;Şs(U) ď&#x192;? clď&#x20AC;Şs(int(A)). Thus we obtain A ď&#x192;? clď&#x20AC;Şs(int(A)). Theorem3.3. If A is semi- -open set in a space (X, ď ´, ) and A ď&#x192;? B ď&#x192;? clď&#x20AC;Şs(A), then B is semi- -open in (X, ď ´, ). Proof. Since A is semi- -open, there exists an open set U such that U ď&#x192;? A ď&#x192;? clď&#x20AC;Şs(U). Then we have U ď&#x192;? A ď&#x192;? B ď&#x192;? clď&#x20AC;Şs(A) ď&#x192;? clď&#x20AC;Şs(clď&#x20AC;Şs(U)) = clď&#x20AC;Şs(U) and hence U ď&#x192;? B ď&#x192;? clď&#x20AC;Şs(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 ď&#x192;&#x17D; SISO(X, ď ´) for each ď Ą ď&#x192;&#x17D; ď &#x201E;, then ď&#x192;&#x2C6;{Uď Ą : ď Ą ď&#x192;&#x17D; ď &#x201E;} ď&#x192;&#x17D; SISO(X, ď ´), 2. If A ď&#x192;&#x17D; SISO(X, ď ´) and B ď&#x192;&#x17D; ď ´, then A ď&#x192;&#x2021; B ď&#x192;&#x17D; SISO(X, ď ´). Proof. 1. Since Uď Ą ď&#x192;&#x17D; SISO(X, ď ´), we have Uď Ą ď&#x192;? clď&#x20AC;Şs(int(Uď Ą)) for each ď Ą ď&#x192;&#x17D; ď &#x201E;. Thus by using Lemma 2.3, we obtain clâ&#x2C6;&#x2014;s (int(UÎą )) ď&#x192;?

UÎą ď&#x192;? đ?&#x203A;źâ&#x2C6;&#x2C6;â&#x2C6;&#x2020;

ď Ąď&#x192;&#x17D;ď &#x201E;

(int(UÎą ))â&#x2C6;&#x2014; ď&#x192;&#x2C6;(int(UÎą )) ď&#x192;? đ?&#x203A;ź â&#x2C6;&#x2C6;â&#x2C6;&#x2020;

www.iosrjournals.org

ď&#x192;&#x2C6; int

(int(UÎą )) đ?&#x203A;źâ&#x2C6;&#x2C6;â&#x2C6;&#x2020;

â&#x2C6;&#x2014;

UÎą Îąâ&#x2C6;&#x2C6;â&#x2C6;&#x2020;

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On Semi- -Open Sets and Semi- -Continuous Functions ď&#x192;?

int Îąâ&#x2C6;&#x2C6;â&#x2C6;&#x2020;

UÎą = clâ&#x2C6;&#x2014;s int

ď&#x192;&#x2C6; int

UÎą

Îąâ&#x2C6;&#x2C6;â&#x2C6;&#x2020;

â&#x2C6;&#x2014;

UÎą

.

Îąâ&#x2C6;&#x2C6;â&#x2C6;&#x2020;

