some generalized continuous functions via ...

IJMCA; Vol. 4, Nos. 2, July-December 2012, pp. 119-124

SOME GENERALIZED CONTINUOUS FUNCTIONS VIA GENERALIZED SEMIPRE-OPEN SETS Govindappa Navalagi1, R. G. Charantimath2 & Nagarajappa C. S.3 Abstract: In the year 1986, D. Andrijevic introduced and studied the concepts of semipreopen sets, semipreclosed sets, semipreinterior operator and semipreclosure operator. Since then many authors have been studied these sets and their operators. In the year 1970, N. Levine had generalized the concept of closed sets and open sets to generalized closed (in brief, g-closed) sets and generalized open (in brief, g-open) sets in topology for the first time. Then, in the year 1995 Dontchev has generalized semipreopen sets and semipreclosed to generalized semipreopen (in brief, gsp-open) sets and generalized semipreclosed (in brief, gsp-closed) sets. Also, Dontchev in his paper has studied the concepts of gsp-continuous functions and gsp-irresolute functions. The aim of this paper is to study some more properties of gsp-continuous functions and gsp-irresolute functions and also introduce and study some allied continuous functions in terms of gsp-open sets and gsp-closed sets in topology. 2010 M.S.C.: 54 A05, 54 B 05, 54 C08. Keywords and Phrases: Semipreopen sets, Semipreclosed sets, gsp-open sets, gsp-closed sets, Semiprecontinuous, gsp-continuous functions.

1. INTRODUCTION

The concepts of generalized closed sets and generalized continuous functions were intensively studied in recent years by Balachandran, Devi, Maki and Sundaram [1]. In 1970, N. Levine [2] first considered the concept of generalized closed (briefly g-closed) sets were defined and investigated. Arya and Nour [3] defined generalized semi-open (briefly, gs-open) sets using semiopenness and obtained some characterization of s normal space. In 1986 D. Andrijevic [4] introduced and studied semi-pre open sets in topology. Later, many authors have been studied these semi-pre open sets and semi-pre closed sets, by defining subsets, generalized closed sets, separation axioms, neighbourhoods and generalized continuous functions. In 1995 Dontchev [5] defined generalized semi-pre open sets (in briefly gsp-open sets) using semi-pre closure operator. The aim of this paper is to define and study 1

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Department of Mathematics, KLE Society’s, G.H. College, Haveri-581 11 0, Karnataka, India, E-mail: [email protected] Departmen t of Mathematics, K.I.T. Tipatu r-572 202 , Dist. Tumku r, Karnataka, India, E-mail: [email protected] Department of mathematics, Reva Institute of Technology, Kattigenahalli, Yalhanka Bengalure-560 064, Karnataka, Indian, E-mail: [email protected]

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Govindappa Navalagi, R. G. Charantimath & Nagarajappa C. S.

the notions of gsp-continuous functions, gsp-irresolute functions, gsp-open functions and pre-gsp-open functions. 2. PRELIMINARIES

Throughout this paper (X, �) (or simply X) denote topological space on which no separation axioms are assumed unless explicitly stated. If A is any subset of space X, then Cl (A) and Int (A) denote the closure of A and the interior of A in X resp. The following definitions are useful in the sequel: Definition 2.1: A subset A of space X is called (i) a semi-open set [8] if A � Cl (Int (A)) (ii) a semi-closed set [6, 5] if Int (Cl (A)) � A. (iii) a semi-pre open set [1] if A � Cl (Int(Cl (A))) (iv) a semi-pre closed set [1] if Int (Cl (Int (A))) � A (v) a preopen set [12] if A � Int Cl (A) (vi) a preclosed set [ 7 ] if Cl Int (A) � A (vii) an �-open [ 13] if A � Int Cl Int (A) (viii) a �-closed [13] if Cl Int Cl (A) � A. The family of all semi open (resp. semi-pre open) sets of X will be denoted by SO (X ) SPO (X ). The semi-closure (resp. the semi-pre closure, the preclosure, the �-closure) of a subset A of a space X is the intersection of all semi-closed (resp. semi pre closed, preclosed, �-closed) sets that contain A and is denoted by sCl (A) [4, 5] (resp. spCl (A [1], pCl (A), �Cl (A) [13]). The semi-interior (resp. semipreinterior) of a subset A of a space X. Is the union of all semiopen (resp. semipreopen) sets which are contained in A and is denoted by s Int (A) [5] (resp. sp Int (A) [1]). Lemma 2.2[1]: A sub set A in a space X is semi-pre closed if and only if A = spCl (A). Definition 2.3: A subset A of a space X is called: (i) a generalized semi-closed (briefly gs-closed) set [2] if sCl (A) � U whenever A � U and U is open in X (ii) an �-generalized closed (briefly �g-closed) set [10] if �Cl (A) � U whenever A � U and U is open in X. The complement of an �g-closed set is called an �g-open set.

