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Annals of the Alexandru Ioan Cuza University - Mathematics
, Volume 60 (2) – Nov 24, 2014

/lp/de-gruyter/weakly-q-m-continuous-functions-3MOrwC6XKw

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- de Gruyter
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- Copyright © 2014 by the
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- 1221-8421
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- 1221-8421
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- 10.2478/aicu-2013-0015
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In this paper we introduce a new class of functions called weakly (q , m)continuous functions. Some characterizations and several properties concerning weak (q , m)-continuity are obtained. Mathematics Subject Classification 2010: 54C08, 54A05, 54D25. Key words: m-structure, (q , m)-continuity, weak (q , m)-continuity, strongly clpm-closed graph, mX -regular set, ultra Hausdorff space, ultraregular space. 1. Introduction Semi-open sets, preopen sets, -sets, b-open sets, -open sets play an important role for generalization of continuity in topological spaces. By using these sets several authors introduced and studied various modifications of continuity such as weak continuity, almost s-continuity ([22]), p()continuity ([7]). Popa and Noiri [30] introduced the notions of minimal structures. After this work, various mathematicians turned their attention in introducing and studying diverse classes of sets and functions defined on an structure, because this notions are a natural generalization of many well known results related with generalized sets and several weaker forms of continuity such as ([20], [21], [32], [33], [40]). The notion of weakly M -continuous and weakly (, m)-continuous functions are introduced and studied by Popa and Noiri ([28], [29]) for unifying weak continuity types using minimal conditions. They also defined weakly (, )-continuous functions as a special case of weak (, m)-continuity. Weak (, )-continuity is also studied by present author [39] and by Basu and Ghosh [4] (under ¨ UGUR SENGUL ¸ the name of (, )-continuous functions). Recently Son, Park and Lim introduced and studied weakly clopen functions ([36]). In fact this type of functions can be unified as weakly M -continuous function from a space with quasi-topology q , to a space with an m-structure, that is a function f : (X, q ) (Y, mY ) which can be named as weakly (q , m)-continuous functions. The purpose of this paper is to introduce and investigate the notion of weakly (q , m)-continuous functions. In addition we also discuss possible generalizations of the concept of almost clopen functions due to Ekici [11], which is recently studied in detail by various authors ([15],[16]). 2. Preliminaries Throughout this paper (X, ) and (Y, ) (or simply X and Y ) represent nonempty topological spaces on which no separation axioms are assumed, unless otherwise mentioned. For a subset S of (X, ), cl(S) and int(S) represent, the closure of S and the interior of S, respectively. A subset S of a space (X, ) is said to be regular open ([38]) (resp. regular closed ([38])) if S = int(cl(S)) (resp. S = cl(int(S))). A point x of X is called a -cluster ([41]) point of A if cl(U )A = for every open set U of X containing x. The set of all -cluster points of A is called the -closure ([41]) of A and is denoted by cl (A). A set A is said to be -closed if A = cl (A). The complement of a -closed set is said to be -open. A subset S of a space (X, ) is said to be semi-open ([17]) (resp. preopen ([19]), -open ([27]), semi-preopen ([2]) or -open ([1]), b-open ([3])) if S cl(int(S)) (resp. S int(cl(S)), S int(cl(int(S))), S cl(int(cl(S))), S cl(int(S)) int(cl(S))). The family of all semi-open (resp.preopen, -open, -open, b-open) sets of X is denoted by SO(X) (resp. P O(X), O(X), O(X), BO(X)). The complement of a semi-open (resp.preopen, -open, -open, b-open) set is said to be semiclosed (resp. preclosed, -closed, -closed, b-closed). If S is a subset of a space X, then the b-closure of S, denoted by bcl(S), is the smallest b-closed set containing S. The semiclosure (resp. preclosure, -closure, b-closure) of S is similarly defined and is denoted by scl(S) (resp. pcl(S), Cl(S), bCl(S)). A point x X is said to be in the semi--closure ([9]) (resp. -closure or sp--closure ([24])) of A, denoted by scl (A) (resp. by cl (A)), if A scl(V ) = (resp. A cl(V ) = ) for every V SO(X, x) (resp. V O(X, x)). If scl (A) = A (resp. cl (A) = A), then A is said to be semi--closed (resp. --closed or sp--closed ([24])). The complement of a semi--closed (resp. --closed) set is said to be semi--open (resp. WEAKLY (q , m)-CONTINUOUS FUNCTIONS --open). The quasi-component ([10]) of a point x X is the intersection of all clopen