This shows that Îąâ&#x2C6;&#x2C6;â&#x2C6;&#x2020; UÎą ď&#x192;&#x17D; SISO(X, ď ´). 2. Let A ď&#x192;&#x17D; SISO(X, ď ´) and B ď&#x192;&#x17D; ď ´. Then A ď&#x192;? clď&#x20AC;Şs(int(A)) and by using Lemma 2.3, we have A ď&#x192;&#x2021; B ď&#x192;? clď&#x20AC;Şs(int(A)) ď&#x192;&#x2021; B = ((int(A))ď&#x20AC;Ş ď&#x192;&#x2C6; (int(A))) ď&#x192;&#x2021; B = ((int(A))ď&#x20AC;Ş ď&#x192;&#x2021; B) ď&#x192;&#x2C6; (int(A) ď&#x192;&#x2021; B) ď&#x192;? (int(A) ď&#x192;&#x2021; B)ď&#x20AC;Ş ď&#x192;&#x2C6; int(A ď&#x192;&#x2021; B) = (int(A ď&#x192;&#x2021; B))ď&#x20AC;Ş ď&#x192;&#x2C6; int(A ď&#x192;&#x2021; B) = clď&#x20AC;Şs(int(A ď&#x192;&#x2021; B)). This shows that A ď&#x192;&#x2021; B ď&#x192;&#x17D; 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 Xâ&#x2C6;&#x2019; int(clď&#x20AC;Şs(A)) ď&#x201A;š clď&#x20AC;Şs(int(X â&#x2C6;&#x2019; 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 clď&#x20AC;Şs(int(X â&#x2C6;&#x2019; A)) = clď&#x20AC;Şs({a}) = {a} and X â&#x2C6;&#x2019; int(clď&#x20AC;Şs(A)) = X â&#x2C6;&#x2019; int({b}) = {a, c, d}. Theorem 3.8. If a subset A of a space (X, ď ´, ) is semi- -closed, then int(clď&#x20AC;Şs(A)) ď&#x192;? A. Proof. Since A is semi- -closed, X â&#x2C6;&#x2019; A ď&#x192;&#x17D; SISO(X, ď ´). Since ď ´ď&#x20AC;Şs(I) is finer than ď ´, we have X â&#x2C6;&#x2019; A ď&#x192;? clď&#x20AC;Şs(int(X â&#x2C6;&#x2019; A)) ď&#x192;? cl(int(X â&#x2C6;&#x2019; A)) = X â&#x2C6;&#x2019; int(cl(A)) ď&#x192;? X â&#x2C6;&#x2019; int(clď&#x20AC;Şs(A)). Therefore we obtain int(clď&#x20AC;Şs(A)) ď&#x192;? A. Corollary3.9. Let A be a subset of a space (X, ď ´, ) such that X â&#x2C6;&#x2019; int(clď&#x20AC;Şs(A)) = clď&#x20AC;Şs(int(X â&#x2C6;&#x2019; A)). Then A is semi- -closed if and only if int(clď&#x20AC;Şs(A)) ď&#x192;? A. Proof. This is an immediate consequence of Theorem 3.8. Theorem3.10. [8] Let (X, ď ´, Aď&#x20AC;Ş( , ď ´|Y) = Aď&#x20AC;Ş( , ď ´) ď&#x192;&#x2021; Y.

) be an ideal space and A ď&#x192;? Y ď&#x192;? X, where Y is ď Ą-open in X. Then

Theorem3.11. Let (X, ď ´, ) be an ideal topological space. If Y ď&#x192;&#x17D; ď ´ and W ď&#x192;&#x17D; SISO(X), then Y ď&#x192;&#x2021; W ď&#x192;&#x17D; SISO(Y, ď ´|Y, ). Proof. Since Y is open, we have intY(A) = int(A) for any subset A of Y. Now Y ď&#x192;&#x2021; W ď&#x192;? Y ď&#x192;&#x2021; clď&#x20AC;Şs(int(W)) = Y ď&#x192;&#x2021; ((int(W))ď&#x20AC;Ş ď&#x192;&#x2C6; int(W)) = [(Y ď&#x192;&#x2021; (int(W))ď&#x20AC;Ş ) ď&#x192;&#x2C6; (Y ď&#x192;&#x2021; int(W))] ď&#x192;&#x2021; Y = [Y ď&#x192;&#x2021; (Y ď&#x192;&#x2021; (int(W)))ď&#x20AC;Ş] ď&#x192;&#x2C6; [(Y ď&#x192;&#x2021; int(W)) ď&#x192;&#x2021; Y] = Y ď&#x192;&#x2021; [intY (Y ď&#x192;&#x2021;W)]ď&#x20AC;Ş ď&#x192;&#x2C6; (Y ď&#x192;&#x2021; [intY (Y ď&#x192;&#x2021; W)]) = [intY (Y ď&#x192;&#x2021;W)]ď&#x20AC;Ş ( , ď ´|Y) ď&#x192;&#x2C6; [intY (Y ď&#x192;&#x2021; W)] = clâ&#x2C6;&#x2014;s Y (intY(Y ď&#x192;&#x2021; W)). This shows that Y ď&#x192;&#x2021; W ď&#x192;&#x17D; SISO(Y, ď ´|Y, ).

IV.