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(iii) a generalized �-closed (briefly g�-closed) set [11] if �Cl (A) � U whenever A � U and U is �-open in X. (iv) a generalized pre-closed (briefly gp-closed set [16] if pCl (A) � U whenever A � U and U is open in X. (v) a generalized semi-pre closed (briefly gsp-closed) set [6] if spcl (A) � U whenever A � U and U is open in X. Definition 2.4: A function f : X � Y is called (i) semi-continuous [8] if the inverse image of each open set of Y is semiopen in X. (ii) semi pre continuous [15] if the inverse image of each open set of Y is semipreopen in X. (iii) g-continuous [9] if the inverse image of each open set of Y is g-open in X. (iv) gp-continuous [1] if the inverse image of each open set of Y is gp-open in X. (v) gsp-continuous [5] if the inverse image of each open set of Y is gsp-open in X. (vi) precontinuous [14] if the inverse image of each open set of Y is preopen in X. (vii) preirresolute [13] if the inverse image of each preopen set of Y is preopen in X. 3. PROPERTIES OF gsp-CONTINUOUS FUNCTIONS

We, recall the following. Definition 3.1: A function f : X � Y is called gsp-continuous [5] if f – 1(V) if gsp-closed in X for every closed set V of Y. Every g-continuous function is gsp-continuous function Every gp-continuous function is gsp-continuous function We recall the following: Definition 3.2: A function f : X � Y is called generalized semi-pre irresolute [5] (briefly gsp-irresolute) if f – 1(V) is gsp-closed in X for every gsp-closed sert V of Y. Every gp-irresolute is gsp-irresolute. Theorem 3.3: If the bijective function f : X � Y is semi-pre continuous and open then f is gsp-irresolute. Proof: Let V be gsp-closed in Y and let f – 1(V) � U, where U is open set in X. Clearly V � f (U ). Since f (U ) is open set in Y as f is open and as V is gsp-closed in Y, then spCl (V ) � f (U ) and thus f – 1(spCl (V)) � U. since f is semi-pre irresolute and spCl (V ) is a semi-irresolute closed set, then f – 1(spCl (V )) is semi-pre closed in X. thus spCl ( f – 1(V )) � spCl ( f – 1(spCl(V))) = f – 1(spCl (V )) � U. So f – 1(V ) is gsp-closed set and f is gsp-irresolute.