subsets of X which contain the point x. The quasi-topology q on X is the topology having as base clopen subsets of (X, ). The closure of each point in quasi-topology is precisely the quasi-component of that point. The open (resp. closed) subsets of the quasi-topology is called quasi-open ([10]) (resp. quasi-closed ([10])). For a space (X, ) the space (X, q ) is called by Staum [37] the ultraregular kernel of X and denoted by Xq for simplicity. A space (X, ) is called ultraregular ([37]) if = q . For a subset A of a space X, we define the quasi-interior (resp. quasi-closure) of A, denoted by intq (A) (resp. clq (A)), defined by intq (A) = {U is quasiopen: U A},(resp. clq (A) = {F is quasi-closed: A F }). Definition 1. A subfamily mX of the power set (X) of a nonempty set X is called a minimal structure (briefly m-structure) ([30]) on X if mX and X mX . By (X, mX ), we denote a nonempty subset X with a minimal structure mX on X. Each member of mX is said to be mX -open and the complement of mX -open set is said to be mX -closed. Definition 2. A subset S is said to be mX -regular if it is mX -open and mX -closed. The family of all mX -regular sets of X is denoted by mR(X) and the family of all mX -open (resp. mX -regular or mX -clopen) sets of X containing a point x X is denoted by mO(X, x) (resp. mR(X, x)). Remark 1. Let (X, ) be a topological space. Then the families , q , SO(X), P O(X), O(X), O(X), SR(X), R(X) are all m-structures on X. Definition 3. Let X be a nonempty set and mX an m-structure on X. For a subset A of X, the mX -closure of A and the mX -interior of A are defined in [18] as follows: (a) mX -Cl(V ) = {F : A F, X - F mX } (b) mX -Int(V ) = {U : U A, U mX }. Remark 2. Let (X, ) be a topological space and A a subset of X. If mX = , (resp. q , SO(X), P O(X), O(X), O(X), SR(X), R(X)) then we have: (a) mX -Cl(V ) = cl(V ) (resp. clq (V ), scl(A), pcl(A),Cl(A), cl(A), scl (X), cl (X)). ¨ UGUR SENGUL ¸ (b) mX -Int(V ) = int(V ) (resp. intq (V ), sint(A), pint(A),Int(A), int(A), sint (X), int (X)). Lemma 1 ([18]). Let X be a nonempty set and mX a minimal structure on X. For subsets A and B of X, the following hold: (a) mX -Cl(X - A) = X - (mX -Int(A)) and mX -Int(X - A) = X - (mX - Cl(A)). (b) If X - A mX , then mX -Cl(A) = A and if A mX , then mX - Int(A) = A. (c) mX -Cl() = , mX -Cl(X) = X,mX -Int() = and mX -Int(X) = X. (d) If A B, then mX -Cl(A) mX -Cl(B) and mX -Int(A) mX Int(B). (e) If A mX -Cl(A) and mX -Int(A) A. (f ) mX -Cl(mX -Cl(A)) = mX -Cl(A) and mX -Int(mX -Int(A)) = mX Int(A). Lemma 2 ([30]). Let X be a nonempty set with a minimal structure mX and A a subset of X. Then x mX -Cl(A) if and only if U A = , for every U mX containing x. A point x X is called a m -adherent point ([29]) of S if mX -Cl(U ) S = for every mX -open set U containing x. The set of all m -adherent points of S is denoted by mCl (S). A subset S is said to be m -closed if S = mCl (S). The complement of a m -closed set is said to be m -open. Definition 4 ([18]). A minimal structure mX on a nonempty set X is said to have property (B) if the union of any family of subsets belonging to mX belongs to mX . Lemma 3 ([30]). Let X be a nonempty set and mX a minimal structure on X satisfying the property (B). For a subset A of X, the following properties hold: (a) A mX if and only if mX -Int(A) = A. WEAKLY (q , m)-CONTINUOUS FUNCTIONS (b) A is mX -closed if and only if mX -Cl(A) = A. (c) mX -Int(A) mX and mX -Cl(A) is mX -closed. Definition 5. A minimal structure mX on a nonempty set X satisfying the property (B) is said to have property (mR) if for any subset A of X the following two conditions are true: (a) mX -Cl(mX -Int(mX -Cl(A))) = mX -Int(mX -Cl(A)). (b) mX -Int(mX -Cl(mX -Int(A))) = mX -Cl(mX -Int(A)). Remark 3. Let X be a nonempty set and mX a minimal structure on X. For the case mX = SO(X), mX satisfies equalities in Definition 5 by [1], for mX {BO(X), O(X)} mX -Cl(mX -Int(A)) = mX -Int(mX -Cl(A)) is true. This statement implies conditions of Definition 5. Lemma 4. Let X be a nonempty set and mX a minimal structure on X satisfying the property (B) and have property (mR), then the following properties hold: (a) If V mX then mX -Cl(V ) is mX -regular. (b) If F is mX -closed then mX -Int(A) is mX -regular. Proof. (a) If V mX then by property (mR) (b), mX -Int(mX Cl(V )) = mX -Cl(V ), that is mX -Cl(V ) is both mX -open and mX -closed. That is the mX -closure of every mX -open set is mX -open, then mX is m-extremely disconnected (see [40] Definition 3.14). (b) If F is mX -closed then by property (mR) (a), mX -Cl(mX -Int(F )) = mX -Int(F ), that is mX -Int(F ) is both mX -open and mX -closed. Definition 6. A function f : (X, mX ) (Y, mY ), where X and Y are nonempty sets with minimal structures mX and mY , respectively, is said to be weakly M -continuous ([28]) (M -continuous, ([30]), almost M -continuous ([5])) at x X if for each V mY containing f (x) there exist U mX containing x such that f (U ) mY -Cl(V ) (resp. f (U ) V , f (U ) mY Int(mY -Cl(V ))). A function f : (X, mX ) (Y, mY ) is said to be weakly M -continuous (resp. M -continuous, almost M -continuous) if it has the property at each point x X. Definition 7. A function f : (X, mX ) (Y, mY ), is said to be M continuous ([20]) if for every V mY , f -1 (V ) mX . ¨ UGUR SENGUL ¸ Remark 4. Let Y be a nonempty set and mY a minimal structure on Y for which satisfying the property (mR). For a function f : (X, mX ) (Y, mY ) the following properties are equivalent: (a) f is weakly M -continuous. (b) f is almost M -continuous. (c) For m = mR(Y ), f : (X, mX ) (Y, m ) is M -continuous. Y Y Proof. Let V mY , then by property (mR)(b), mX -Int(mX -Cl(V )) = mX -Cl(V ), that is mX -Int(mX -Cl(V )) and mX -Cl(V ) are both mX -regular and equal. Lemma 5 ([30]). For a function f : (X, mX ) (Y, mY ) the following properties are equivalent: (a) f is M -continuous. (b) f -1 (V ) = mX -Int(f -1 (V )) for every V mY . (c) f (mX -Cl(A)) mY -Cl(f (A)) for every subset A of X. (d) mX -Cl(f -1 (B)) f -1 (mY -Cl(B)) for every subset B of Y. (e) f -1 (mX -Int(B)) mX -Int(f -1 (B)) for every subset B of Y. (f ) mX -Cl(f -1 (K)) = f -1 (K) for every mY -closed set K of Y . Definition 8. A function f : X Y is (, m)-continuous ([29]), (resp. weakly (, m)-continuous ([29]), almost (, m)-continuous) for each x X and each mY -open set V containing f (x), there exists an open set U containing x, such that f (U ) V (resp. f (U ) mY -cl(V ), f (U ) mY Int(mY -cl(V ))). Definition 9. A function f : (X, ) (Y, ) is said to be; (a) almost continuous ([35]) (resp. (, b)-continuous, almost s-continuous ([22]), weakly (, )-continuous ([29]), p()-continuous ([7])), if for each x X and each open (resp. b-open, semiopen, -open, preopen) set V containing f (x), there exists an open set U containing x such that f (U ) int(cl(V )) (resp.f (U ) bcl(V ), f (U ) scl(V ), f (U ) cl(V ), f (U ) pcl(V )), WEAKLY (q , m)-CONTINUOUS FUNCTIONS (b) almost clopen ([11]) if for each x X and each open set V in Y containing f (x), there exists a clopen set U containing x such that f (U ) int(cl(V )). Remark 5. Let (X, ) and (Y, ) be topological spaces. (a) We put mX = and mY = (resp. SO(Y ), P O(Y ), O(Y )). Then, a weakly M -continuous function f : (X, ) (Y, mY ) is weakly continuous (resp. almost s-continuous, p()-continuous, weakly (, )-continuous). (b) We put mX = q and mY = . Then, an almost M -continuous f : (X, q ) (Y, mY ) is almost clopen. Definition 10. Let Y be a nonempty set and mY a minimal structure on Y . A function f : X Y , is said to be (q , m)-continuous (resp. weakly (q , m)-continuous, almost (q , m)-continuous) at x X, if for each V mY containing f (x) there exists a clopen set U containing x such that f (U ) V (resp. f (U ) mY -Cl(V ), f (U ) mY -Int(mY -Cl(V ))). A function f : (X, q ) (Y, mY ) is said to be (q , m)-continuous (resp. weakly (q , m)-continuous, almost (q , m)-continuous) if it has the property at each point x X. We will write for (q , m)-continuous (resp. weakly (q , m)-continuous, almost (q , m)-continuous) briefly (q , m).c (resp. w.(q , m).c, a.(q , m).c). Proposition 1. A function f : (X, q ) (Y, mY ) is (q , m)-continuous (resp. weakly (q , m)-continuous, almost (q , m)-continuous) if and only if f : (X, q ) (Y, mY ) is M -continuous (resp. weakly M -continuous, almost M -continuous). Proof. () Let x X and V be a mY -open set in Y containing f (x). Then by definition there exists a clopen set U containing x such that f (U ) V (resp. f (U ) mY -Cl(V ), f (U ) mY -Int(mY -Cl(V ))). Since every clopen set is quasi-open we have f : (X, q ) (Y, mY ) is M continuous (resp. weakly M -continuous, almost M -continuous). () Let x X and V is a mY -open set containing f (x) then there exists a quasi-open set U containing x, such that f (U ) V (resp. f (U ) mY Cl(V ), f (U ) mY -Int(mY -Cl(V ))). Since U is quasi open there exists a clopen set W in U containing x such that f (W ) V (resp. f (W ) mY Cl(V ), f (W ) mY -Int(mY -Cl(V ))) and by Definition 10, f is (q , m).c. (resp. w.(q , m).c., a.