AMERICAN JOURNAL OF MATHEMATICAL SCIENCE AND APPLICATIONS 2(1) • January-June 2014 • ISSN : 2321-497X • pp. 29-32
SOME CONTINUOUS FUNCTIONS VIA SEMI-OPEN SETS IN TOPO LOGY Md. Hanif PAGE1 and V. T. Hosamath2 1
Department of Mathematics, B. V. B. College of Engg. and Technology, Hubli-580031, Karnataka, India E-mail:
[email protected] 2 Department of Mathematics, K. L. E. Institute of Technology, Hubli-580030, Karnataka, India E-mail:
[email protected]
ABSTRACT: In this paper we introduce new class of functions called semi-g-continuous functions in topological spaces and study some more its properties and its relationship with some existing functions. Math. Subject classification: 54A05 Key words: g-closed set, semi-open set, g-open set, semi-closed set.
1. INTRODUCTION Levine [7] introduced the concept of generalized closed sets in topological space and a class of topological space called T1/2-space. In 1963, Levine[6] introduced weaker forms of open sets, called semi-open sets. A weak form of continuous function called g-continuous was introduced and studied by Balachandran, Sundaram and H.Maki[1]. In this paper we introduce and study a new type of continuous function called semi-g-continuous using semi-open sets. 2. PRELIMINARIES Throughout this paper (X, τ) and (Y, σ) (or simply X and Y) denote topological spaces 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 respectively. The following definitions are useful in the sequel. Definition 2.1: A subset A of space X is called (i) a semi-open set [6] if A � Cl(Int(A)) (ii) a semi-closed set [3] if Int(Cl(A)) � A Definition 2.2: A subset A of a topological space (X, τ) is called a generalized closed set (briefly gclosed) [7] if Cl(A) � U whenever A � U and U is open in (X, τ). The complement of g-closed set is called g-open set. Definition 2.3: A function f : (X, τ) � (Y, σ) is called (i)
semi-continuous [6] if f -1(V) is a semi-open set of (X, �) for every open set V of (Y, �).
(ii) g-continuous [1] if f -1(V) is a g-closed set of (X, τ) for every closed set V of (Y, σ). (iii) gc-irresolute [1] if f -1(V) is a g-open set of X for every g-open set V of Y. Definition 2.4: A space X is called
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(i)
T1/2-space[2] if every g-closed set is closed set.
(ii) Td-space[4] if every gs-closed set is g-closed set. (iii) Tb-space[4] if every gs-closed set is closed set. Definition 2.5: A function f : X�Y from a topological space X into a topological space Y is said to be strongly continuous[5] if f(Ā) ��A for all A ��X. Theorem 2.6[8]: Every restriction of θgs-continuous function is θgs-continuous. Theorem 2.7 (Sum Theorem)[8]: Let f : X � Y be a function and X = X1 � X2 where X1 and X2 are closed and (f/X1) and (f/X2) are θgs-continuous. Then f is θgs-continuous. 3. SEMI-G-CONTINUOUS FUNCTION Definition 3.1: A function f : (X, τ) ��(Y, σ) is called semi-g-continuous if f -1(V) is a semi-closed set of (X, τ) for every g-closed set V of (Y, σ). Theorem 3.2: Every semi-g-continuous function is semi-continuous function. Proof: Let f be a function f : X � Y. Let V be an open set in Y. Since every open set is g-open set and f is semi-g-continuous. Hence f is semi-continuous function. Remark 3.3: Converse of the above theorem is not true in general as shown by the following examples. Example 3.4: Let X = Y = {a, b, c} and τ = {X, {a},{b},{a, b}}, σ = {X, Ø, {a},{a,b},{a,c}} be the topologies on X and Y respectively. Define f : X � Y by f(a)=b, f(b) = c, f(c) = a is semi-continuous but not semi-g-continuous since