Semi-

-continuous functions

Definition4.1. A function f : (X, ď ´, ) ď&#x201A;Ž (Y, ď ł) is said to be semi- -continuous [13] ( resp. semi- -continuous [4], semi-continuous [10]) if đ?&#x2018;&#x201C; â&#x2C6;&#x2019;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&#x201A;Ž (Y, ď ł, ) is said to be -irresolute ( resp. -irresolute [5]) if đ?&#x2018;&#x201C; â&#x2C6;&#x2019;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&#x201A;Ž (Y, ď ł), the following are equivalent: 1. f is semi- -continuous, 2. for each x ď&#x192;&#x17D; X and each V ď&#x192;&#x17D; ď ł containing f (x), there exists W ď&#x192;&#x17D; SISO(X, ď ´) containing x such that f (W) ď&#x192;? V, 3. the inverse image of each closed set in Y is semi- -closed. Proof. (1) ď&#x192;&#x17E; (2). Let x ď&#x192;&#x17D; X and V be any open set of Y containing f (x). Set W = đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V), then by Definition 4.1, W is a semi- -open set containing x and f (W) ď&#x192;? V. (2) ď&#x192;&#x17E; (3). Let F be a closed set of Y. Set V = Y â&#x2C6;&#x2019; F, then V is open in Y. Let x ď&#x192;&#x17D; đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V), by (2), there exists a semi- -open set W of X containing x such that f (W) ď&#x192;? V. Thus, we obtain x ď&#x192;&#x17D; W ď&#x192;? clď&#x20AC;Şs(int(W)) ď&#x192;? www.iosrjournals.org

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On Semi- -Open Sets and Semi- -Continuous Functions clď&#x20AC;Şs(int(đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V))) and hence đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V) ď&#x192;? clď&#x20AC;Şs(int(đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V))). This shows that đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V) is semi- -open in X. Hence đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (F) = X â&#x2C6;&#x2019; đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (Y â&#x2C6;&#x2019; F) = X â&#x2C6;&#x2019; đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V) is semi- -closed in X. (3) ď&#x192;&#x17E; (1). Let V be a open set of Y. Set F = Y â&#x2C6;&#x2019; V, then F is closed in Y. đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V) = X â&#x2C6;&#x2019; đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (Y â&#x2C6;&#x2019; V) = X â&#x2C6;&#x2019; đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (F). By (3) đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V) is semi- -open in X. Theorem 4.5. Let f : (X, ď ´, ) ď&#x201A;Ž (Y, ď ł), be semi- -continuous and U ď&#x192;&#x17D; ď ´. Then the restriction f |U : (U, ď ´|U, ) ď&#x201A;Ž (Y, ď ł) is semi- -continuous. Proof. Let V be any open set of (Y, ď ł). Since f is semi- -continuous, đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V) ď&#x192;&#x17D; SISO(X, ď ´) and by Theorem 3.11, (f |U)â&#x2C6;&#x2019;1(V) = đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V) ď&#x192;&#x2021; U ď&#x192;&#x17D; SISO(U, ď ´|U). This shows that f |U: (U, ď ´|U, ) ď&#x201A;Ž (Y, ď ł) is semi- -continuous. Theorem 4.6. For function f : (X, ď ´, ) ď&#x201A;Ž (Y, ď ł, ) and g : (Y, ď ł, ) ď&#x201A;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&#x201A;Ž (Y, ď ł) is semi- -continuous if and only if the graph function g : X ď&#x201A;Ž X Ă&#x2014; Y, defined by g(x) = (x, f (X)) for each x ď&#x192;&#x17D; X, is semi- -continuous. Proof. Necessity. Suppose that f is semi- -continuous. Let x ď&#x192;&#x17D; X and W be any open set of X Ă&#x2014; Y containing g(x). Then there exists a basic open set U Ă&#x2014; V such that g(x) = (x, f (x)) ď&#x192;&#x17D; U Ă&#x2014; V ď&#x192;? W. Since f is semi- continuous, then there exists a semi- -open set U of X containing x such that f (U ) ď&#x192;? V. By Theorem 3.4 U ď&#x192;&#x2021; U ď&#x192;&#x17D; SISO(X, ď ´) and g(U ď&#x192;&#x2021; U) ď&#x192;? U Ă&#x2014; V ď&#x192;? W. This shows that g is semi- -continuous. Sufficiency: Suppose that g is semi- -continuous. Let x ď&#x192;&#x17D; X and V be any open set of Y containing f (x). Then X Ă&#x2014; V is open in X Ă&#x2014; Y and by semi- -continuity of g, there exists U ď&#x192;&#x17D; SISO(X, ď ´) containing x such that g(U) ď&#x192;? X Ă&#x2014; V. Therefore we obtain f (U) ď&#x192;? V. This shows that f is semi- -continuous. Theorem 4.8. Let f : (X, ď ´, ) ď&#x201A;Ž (Y, ď ł, ) be semi- -continuous and đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V*) ď&#x192;? (đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V))* for each V ď&#x192;&#x17D; 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 ď&#x192;&#x17D; ď ł such that V ď&#x192;? B ď&#x192;? clď&#x20AC;Şs(V). Therefore, we have đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V) ď&#x192;? đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (B) ď&#x192;? đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (clď&#x20AC;Şs(V)) ď&#x192;? clď&#x20AC;Şs(đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V)). Since f is semi- continuous and Vď&#x192;&#x17D; ď ł, đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (V) ď&#x192;&#x17D; SISO(X, ď ´) and hence by Theorem 3.3, đ?&#x2018;&#x201C; â&#x2C6;&#x2019;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&#x201A;Ž (Y, ď ł, ) is called semi- -open (resp. semi- -closed) if for each U ď&#x192;&#x17D; ď ´ (resp. U is closed) f (U) ď&#x192;&#x17D; SISO(Y, ď ł, ) (resp. f (U) is semi- -closed set). Definition 5.2. [5] A function f : (X, ď ´) ď&#x201A;Ž (Y, ď ł, ) is called semi- -open (resp. semi- -closed) if for each U ď&#x192;&#x17D; 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&#x201A;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&#x201A;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.