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We recall the following. Definition 3.4: A function f : X � Y is called �g-continuous [16] if f – 1(V ) is �g-closed in X for every closed set V of Y. Clearly, every �g-continuous � gp-continuous � gsp-continuous. We define the following. Definition 3.5: A topological space (X,Y ) is called a T g*-space if every gsp-closed set is g-closed. Theorem 3.6: Every gsp-continuous function f : X � Y is g-continuous if X is a T g*-space. Proof: Obvious. We, define the following. Definition 3.7: A function f : X � Y is said to be pre-gsp-continuous if inverse image of each semipre open set of Y is gsp-open in X. Definition 3.8: A function f : X � Y is said to be p-gsp-continuous if inverse image of each preopen set of Y is gsp-open in X. Next, we prove the following. Theorem 3.9: For any gsp-irresolute function f : X � Y and p-gsp-continuous function g : Y � Z, the composition g o f : X � Z is p-gsp-continuous. Proof: Let U be any preopen set in Z. Since g is p-gsp-continuous, then g– 1(U ) is gsp-open in Y. Hence, f – 1(g– 1(U )) is gsp-open in X since f is gsp-irresolute. But f – 1(g– 1(U )) = (g o f )– 1(U ). So g o f is p-gsp-continuous. We, give the following. Theorem 3.10: Let f : X � Y is a p-gsp-continuous function and g : Y � Z is precontinuous function, then their composition is a gsp-continuous. Theorem 3.11: Let f : X � Y is a pre-gsp-continuous and g : Y � Z is semiprecontinuous, then their composition g o f is gsp-continuous. Theorem 3.12: Let f : X � Y is a pre-gsp-continuous and g : Y � Z is semipreirresolute, then their composition g o f is pre-gsp-continuous. Theorem 3.13: Let f : X � Y is a pre-gsp-continuous and g : Y � Z is p-gsp-continuous, then their composition g o f is p-gsp-continuous. Theorem 3.14: Let f : X � Y is a gsp-irresolute and g : Y � Z is pre-gsp-continuous, then their composition g o f is pre-gsp-continuous. We, define the following. Definition 3.15: A function f : X � Y is called strongly gsp-continuous if the inverse image of every gsp-open set of Y is open in X.

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Clearly, every strongly gsp-continuous function is continuous. We, recall the following. Definition 3.16: A function f : X � Y is called strongly continuous if the inverse image of every subset in Y is clopen in X. We, give the following: Theorem 3.17: If the function f : X � Y is strongly continuous, then f is strongly gsp-continuous. We, prove the following: Theorem 3.18: If the function f : X � Y is strongly gsp-continuous and the function g : Y � Z is gsp-continuous, then g o f : X � Z is continuous. Proof: let U be any open set in Z. Since g is gsp-continuous, then g– 1(U ) is gsp-open in Y. Hence, f – 1(g– 1(U )) is open in X because f is strongly gsp-continuous. But f – 1(g– 1(U )) = (g o f )– 1(U ). So gof is continuous. We, give the following. Theorem 3.19: If the function f : X � Y is gsp-continuous and the function g : Y � Z is strongly gsp-continuous, then g o f : X � Z is gsp-irresolute. Theorem 3. 20: If the function f : X � Y is strongly gsp-continuous and the function g : Y � Z is gsp-irresolute, then g o f : X � Z is strongly gsp-continuous. We, define the following. Definition 3.21: A function f : X � Y is said to be contra gsp-continuous if the inverse image of each open set of Y is gsp-closed set in X. Definition 3.22: A function f : X � Y is said to be contra pre-gsp-continuous if the inverse image of each semipreopen set of Y is gsp-closed set in X. Definition 3.23: A function f : X � Y is said to be contra gsp-irresolute if the inverse image of each gsp-open set of Y is gsp-closed set in X. Definition 3.24: A function f : X � Y is said to be s-gsp-continuous if the inverse image of each semiopen set of Y is gsp-open set in Y. We, give the following Theorem 3.25: If the function f : X � Y is contra gsp-irresolute and the function g : Y � Z is gsp-continuous, then g o f : X � Z is contra gsp-continuous. Theorem 3.26: If the function f : X � Y is strongly gsp-continuous and the function g : Y � Z is contra gsp-irresolute, then g o f : X � Z is strongly gsp-continuous. Theorem 3. 27: Let f : X � Y is a contra pre-gsp-continuous and g : Y � Z is semiprecontinuous, then their composition g o f is contra gsp-continuous. Theorem 3.28: Let f : X � Y is a gsp-continuous and g : Y � Z is strongly gsp-continuous, then their composition g o f is gsp-irresolute.

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Theorem 3.29: Let f : X � Y is a contra gsp-irresolute and g : Y � Z is p-gsp-continuous, then their composition gof is p-gsp-continuous. Theorem 3.30: Let f : X � Y is a gsp-irresolute and g : Y � Z is contra pre-gsp-continuous, then their composition g o f is contra pre-gsp-continuous. REFERENCES

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