f -1({b,c}) = {a,b} is not a semi-closed set in X. Theorem 3.5: The notion of semi-g-continuous and gc-irresolute are independent of each other. Example 3.6: Let X = Y = {a, b, c} and τ = {Ø, X,{b},{c},{b,c}}, σ = {Ø,Y,{a},{c},{a,c}} be the topologies on X and Y respectively. Define f : X � Y by f(a) = b, f(b)=c, f(c) = a is semi-g-continuous but not gc-irresolute since f -1({b}) = {c} is not a g-closed in X. Example 3.7: Let X = Y = {a, b, c} and τ = {Ø, X,{a}, {a,c}}, σ = {Ø, Y,{a},{b},{a,b}} be the topologies on X and Y respectively. Define f : X � Y by f(a) = a, f(b)=c, f(c) = b is gc-irresolute but not semi-gcontinuous since f -1({b}) = {c} is not a semi-closed in X. Theorem 3.8: A function f : X � Y be semi-g-continuous if and only if , for every g-open set A of Y, f (A) is semi-open in X. –1
Proof: Necessity. Let f : X � Y be semi-g-continuous and A be a g-open set in Y. Then Y–A is g-closed set in Y and since f is semi-g-continuous, f –1(Y-A) is semi-open set in X. But f -1(Y-A) = X - f -1(A) and so that f -1(A) is semi-open in X. Sufficiency: Assume that f –1(A) is semi-open in X for each g-open set A inY. Let F be a g-closed set in Y. Then Y–F is g-open in Y and by assumption f –1(Y-F) is semi-open in X, since f –1(Y-F) = X – f –1(F), we have f –1(F) is semi-closed in X and so f is semi-g-continuous. Theorem 3.9: Every restriction on semi-g-continuous is semi-g-continuous. Proof: Let f : X � Y be semi-g-continuous function and A be any subset of X. For any g-open subset of Y, (f/A)–1(S) = A �� f –1(S). But f is semi-g-continuous, f –1(S) is semi-open and hence A �� f –1(S) is relatively semi-open subset of A. That is, (f/A)–1(S) is semi-open subset of A. Hence (f/A) is semi-gcontinuous.
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Theorem 3.10 (Sum Theorem): Let f :X � Y be a function and X = X1 � X2 where X1 and X2 are semi-closed and (f/X1) and (f/X2) are semi-g-continuous. Then f is semi-g-continuous. Proof: Let A be a g-closed subset of Y. Then, since (f/X1) and (f/X2) are both semi-g-continuous, therefore (f/X1)–1(A) and (f/X2)–1(A) are both semi-closed in X1 and X2 respectively. Since X1 and X2 are semi-closed subsets of X, therefore (f/X1)–1(A) and (f/X2)–1(A) are g-closed subsets of X. Also f –1(A)= (f/X1)–1(A) � (f/X2)–1(A). Thus f –1(A) is union of two semi-closed sets and therefore semi-closed. Hence f is semi-g-continuous. Definition 3.11: A function f : X � Y is called (g, sg)-continuous if f –1(V) is a gs-closed set of X for every g-closed set V of Y. Definition 3.12: A function f : X � Y is called (gs, g)-continuous if f –1(V) is a g-closed set of X for every gs-closed set V of Y. Definition 3.13: A space X is said to be (i) sg-T1/2-space if every semi-closed is g-closed. (ii) gs-T1/2-space if every g-closed is semi-closed. Theorem 3.14: Let X and Z be topological spaces and Y be sg-T 1/2-space. Then the composition gof : X � Z of semi-g-continuous functions f : X � Y and g : Y � Z are semi-g-continuous. Proof: Let F be any g-closed set in Z. Since g is semi-g-continuous, g –1(F) semi-closed in Y, since Y is sg-T1/2-space therefore g –1(F) is g-closed in Y. Hence g –1(F) is g-closed and f is semi-g-continuous, f -1(g -1(F)) is semi-closed in X. But f -1(g –1(F)) = (gof)–1(F) and so, gof is semi-g-continuous. Theorem 3.15: If f : X � Y is semi-g-continuous and g : Y � Z is g-continuous, then their composition gof : X � Z is semi-g-continuous. Proof: Let F be any g-closed set in Z. Since g is g-continuous, g –1(F) is g-closed in Y. Since f is semig-continuous and