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CONCERNING SEMI-CONTINUOUS FUNCTIONS Dragan S. Jankovic (received 22 April, 1981) In 1963, N. Levine [4] introduced the concept of semi-open sets and semi-continuous functions. A subset A of a topological space said to be semi-open if there exists an open set U c A c cl(U) , f function cl(U) where U X X . in X . A V Y , in Y f 1(7) S. Crossley and S. Hildebrand [l] defined a set to be semi-closed, if its complement is semi-open. semi-closure of a set U into a topological space is said to be semi-continuous if for any open set is semi-open in is such that denotes the closure of from a topological space X A They also defined the in a topological space, denoted be the intersection of all semi-closed sets that contain sc04) , A . to In [2] T. Hamlett investigated some questions concerning semi-continuous functions into Hausdorff spaces. The purpose of the present note is to generalise and improve the following theorem in [2]. Theorem A. Let X Hausdorff space, (1) (2) If f be an arbitrary topological space and let Let f and g be functions from is continuous and g is semi-continuous, then {x € X : f(x) = g(x)} is semi-closed. If g f is continuous, dense subset D of X , Throughout this note and f : X —y Y X is semi-continuous, and then and f = g Y on f Our main result is the following theorem. 109 f = g Y . on a X . from a topological space Y . Math. Chronicle 12(1983) 109-111. into be a will represent topological spaces will denote a function into a topological space X Y Theorem. of If g : X — + Y X x y, then is semi-continuous and p^[S fl G(g)) S is a closed subset is semi-closed in denotes the graph of a function g and X , where p^ : X * Y — * X G{g) represents the projection. Proof. Let X x Y , V let x € sclp^lS n G{g))) , let U S where is a closed subset of x be an arbitrary open set containing be an arbitrary open set containing g is semi-continuous, (10 is semi-open in g{x) X . in Y . in X , and Since g It is shown in Theorem 1.9 of [l] that an intersection of an open set with a semi-open set is semi-open. u € [u fl G 1 (7)) (1 g(u) € V . that U fl g 1 (10 Therefore (5 (1 G(g)) . Since S is closed, and the result follows. Corollary 1. f : X — *■ (u,g{u)) € S This implies that x € Pj (5 fl G{g)) If Since and (U x V) fl S t <J> and, consequently, Therefore [x,g(x)) € cl{S) . is semi-open. [U n g'hv)) n P l ( s n G(g)) t * • Let a e sc(P l (5 n <;(£))) , Y (a:,<7 (x)) 6 S fl G{g) . has a closed graph and g : X — *■ Hence Y is semi-continuous} then (1) {x t X : /Or) = g(.x)} (2) If f = g Proof. on a dense subset X space D of X , then f = g {a: € X : f(x) = g{x)} = Pj [G(f) H G(g)) Since is a closed subset of Theorem. is semi-closed. X x y , on X . and since £(/) the first result follows from our It is shown in Theorem 2.4 of [2 ] that the subset is dense if and only if D of a sc(D) = X . Therefore, the second assertion follows from the first. A function every x € X f : X — >■ Y is said to be weakly-continuous [3] if for and every open set V containing 110 fix) in Y there exists an open set U containing x in X such that /(£/) £ cl(V) . It is shown in Theorem 10 of [5] that a weakly-continuous function into a Hausdorff space has a closed graph. Corollary 2. If f : X — ► Y continuous, and Y Therefore, by Corollary 1 follows is weakly-continuous, g : X — *■ Y is semi- is Hausdorff, then (1) {x € X : f(_x) = g{x)} (2) If f = g is semi-closed. on a dense subset D of X , then f = g on X . Finally, the fact that continuity implies weak-continuity gives that the Theorem A is the consequence of Corollary 2. REFERENCES 1. S.G. Crossley and S.K. Hildebrand, Semi-closure, Texas J. Sci. 22 (1971), 99-112. 2. T.R. Hamlett, Semi-continuous functions, Math. Chronicle 4(1976), 101-107. 3. N. Levine, A decomposition of continuity in topological spaces, Amer. Math. Monthly 68(1961), 44-46. 4. N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70(1963), 36-41. 5. T. Noiri, Between Continuity and Weak-continuity, Boll. U. M. I. (4) 9(1974), 647-654. University of Belgrade 111