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JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(2): Pg.219-222 On Some New Open Functions in Topological Spaces R. V. Pattanashetti Department of Mathematics, PC Jabin Science College, Hubli, Karnataka, INDIA. (Received on: April 9, 2014) ABSTRACT A set A in a topological space (X , T) is said to be a g*p-closed (briefly, g*p-closed) if pcl(A) ⊂ U whenever A ⊂ U and U is gopen in X. In this paper, we introduce quasi g*p-open map from a space X into a space Y as the image of every g*p-open set is open. Also, we obtain its characterizations and its basic properties. Mathematics Subject classification: 54C10, 54C08, 54C05. Keywords: Topological spaces, g*p-open set, g*p-closed set, g*p-interior, g*p closure, quasi g*p-open function. 1. INTRODUCTION The notion of pre-open set5 plays a significant role in general topology. The most important generalizations of regularity (resp. normality) are the notions of preregularity1 and strong regularity (resp. prenormality6, strong normality6. Levine3 in 1963, started the study of generalized open sets with the introduction of semi-open sets. Then Njatsad7 studied a-open sets; In 1982, Mashhour et al.5 introduced preopen sets and pre-continuity in topology. In 2002, Veerakumar12 have defined the notion of g*p-closed sets, g*p-continuity and g*p-irresolute maps. In this paper, we will continue the study of related functions by involving g*p-open sets. We introduce and characterize the concept of quasi g*popen functions. 2. PRELIMINARIES Throughout this paper, by spaces we mean topological spaces on which no separation axioms are assumed unless otherwise mentioned and f: (X, τ)→(Y, a) (or simply f: X→Y ) denotes a function f of a space (X , τ) into a space (Y, a). For a subset A of X, cl(A) and int(A) denote the closure and interior of A, respectively. 2.1 Definition A subset A of X is said to be pre-open5 if A ⊂int(cl(A)). The complement Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257) 220 R. V. Pattanashetti, J. Comp. & Math. Sci. Vol.5 (2), 219-222 (2014) of pre-open set is said to be pre-closed5. The family of all pre-open sets (respectively preclosed sets) of (X , τ) is denoted by pO(X, τ) [respectively pCL (X, τ)]. 2.7 Definition A function f: (X, T) →(Y, σ) is called a g*p-open function12 if the image f (A) is g*p-open in (Y, σ) for each open set A in (X, τ). 2.2 Definition Let A be a subset of X . Then (i) pre-interior2,5 of A is the union of all pre-open sets contained in A. (ii) preclosure2,5 of A is the intersection of all pre-closed sets containing A. The pre-interior [respectively preclosure] of A is denoted by pint(A) [respectively pcl(A)]. 2.8 Definition A function f: (X, τ) →(Y, σ) is called a g*p-continuous12 if the inverse image of every closed set in (Y, σ) is g*pclosed in (X, T). 2.3 Definition A subset A of a space X is called a g-closed set4 (briefly, g-closed) if cl (A) ⊂U whenever A ⊂ U and U is open in X. The complement of g-open set is said to be g-open set4. 2.4 Definition A subset A of a space X is called a g*p-closed set12 (briefly, g*pclosed) if pcl (A)⊂U whenever A ⊂ U and U is g-open in X. The complement of g*popen set is said to be g*p-closed set8. The family of all g*p-open sets (respectively g*p-closed sets) of (X, T) is denoted by G*pO (X, T) [respectively G*pCL(X, T)]. Definition 2.9. A function f: (X, T) →(Y, a) is called a g*p-irresolute [12] if the inverse image of every g*p-closed in (Y, a) is g*pclosed in (X, T). 3. QUASI g*p-open FUNCTIONS 3.1 Definition A function f: X→Y is said to be quasi g*p-open if the image of every g*p-open set in X is open in Y. 3.2 Theorem A function f: X →Y is quasi g*p-open iff for every subset U of X, f(g*pint(U)) ⊂int(f(U)). 