g –1(F) is g-closed in Y. But f –1(g -1(F))=(gof)-1(F) is semi-closed in X and hence gof is semi-g-continuous. Theorem 3.16: Let f : X � Y be a semi-g-continuous and X is sg-T1/2-space then f is gc-irresolute. Proof: Let A be a g-closed subset of Y. Then f –1(A) is semi-closed as f is semi-g-continuous. Since X is sg-T1/2-space, so f –1(A) is g-closed. Therefore f is gc-irresolute. Theorem 3.17: If f : X � Y is semi-g-continuous and g : Y � Z is (gs, g)-continuous, then their composition gof : X � Z is semi-g-continuous. Proof: Let V be any gs-closed in Z. Since g is (gs, g)-continuous, g –1(V) is g-closed in Y. Since f is semi-g-continuous and g –1(V) is g-closed in Y. But f –1(g –1(V)) = (gof)–1(V) is semi-closed in X and hence gof is semi-g-continuous. Theorem 3.18: If a function f : X � Y is gc-irresolute and X is a gs-T1/2-space then f is semi-gcontinuous. Proof: Let B be any g-closed in Y. Since f is gc-irresolute, then f –1(B) is g-closed in X. But X is gs-T1/2-space. Therefore f –1(B) is semi-closed in X, which implies f is semi-g-continuous. Theorem 3.19: If f : X � Y is semi-g-continuous and g : Y � Z is a gs-continuous and Y is Td-space then gof is semi-g-continuous.
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Proof: Let V be any closed set in Z. Since g is gs-continuous then g –1(V) is gs-closed in Y. As, Y is Td-space, g -1(V) is g-closed in Y. But f -1(g -1(V)) = (gof)-1(V) is semi-closed in X and hence gof is semi-gcontinuous. Theorem 3.20: If f : X � Y is (g, sg)-continuous and X is Td-space then f is gc-irresolute. Proof: Let F be g-closed set in Y. Since f is (g, sg)-continuous, g –1(F) is gs-closed set in X. But X is Td-space. Therefore g –1(F) is g-closed. Hence f is gc-irresolute. Remark 3.21: Composition of two semi-g-continuous function need not be semi-g-continuous. Theorem 3.22: If f : X � Y is irresolute and g : Y � Z is semi-g-continuous then gof : X � Z is semig-continuous. Proof: Let V be any g-closed set in Z. Since g is semi-g-continuous, g –1(V) is semi-closed in Y. Since f is irresolute and g –1(V) is semi-closed in Y. But f –1(g -1(V)) = (gof)–1(V) is semi-closed in X and hence gof is semi-g-continuous. Theorem 3.23: If f : X � Y is strongly continuous and g : Y � Z is semi-g-continuous then gof : X � Z is g-continuous. Proof: Let F be any g-closed set in Z. Since g is semi-g-contionuous, g –1(F) is semi-closed set in Y. Since f is strongly continuous and g –1(F) is semi-closed. But f –1(g –1(F)) = (gof)–1(F) is closed in X and hence gof is g-continuous. References [1] K. Balachandran, P. Sundaram and H. Maki, On generalized continuous maps in topological spaces, Mem. Fac. Kochi Univ. Ser. A, Math., 12, (1991), 5-13. [2] P. Bhattacharya and B. K. Lahiti, Semi-generalized closed sets in topology, Indian J. Math., 29(3), (1987), 375-382. [3] S. G. Crossely and S. K. Hildbrand, On semi-closure. Texas J. Sci., 22, (1971), 99-112. [4] R. Devi, H. Maki and K Balachandran, Semi-generalized closed maps and generalized semi closed maps, Mem. Fac. Sci. Kochi Univ. Ser. A. Math., 14, (1993), 41-54. [5] N. Levine, Strong continuity in topological spaces, Amer. Math. Monthly 67, (1960), 269. [6] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70, (1963), 36-41. [7] N. Levine, Generalized closed sets in topology, Rend. Circ. Math. Palermo, 19(2), (1970), 89-96. [8] G. B. Navalagi and Md. Hanif Page, On �gs-continuity and �gs-irresoluteness, International Journal of Mathematics, Computer Sciences and Information Technology, Vol. I, No.1, January-June, pp. 95-101.