(q , m).c.). ¨ UGUR SENGUL ¸ Theorem 1. For a function f : (X, q ) (Y, mY ) the following properties are equivalent: (a) f is (q , m)-continuous. (b) f -1 (V ) = intq (f -1 (V )) for every V mY . (c) f (clq (A)) mY -Cl(f (A)) for every subset A of X. (d) clq (f -1 (B)) f -1 (mY -Cl(B)) for every subset B of Y. (e) f -1 (mY -Int(B)) intq (f -1 (B)) for every subset B of Y. (f ) clq (f -1 (K)) = f -1 (K) for every mY -closed set K of Y . Proof. Here we will use same techniques with the proof of Lemma 5 ([30]). (a)(b) Let V mY and x f -1 (V ). Then f (x) V . There exists U q containing x such that f (U ) V . Thus x U f -1 (V ). This implies that x intq (f -1 (V )). This shows that f -1 (V ) intq (f -1 (V )). Hence we have f -1 (V ) = intq (f -1 (V )). (b)(c) Let A be any subset of X. Let x clq (A) and V mY containing f (x). Then x f -1 (V ) = intq (f -1 (V )). There exists U q such that x U f -1 (V ). Since x clq (A) , U A = and = f (U A) f (U ) f (A) V f (A). Since V mY containing f (x), f (x) mY -Cl(f (A)) and hence f (clq (A)) mY -Cl(f (A)). (c)(d) Let B be any subset of Y . Then, we have f (clq (f -1 (B))) mY Cl(f (f -1 (B))) mY -Cl(B). Therefore we obtain clq (f -1 (B)) f -1 (mY Cl(B)). (d)(e) Let B be any subset of of Y . Then, we have X - intq (f -1 (B)) = clq f -1 (Y - B) f -1 (mY -Cl(Y - B)) = f -1 (Y - mY -Int(B)) = X - f -1 (mY -Int(B)). Therefore, we obtain f -1 (mY -Int(B)) intq (f -1 (B)). (e)(f) Let K be any subset of Y such that Y - K mY . By (e), we have X - f -1 (K) = f -1 (mY -int(Y - K)) intq (f -1 (Y - K)) = intq (X - f -1 (K)) = X -clq f -1 (K) . Therefore, we have clq (f -1 (K)) f -1 (K) clq (f -1 (K)). Thus we obtain clq (f -1 (K)) = f -1 (K). (f)(a) Let x X and V mY containing f (x). By (f) we have X - f -1 (V ) = f -1 (Y - V ) = clq (f -1 (Y - V )) = clq (X - f -1 (V )) = X-intq f -1 (V ) . Hence, we have x f -1 (V ) = intq f -1 (V ) . Therefore, there exists U q such that x U f -1 (V ). Therefore, x U q and f (U ) V . This shows that f is (q , m)-continuous. WEAKLY (q , m)-CONTINUOUS FUNCTIONS Definition 11 ([28]). Let X be a nonempty set with a minimal structure mX is said to be m-regular if for each mX -closed set F and each x F , / there exist disjoint mX -open sets U and V such that x U and F V . Lemma 6 ([28]). Let X be a nonempty set with a minimal structure mX and mX satisfy property B. Then (X, mX ) is said to be m-regular if and only if for each x X and each mX -open set U containing x, there exists an mX -open set V such that x V mX -Cl(V ) U . Theorem 2. Let (Y, mY ) be m-regular and satisfying property (B). Then for a function f : (X, q ) (Y, mY ) the following properties are equivalent: (a) f is (q , m).c. (b) f -1 (mCl (B)) clq (f -1 (mCl (B))) for every subset B of Y. (c) f is weakly (q , m)-continuous. (d) f -1 (F ) = clq (f -1 (F )) for every m -closed set F of Y . (e) f -1 (V ) = intq (f -1 (V )) for every m -open set V of Y . Proof. Consider a function f : (X, mX ) (Y, mY ), where X and Y are nonempty sets with minimal structures mX and mY , respectively, and let (Y, mY ) be m-regular and satisfying the property (B). Then put mX = q in Theorem 4.2 of [28]. Theorem 3. Let (Y, mY ) satisfy the property (B). For a function f : (X, q ) (Y, mY ), the following are equivalent: (a) f is w.(q , m).c. (b) For each x X and each mY -open set V of Y containing f (x), there exists a quasi-open set U of X containing x such that f (U ) mY Cl(V ). (c) f -1 (V ) intq (f -1 (mY -Cl(V ))) every mY -open set V of Y . (d) clq (f -1 (mY -Int(mY -Cl(B))) f -1 (mY -Cl(B)) for every subset B of Y . (e) clq (f -1 (mY -Int(F ))) f -1 (F ) every mY -closed set F of Y . ¨ UGUR SENGUL ¸ (f ) clq (f -1 (V )) f -1 (mY -Cl(V )) every mY -open set V of Y . (g) f (clq (A)) mCl (f (A)) for each subset A of X. (h) clq (f -1 (B)) f -1 (mCl (B)) for each subset B of Y . Proof. (a)(b): These implications are clear from the definition of quasi topology. (b)(c): Let V be a mY -open set of Y and x f -1 (V ). Then f (x) V and by (b), there exists a quasi-open set U of X containing x such that f (U ) mY -Cl(V ). Then x U f -1 (mY -Cl(V )) and hence x intq (f -1 (mY -Cl(V ))). (a)(c): It follows from Theorem 3.2 of [28]. (a)(d)(e)(a): It follows from Theorem 2.1 of [25]. (f)(a): It follows from Theorem 3.4 of [28]. (a)(g)(h)(a): It follows from Theorem 3.3 of [28]. The above proofs do not use the property (B), except for the case (f)(a). Definition 12. Let Y be a nonempty set and mY a minimal structure on Y . A function f : X Y , is said to be (q , m )-continuous if for each V mY , f -1 (V ) is clopen in X. Remark 6. Let Y be a nonempty set and mY a minimal structure on Y . We put mY = (resp. RO(Y ), SR(Y )). Then, a (q , m )-continuous function f : (X, q ) (Y, mY ) is perfectly continuous ([23]) (resp. regular set connected ([10]), almost s-continuous) Proposition 2. Let Y be a nonempty set and mY a minimal structure on Y . Then, the following are quivalent: (a) f : X Y is (q , m )-continuous. (b) For each Y - F mY , X - f -1 (F ) is clopen in X. Proof. (a)(b) Let Y - F mY , since f : X Y is (q , m )continuous f -1 (Y - F ) = X - f -1 (F ) is clopen in X and hence f -1 (F ) is clopen in X (b)(a) Let U mY then Y - (Y - U ) mY and by (b), f -1 (Y - (Y - U )) = X - f -1 (Y - U ) = X - X - f -1 (U ) = f -1 (U ) is clopen in X, hence f : X Y is (q , m )-continuous. Note that property (mR) requires property (B). WEAKLY (q , m)-CONTINUOUS FUNCTIONS Theorem 4. Let Y be a nonempty set and mY a minimal structure on Y for which satisfying the property (mR). Then the following properties are equivalent for a function f : (X, q ) (Y, mY ) : (a) f is weakly (, m)-continuous. (b) For each x X and each V mR(Y, f (x)), there exists an open set U containing x such that f (U ) V . (c) For each x X and each V mR(Y, f (x)), there exists an -open set U containing x such that f (U ) V . (d) f -1 (V ) is -open in X for every V mR(Y ). (e) f -1 (V ) is clopen in X for every V mR(Y ). (f ) f is w.