2.5 Definition Let A be a subset of X. Then (i) g*p-fnterior12 of A is the union of all g*p-open sets contained in A. (ii) g*p-closure12 of A is the intersection of all g*p-closed sets containing A. Proof. Let f be a quasi g*p-open function. Now, we have int(U)⊂U and g*p-int(U) is a g*p-open set. Hence we obtain that f(g*pint(U)) ⊂f (U). As f(g*p-int(U)) is open, f(g*p-int(U)) ⊂ int (f (U)). Conversely, assume that U is a g*p-open set in X. Then f (U) = f(g*pint(U)) ⊂ int (f (U)) but int (f (U)) ⊂f (U). Consequently f (U) = int (f (U)) and hence f is quasi g*p-open. The g*p-interior [respectively g*pclosure] of A is denoted by g*pint(A) [respectively g*pcl(A)]. 3.3 Lemma If a function f: X→Y is quasi g*p-open, then g*p-int (f-1 (G)) ⊂f-1 (int (G)) for every subset G of Y. 2.6 Definition A subset S is called a g*pneighbourhood12 of a point x of X , if there exists a g*p-open set U such that x ∈U ⊂ S. Proof. Let G be any arbitrary subset of Y. Then, g*p-int(f -1(G)) is a g*p- open set in X and f is quasi g*p-open, then f(g*p-int(f- Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257) R. V. Pattanashetti, J. Comp. & Math. Sci. Vol.5 (2), 219-222 (2014) 1 (G))) ⊂ int(f(f-1(G))) ⊂ int(G). Thus g*pint(f -1(G)) ⊂ f-1(int(G))). 3.4 Theorem For a function f: X → Y, the following are equivalent. (i) f is quasi g*p-open. (ii) For each subset U of X , f(g*p-int(U)) ⊂int (f (U)). (iii) For each x ∈X and each g*pneighbourhood U of x in X, there exists a neighbourhood V of f (∅) in Y such that V ⊂f (U). Proof (i) => (ii): It follows from Theorem 3.2. (ii) => (iii) : Let x∈X and U be an arbitrary g*p-neighbourhood of X in X. Then there exist a g*p-open set V in X such that x∈V ⊂U. Then by (ii),we have f (V) = f(g*pint(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). (iii) => (i) : Let U be an arbitrary g*p-open set in X. Then for each y ∈f (U), by (iii) there exist a neighbourhood Vy of y in Y such that Vy ⊂ f (U). As Vy is a neighbourhood of y, there exist an open set Wy in Y such that y ∈Wy ⊂Vy. Thus f(U) = U{Wy: y ∈ f(U)} which is a open set in Y . This implies that f is quasi g*p-open function. 3.5 Theorem A function f: X→Y is quasi g*p-open iff for any subset B of Y and for any g*p-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 g*p-open. Let B ⊂Y and F be a g*p-closed set of X 221 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 g*p-open, we obtain G as a closed set of Y. Moreover, we have f-1 (G) ⊂F. Conversely, Let U be a g*p-open set of X and put B = Y \f (U). Then X \ U is a g*p-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) c Y \F. On the otherhand, 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 g*p-open function. 3.6 Theorem A function f: X→Y is quasi g*p-open iff f-1 (cl (B)) ⊂g*pcl (f-1 (B)) for every subset B of Y. Proof. Suppose that f is quasi g*p-open. For any subset B of Y, f-1 (B) ⊂g*p-cl (f1 (B)). Therefore by Theorem 3-5., there exists a closed set F in Y such that B ⊂ F and f-1 (F) ⊂ g*p-cl (f-1 (B)). Therefore, we obtain f-1 (cl (B)) ⊂f-1 (F) ⊂g*p-cl (f-1 (B)). Conversely, let B ⊂ Y and F be a g*p-closed set of X containing f-1 (B).Put W = clY (B), then we have B ⊂W and W is closed set and f-1 (W) ⊂g*pcl (f-1 (B)) ⊂F. Then by Theorem 3.5., f is quasi g*p-open. 3.7 Lemma Let f: X →Y and g :. Y → Z be two functions and g°f: X →Z is quasi g*popen. If g is continuous injective, then f is quasi g*p-open. Proof. Let U be a g*p-open set in X, then (g°f)(U) is open in Z. Since g o f is quasi g*p-open. Again g is an injective continuous function, f(U) = g-1(g o f(U))) is open in Y . This shows that f is quasi g*p-open. Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257) 222 R. V. Pattanashetti, J. Comp. & Math. Sci. Vol.5 (2), 219-222 (2014) ACKNOWLEDGMENTS The author is grateful to the University Grants Commission, for its Financial support under UGC Minor Research Project and also to PC Jabin Science College, Hubli, Karnataka state, India. REFERENCES 1. H. Corson and E. 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