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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&#x201A;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&#x201A;Ž (Y, ď ł, ) is semi-closed, but it is not semi- -closed. Theorem 5.8. A function f : (X, ď ´) ď&#x201A;Ž (Y, ď ł, ) is semi- -open if and only if for each x ď&#x192;&#x17D; X and each neighbourhood U of x, there exists V ď&#x192;&#x17D; SISO(Y, ď ł) containing f(x) such that V ď&#x192;? f (U). Proof. Suppose that f is a semi- -open function. For each x ď&#x192;&#x17D; X and each neighbourhood U of x, there exists U ď&#x192;&#x17D; ď ´ such that x ď&#x192;&#x17D;U ď&#x192;? U. Since f is semi- -open, V = f (U ) ď&#x192;&#x17D; SISO(Y, ď ł) and f (x) ď&#x192;&#x17D; V ď&#x192;? f (U). Conversely, let U be an open set of (X, ď ´) . For each x ď&#x192;&#x17D; U, there exists Vx ď&#x192;&#x17D; SISO(Y, ď ł) such that f (x) ď&#x192;&#x17D; Vx ď&#x192;? f (U). Therefore we obtain f (U) = ď&#x192;&#x2C6;{Vx : x ď&#x192;&#x17D; U} and hence by Theorem 3.4, f (U) ď&#x192;&#x17D; SISO(Y, ď ł). This shows that f is semi- -open. Theorem 5.9. Let f : (X, ď ´) ď&#x201A;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?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (W), then there exists a semi- -closed (resp. semi- open) subset H of Y containing W such that đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (H) ď&#x192;? 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?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (W). Then X â&#x2C6;&#x2019; F is open and since f is semi- -open, f (X ď&#x20AC;­ F) is semi- -open. Hence H = Y â&#x20AC;&#x201C; f (X â&#x2C6;&#x2019; F) is semi- -closed. It follows from đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (W) ď&#x192;? F that W ď&#x192;? H. Moreover we obtain đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (H) ď&#x192;? F. For a semi- -closed function can be proved similarly. Theorem 5.10. For any bijective function f : (X, ď ´) ď&#x201A;Ž (Y, ď ł, ), the following are equivalent: 1. đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 : (X, ď ł, ) ď&#x201A;Ž (X, ď ´ ) is semi- -continuous, 2. f is semi- -open, 3. f is semi- -closed, Proof. Obvious.

References

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