(q , m).c. (g) For each x X and each V mR(Y, f (x)), there exists an clopen set U containing x such that f (U ) V . (h) For each x X and each V mR(Y, f (x)), there exists a quasi-open set U of X containing x such that f (U ) V . (i) f : (X, q ) (Y, mY ) is weakly M -continuous. Proof. (a)(b): Let x X and V mR(Y, f (x)). There exists an open set U containing x such that f (U ) mY -Cl(V ) = V . (b)(c): This is clear. (c)(d): Let V mR(Y ) and x f -1 (V ). Then f (x) V mY . There exists an -open set Ux containing x such that f (Ux ) mY -Cl(V ) = V . Therefore, x Ux f -1 (V ) and hence Ux = f -1 (V ) is -1 xf (V ) -open in X. (d)(e): Let V mR(Y ) Since Y - V mR(Y ) by (d) X - f -1 (V ) = -1 (Y - V ) is -open. Therefore f -1 (V ) -closed and -open in X. Hence f by Lemma 3.1 of [14], f -1 (V ) is clopen. Note that (e) can be rephrased as, f : (X, q ) (Y, m1 ) is (q , m1 )-continuous or M -continuous, where Y m1 = mR(Y ). Y (e)(f): Let x X and V be any mY -open set of Y containing f (x). By Lemma 4, mY -Cl(V ) is mY -clopen and hence f -1 (mY -Cl(V )) is clopen ¨ UGUR SENGUL ¸ in X. Put U = f -1 (mY -Cl(V )), then U is clopen set containing x and f (U ) mY -Cl(V ). (f)(g): Let x X and V mR(Y, f (x)). There exists a clopen set U containing x such that f (U ) mY -Cl(V ) = V . (g)(h): It follows from the definition of quasi topology. (h)(i): Let x X and V mR(Y, f (x)).Then by (h), there exists a quasi-open set U containing x such that f (U ) V . Since every mY regular set is mY -open, f is (q , m)-continuous.Then f : (X, q ) (Y, mY ) is M -continuous, hence weakly M -continuous. (i)(a): This is clear. Remark 7. Let Y be a nonempty set and mY a minimal structure on Y if mY {SO(X), BO(X), O(X)} then mY has property (mR), so a weakly (q , m)-continuous function f : (X, q ) (Y, mY ) is a generalization and unification of almost s-continuity (resp. (, b)-continuity, weakly (, )continuity). Remark 8. We have the following implications for a function f : XY : (, m)-continuous almost (, m)-continuous weakly (, m)-continuous ( q , m)-continuous almost ( q , m)-continuous weakly ( q , m)-continuous ( q , m )-continuous Note that these implications cannot be reversed in general as the following examples shows: Example 1. Let X be the real numbers with the upper limit topology and Y be the real numbers with the usual topology . Let f : (X, ) (Y, ) be the identity function. Then f is (q , )-continuous function but not (q , )-continuous since f -1 ((0, 1)) is not clopen in (X, ). Example 2 ([11]). Let X be the real numbers with the usual topology and f : X X be the identity function. Then f is an almost (, )continuous function which is not almost (q , )-continuous. Example 3 ([11]). Let R and Q be the real and rational numbers, respectively. Let A = {x R : x is rational and 0 < x < 1}. We define two topologies on R as = {R, , A, R-A} and = {R, , {0}}. Let f : (R, ) WEAKLY (q , m)-CONTINUOUS FUNCTIONS (R, ) be a function which is defined by f (x) = 1 if x Q and f (x) = 0 if x Q. Then f is almost (q , )-continuous and weakly (q , )-continuous, / but f is not (q , )-continuous since for f (x) = 0 {0} (x Q), there / is no clopen set U containing x such that f (U ) {0}. Example 4 ([36]). Let X = {a, b, c, d}, = {X, , {d}, {a, b, c}} and = {X, , {c}, {d}, {a, c}, {c, d}, {a, c, d}}. Then the identity function f : (X, ) (X, ) is weakly (q , )-continuous but not almost (, )continuous (hence not almost (q , )-continuous) since there exists a regular open set {a, c} of (X, ) such that f -1 ({a, c}) is not clopen in (X, ). Example 5 ([36]). Let X be the real numbers and be the usual topology on X. Then the identity function f : (X, ) (X, ) is almost (, )-continuous (hence weakly (, )-continuous) but not weakly (q , )continuous since the only clopen set of X is itself. Definition 13. A filter base F is said to be: (a) mX --convergent ([21]) to a point x in X, if for any mX open set U containing x there exist B F such that B mX -Cl(U ). (b) clopen convergent ([12]) to a point x in X, if for any clopen set U containing x, there exist B F such that B U . Definition 14. A net (x ) in a space X, -converges ([8]) (resp. clopen converges ([15]), mX --converges ([21])) to x if and only if for each open (resp. clopen, mX -open) set U containing x, there exists a 0 such that x cl(U ) (resp. x U , x mX -Cl(U )) for all 0 . Lemma 7. For a net (x ) in a space X: (a) if (x ) converges to x, then (x ) -converges to x ([6]). (b) if (x ) converges or -converges to x, then (x ) clopen converges to x ([15]). Theorem 5. A function f : X Y is weakly (q , m)-continuous if and only if for each point x X and each filter base F in X that clopen converging to x the filter base f (F) is mY --convergent to f (x). ¨ UGUR SENGUL ¸ Proof. Suppose that x X and F is any filter base in X that clopen converges to x. By hypothesis for any mY -open set V containing f (x) there exists a clopen set U containing x in X such that f (U ) mY -Cl(V ). Since F is clopen convergent to x in X then there exists B F such that B U . It follows that f (B) mY -Cl(V ). This means that f (F) is mY --convergent to f (x). Conversely, let x be a point in X and V be a mY -open set containing f (x). If we set F = {U : U is clopen and x U }, then F will be a filter base which clopen converges to x. So there exists U F such that f (U ) mY -Cl(V ). This completes the proof. Theorem 6. The implications (a) (b) (c) (d) (e) hold for the following properties of a function f : (X, q ) (Y, mY ) : (a) f is w.(q , m).c. (b) For each x X and each net (x ) in X which clopen converges to x, the net (f (x )) mY --converges to f (x). (c) For each x X and each net (x ) in X which -converges to x, the net (f (x )) mY --converges to f (x). (d) For each x X and each net (x ) in X which converges to x, the net (f (x )) mY --converges to f (x). (e) f is weakly (, m)-continuous. Proof. (a) (b): Let x X and let (x ) be a net in X such that (x ) clopen converges to x. Let V be a mY -open set containing f (x). Since f is weakly (q , m)-continuous, there exists a clopen set U containing x such that f (U ) mY -Cl(V ). Since (x ) clopen converges to x, there exists 0 such that x U for all 0 . Hence f (x ) mY -Cl(V ) for all 0 . (b) (a): Suppose that f is not weakly (q , m)-continuous. Then there exists x X and a mY open set V containing f (x) such that f (U ) mY -Cl(V ) for all clopen neighborhoods U of x. Thus, for every clopen neighborhood U of x we can find xU U such that f (xU ) mY -Cl(V ). / Let N (x) be the set of clopen neighborhoods of x in X. The set N (x) with the relation of inverse inclusion (that is U1 U2 if and only if U2 U1 ) forms a directed set (Theorem 1.1 of [12]). Clearly the net {xU : U N (x)} WEAKLY (q , m)-CONTINUOUS FUNCTIONS clopen converges to x in X but (f (xU ))U N (x) does not mY -converge to f (x). (b) (c): Let x X and let (x ) be a net in X such that (x ) converges to x. By Lemma 7, (x ) clopen converges to x. By (b), (f (x )) mY --converges to f (x). (c) (d): Let x X and let (x ) be a net in X such that (x ) converges to x. By Lemma 7, (x ) -converges to x. By (c), (f (x )) mY --converges to f (x). (d) (e): Suppose that f is not weakly (, m)-continuous. Then there exists x X and a mY -open set V containing f (x) such that f (U ) mY -Cl(V ) for all open U containing x. Consider the set {xU : U is open set containing x}. Then (xU ) converges to x but (f (xU )) does not mY -converges to f (x). (e) (d): This can be proved similar to (a) (b). If mY satisfies property (mR), (e) (a) is true by Theorem 4. Recall that for a function f : X Y , the subset {(x, f (x)) : x X} X × Y is called the graph of f and is denoted by G(f ). Definition 15. A function f : (X, mX ) (Y, mY ) is said to have a strongly M -closed graph ([29]) if and only if for each (x, y) (X × Y ) - G(f ) there exists an mX -open set U containing x and an mY -open set V containing y such that (U × mY -Cl(V )) G(f ) = . Lemma 8 ([29]). A function f : (X, mX ) (Y, mY ) has a strongly M -closed graph G(f ) if and only if for each (x, y) (X × Y ) - G(f ) there exists an mX -open set U containing x and mY -open set V containing y such that f (U ) mY -Cl(V ) = . Definition 16. A graph G(f ) of a function f : (X, q ) (Y, mY ) is said to be strongly clp-m-closed if for each (x, y) (X × Y ) - G(f ), there exists a clopen set U in X containing x and mY -open set V containing y such that (U × mY -Cl(V )) G(f ) = . Remark 9. If a function f : (X, mX ) (Y, mY ) has the strongly M -closed graph, then for the special case mX = q , G(f ) has strongly clp-m-closed graph. Note that the concepts of strongly M -closed graph and strongly clp-mclosed graph are generalizations of the the following notions. ¨ UGUR SENGUL ¸ Definition 17. A function f : (X, ) (Y, ) has a strongly-closed ([13]) (strongly clp-closed =strongly clopen ([36])) graph if for each (x, y) / G(f ), there exists open sets U (U q ) and V containing x and y, respectively, such that (U × cl(V )) G(f ) = . Note that for a graph G(f ) strongly clp-closednes imply strongly closedness, but the reverse implication is not true in general as the following example shows. Example 6 ([13]). Let X = [0, 1] have the usual topology R1 |X (= mX ) and let Y = [0, 1] have the topology (= mY ) generated by the usual open 3 1 sets together with the set A = {r : r Q ve 4 < r < 4 } as subbase. The identity function i : X, R1 |X (Y, ) , has a strongly-closed graph G(i). But G(i) is not strongly clp-closed. Theorem 7. The following properties are equivalent for a graph G(f ) of a function: (a) G(f ) is strongly clp-m-closed. (b) For each point (x, y) (X × Y ) - G(f ), there exists a clopen set U containing x in X and mY open set V containing y such that f (U ) mY -Cl(V ) = . (c) For each point (x, y) (X × Y ) - G(f ), there exists a quasi-open set U containing x in X and mY -open set in Y containing y such that. f (U ) mY -Cl(V ) = . Proof. (a)(b) It follows from Lemma 8. (b)(c) It is clear since every clopen set is quasi-open. (c)(a) If (c) holds, then the set U in the statement of (c) is quasi open. Then, there exists a clopen set W such that W U and we have f (W ) mY -Cl(V ) f (U ) mY -Cl(V ) = . By Lemma 8 result follows. Definition 18. A nonempty set X with a minimal structure mX , (X, mX ), is said to be m-T2 ([30]) (resp. m-Urysohn ([28])) if for each distinct points x, y X, there exist U , V mX containing x and y, respectively, such that U V = (resp. mX -Cl(U ) mX -Cl(V ) = ). See ([34]) for a study on minimal structures and separation properties. WEAKLY (q , m)-CONTINUOUS FUNCTIONS Theorem 8. If f : X Y is (q , m).c. function and Y is m-T2 , then G(f ) is strongly clp-m-closed in X × Y . Proof. If the condition holds and (x, y) (X × Y ) - G(f ) then, it is true that f (x) = y and there exists V , W mY containing y and f (x), respectively, such that V W = . Then, by Lemma 2, mX -Cl(V )W = . Now, as f is (q , m).c. there exists a clopen set U in X containing x such that f (U ) W . Therefore f (U ) mY -Cl(V ) = and G(f ) is strongly clp-m-closed. Theorem 9. If f : X Y is w.(q , m).c. function and Y is mUrysohn , then G(f ) is strongly clp-m-closed in X × Y . Proof. If the condition holds and (x, y) (X × Y ) - G(f ) then, it is true that f (x) = y and there exists V , W mY containing y and f (x), respectively, such that mY -Cl(V )mY -Cl(W ) = . Now, as f is w.(q , m).c. there exists a clopen set U in X containing x such that f (U ) mY -Cl(W ). Therefore f (U ) mY -Cl(V ) = and G(f ) is strongly clp-m-closed. Theorem 10 ([39]). If f : (X, q ) (Y, mY ) is a w.(q , m).c. and (Y, mY ) is m-T2 , then f has quasi-closed point inverses in X. Theorem 11. If f, g : X Y is w.(q , m).c. function and Y is mUrysohn , then A = {x X : f (x) = g(x)} is quasi-closed in X. Proof. If x X - A, then it follows that f (x) = g(x). Since Y is m-Urysohn, there exists mY -open set U in Y containing f (x) and mY -open set V in Y containing g(x) such that mY -Cl(U ) mY -Cl(V ) = . Since f and g are w.(q , m).c. there exists clopen sets G and H with x G and x H such that f (G) mY -Cl(U ) and g(H) mY -Cl(W ), set O = GH. Then O is clopen, f (O) g(O) = and A O = . Thus every point of X - A has a clopen neighborhood disjoint from A. Hence X - A is a union of clopen sets or equivalently A is quasi-closed. Theorem 12. If f : (X, q )(Y, mY ) is w.(q , m).c. function and Y is m-Urysohn, then A={(x, y)X×X : f (x)=f (y)} is quasi-closed in X×X. Proof. Let (x, y) (X × X) - A, then it follows that f (x) = f (y). Since Y is m-Urysohn, there exist mY -open set U containing f (x) and mY open set V containing f (y) such that mY -Cl(U ) mY -Cl(V ) = . Since f is w.(q , m).c., there exists clopen sets W and O with x O and y W ¨ UGUR SENGUL ¸ such that f (O) mY -Cl(U ) and f (W ) mY -Cl(V ). Then, we have (x, y) O × W f -1 (mY -Cl(U )) × f -1 (mY -Cl(V )). Thus we have O×W is a clopen set containing (x, y) and O×W (X×X)-A. Hence (X ×X)-A is union of clopen sets or equivalently A is quasi-closed in X×X. Definition 19. A space X is said to be ultra Hausdorff ([36]) if every two distinct points of X can be separated by disjoint clopen sets. Note that if a space X is ultra Hausdorff then it is totally disconnected. Theorem 13. Let f : (X, q ) (Y, mY ) have a strongly clp-m-closed graph. Then the following properties hold: (a) If f is injective then X is ultra Hausdorff. (b) If f is surjective then Y is m-T2 . Proof. (a) Suppose that x and y are any two distinct points of X by the injectivity of f , (x, f (y)) G(f ). Since G(f ) is strongly clp-m-closed, / by Theorem 7, there exist a clopen set U containing x and mY -open set V containing f (y) such that f (U ) mY -Cl(V ) = . We have U f -1 (mY Cl(V )) = . Therefore y U . Then U and X - U are disjoint clopen sets / containing x and y, respectively. Hence X is ultra Hausdorff. (b) Let y1 and y2 be any two distinct points of Y . Since f is surjective there exists a point x X such that f (x) = y2 . Since G(f ) is strongly clp-m-closed and (x, y1 ) G(f ) there exists a clopen set U containing x / and mY -open set V in Y containing y1 such that f (U ) mY -Cl(V ) = . Therefore we have y2 f (U ) Y - (mY -Cl(V )) and hence by Lemma 2, Y is m-T2 . Definition 20. A function f : (X, mX ) (Y, mY ) is said to be M closed ([26]) if for each mX -closed set F , f (F ) is mY -closed in Y . Definition 21. A space X is said to be: (a) ultraregular ([37]) if for each closed set F and each x F , there / exist disjoint clopen sets U and V such that x U and F V . (b) ultranormal ([37]) if disjoint closed sets contained in disjoint clopen sets. Note that if a space X is ultraregular then it has a basis consisting of clopen sets. Theorem 14. Let Y be a nonempty set and mY a minimal structure on Y for which satisfying the property (mR). If f : X Y , is a w.(q , m).c. and M -closed injection and Y is m-regular, then X is ultraregular. WEAKLY (q , m)-CONTINUOUS FUNCTIONS Proof. Let F be any quasi-closed set of X and x X - F . Since f is M -closed, f (F ) is mY -closed and f (x) Y - f (F ). Since (Y, mY ) is mregular, there exist disjoint mY -open sets U and V such that f (x) U and f (F ) V . Since U V = by Lemma 2, we have mY -Cl(V )U = . Since mY has property (mR), mY -Cl(V ) is a m-regular set containing f (F ) and disjoint from f (x). Since f is w.(q , m).c. by Theorem 4, the inverse image of mY -Cl(V ), under f is a clopen subset of X containing F and disjoint from x. This shows X is ultraregular. Definition 22. An m-space (X, mX ) is said to be m-normal ([26]) if for each pair of disjoint m-closed sets F1 , F2 of X, there exist U1 , U2 mX such that F1 U1 , F2 U2 and U1 U2 = . Theorem 15. Let Y be a nonempty set and mY a minimal structure on Y for which satisfying the property (mR). If f : X Y , is a w.(q , m).c. and M -closed injection and Y is m-normal, then X is ultranormal. Proof. Let A and B be disjoint closed sets of X. Since f is M -closed injection, f (A) and f (B) are disjoint mY -closed sets of Y . By the mnormality of (Y, mY ), there exist disjoint mY -open sets U and V such that f (A) U and f (B) V . Since U V = by Lemma 2 , we have mY Cl(U ) V = . Since mY has property (mR), mY -Cl(U ) is a m-regular set containing f (A) and disjoint from f (B). Since f is w.(q , m).c. by Theorem 4, the inverse image of mY -Cl(U ) under f is a clopen subset of X containing A and disjoint from B. Thus X is ultranormal. Definition 23. A subset K of a space X is said to be mildly compact ([37]), relative to X if for every cover {V : I} of K by clopen sets of X, there exists a finite subset I0 of I such that K {V : I0 }. Definition 24. A subset K of a nonempty set X with a minimal structure mX is said to be m-compact ([30]) (m-closed ([21])) relative to (X, mX ) if any cover {Ui : i I} of K by mX -open sets, there exists a finite subset I0 of I such that K {Ui : i I0 } (K {mX -Cl(Ui ) : i I0 }). It is clear that (X, mX ) is m-closed if X is m-closed relative to (X, mX ). Let (X, ) be a topological space. Note that, if mX = (resp. SO(X)) the definition of m-closed sets gives the definitions of quasi H-closed ([31]) (resp. of s-closed ([9])) sets. Theorem 16. Let f : (X, q ) (Y, mY ) be a w.(q , m).c. surjection. If X is mildly compact, then Y is m-closed. ¨ UGUR SENGUL ¸ Proof. Let {V : I} be a cover of Y by mY -open sets of Y . For each point x X, there exists (x) I such that f (x) V(x) . Since f is w.(q , m).c., there exists a clopen set Ux of X containing x such that f (Ux ) mX -Cl(V(x) ). The family {Ux : x X}is a cover of X by clopen sets of X and hence there exists a finite subset X0 of X such that X Ux . Therefore, we obtain Y = f (X) mX -Cl(V(x) ). This xX0 xX0 shows that Y is m-closed. Theorem 17 ([21]). Let f : (X, mX ) (Y, mY ) be a function. Assume that mX is a base for a topology. If the graph G(f ) is strongly M -closed, then mX -Cl(f -1 (K)) = f -1 (K) whenever the set K Y is m-closed relative to (Y, mY ). Corollary 1 ([39]). If a function f : (X, q ) (Y, mY ) has a strongly clp-m-closed graph, then f -1 (K) is quasi-closed in (X, q ) for each set K which is m-closed relative to (Y, mY ). Theorem 18. If a function f : (X, q ) (Y, mY ) has a strongly clp-mclosed graph and Y is m-closed, mY has property (mR) then f is w.(q , m).c. Proof. Let V be a mY -open set then by Lemma 4, mY -Cl(V ) mR(Y ) and Y -(mY -Cl(V )) mR(Y ). By the m-closedness of Y , Y -(mY -Cl(V )) is m-closed. By Corollary 1, f -1 (Y - (mY -Cl(V ))) = X - f -1 (mY -Cl(V )) is quasi-closed, hence f -1 (mY -Cl(V )) is quasi open. Then f -1 (V ) intq (f -1 (mY -Cl(V ))) and by Theorem 3, f is w.(q , m).c.

Annals of the Alexandru Ioan Cuza University - Mathematics – de Gruyter

**Published: ** Nov 24, 2014

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