Acta Math. Hungar. 107 (3) (2005), 193–206. GENERALIZATION OF PERFECTLY CONTINUOUS, REGULAR SET-CONNECTED AND CLOPEN FUNCTIONS E. EKICI (Canakkale) Abstract. Noiri in 1984, Dontchev, Ganster and Reilly in recent years and Reilly and Vamanamurthy in 1983 introduced the notion of perfectly continu- ous, regular set-connected and clopen functions, respectively. The aim of this paper is to introduce the notion of a new class of functions which is called al- most clopen functions including the classes of perfectly continuous, regular set- connected and clopen functions. Furthermore, properties of almost clopen func- tions are obtained and relationships among almost clopenness, perfect continuity, regular set-connectedness, clopenness and almost continuity are investigated. 1. Introduction The notion of perfectly continuous function is introduced and studied by Noiri [9], [10] and the notion of clopen functions is introduced by Reilly and Vamanamurthy [11]. Almost continuity is introduced by Singal and Singal [12] and it is studied in several papers by Noiri [6]–[8], and by Nasef and Noiri [5], etc. Recently, a new class of functions called regular set-connected has been introduced by Dontchev, Ganster and Reilly [3]. The purpose of this paper is to obtain a new class of functions includ- ing perfectly continuous, regular set-connected and clopen functions to ob- tain properties of this new class of functions and to investigate relation- ships among almost clopenness, perfect continuity, regular set-connectedness, clopenness and almost continuity. In Section 3, we obtain characterizations and basic properties of almost clopen functions. In Sections 4 and 5, we study and investigate relation- ships between almost clopenness and separation axioms and between almost clopenness and connectedness, respectively. In Section 6, we introduce clopen almost closed graphs and study relationships between almost clopenness and clopen almost closed graphs. In Section 7, we investigate relationships be- Key words and phrases: regular open, clopen, almost clopen, regular set-connected, perfect continuity, clopen almost closed graphs. 2000 Mathematics Subject Classification: 54C10. 0236–5294/$ 20.00 c 2005 Akad´emiai Kiad´o,Budapest 194 E. EKICI tween almost clopenness and compactness. In the last section, we investigate relationships among several functions containing almost clopen, perfectly continuous, regular set-connected, clopen and almost continuous functions. 2. Preliminaries Throughout the present paper, X and Y are always topological spaces. Let A be a subset of X. We denote the interior and the closure of a set A by int (A) and cl (A), respectively. A subset A of a space X is said to be regular open (regular closed) [18] if A = int cl (A) (A = cl int (A)). The family of all regular open (regular closed, clopen) sets of X is denoted by RO(X)(RC(X), CO(X)). For a given topological space (X, τ), the collection of all regular open sets forms a base for a topology τs, coarser than τ, called the semiregularization. In the case when τ = τs, the space (X, τ) is called semi-regular (Stone [18]). Furthermore, the collection of all clopen sets forms a base for a topology τco. Definition 1. A function f : X → Y is called almost continuous if for each x ∈ X and each open set V in Y containing f(x), there exists an open set U in X containing x such that f(U) ⊂ int cl (V ) (Singal and Singal [12]). Definition 2. A function f : X → Y is said to be perfectly continuous if f −1(V ) is clopen in X for every open set V of Y (Noiri [9]). Definition 3. A function f : X → Y is said to be clopen if for each x ∈ X and each open set V of Y containing f(x), there exists a clopen set U of X containing x such that f(U) ⊂ V (Reilly and Vamanamurthy [11]). Definition 4. A function f : X → Y is said to be regular set-connected if f −1(V ) ∈ CO(X) for every V ∈ RO(Y ) (Dontchev, Ganster and Reilly [3]). 3. Almost clopen functions Definition 5. A function f : X → Y is said to be almost clopen if for each x ∈ X and each open set V in Y containing f(x), there exists a clopen set U in X containing x such that f(U) ⊂ int cl (V ). Acta Mathematica Hungarica 107, 2005 PERFECTLY CONTINUOUS, REGULAR SET-CONNECTED AND CLOPEN FUNCTIONS 195 Theorem 6. Let (X, τ) and (Y, υ) be topological spaces. The following statements are equivalent for a function f : X → Y : (1) f is almost clopen; (2) for each x ∈ X and each regular open set G in Y containing f(x), there exists a clopen set U in X containing x such that f(U) ⊂ G; (3) for each x ∈ X and each regular closed set F in Y not contain- ing f(x), there exists a clopen set U in X not containing x such that f −1(F ) ⊂ U; (4) the inverse image of every regular open set of Y is a union of clopen sets of X; (5) the inverse image of every regular closed set of Y is an intersection of clopen sets of X; −1 −1 (6) f (G) ⊂ intτco (f int cl (G) ) for every open subset G of Y ; −1 −1 (7) f (V ) ⊂ intτco f (V ) for every regular open subset V of Y ; −1 −1 (8) clτco (f cl int (F ) ) ⊂ f (F ) for every closed subset F of Y ; −1 −1 (9) clτco f (K) ⊂ f (K) for every regular closed subset K of Y ; (10) f∗ :(X, τco) → (Y, υ), which is defined by f∗(x) = f(x) for each x ∈ X, is almost continuous; (11) f∗∗ :(X, τ) → (Y, υs), which is defined by f∗∗(x) = f(x) for each x ∈ X, is clopen; (12) f‡ :(X, τco) → (Y, υs), which is defined by f‡(x) = f(x) for each x ∈ X, is continuous; (13) for each net (xi)i∈I in X, if xi →τco x, then f(xi) →υs f(x). Proof. (1) ⇒ (2). Let G be any regular open set in Y containing f(x). Since G is open, then by (1), it follows that there exists a clopen set U in X containing x such that f(U) ⊂ int cl (G) = G. (2) ⇒ (1). Let V be any open set in Y containing f(x). Since int cl (V ) is regular open, then by (2), it follows that there exists a clopen set U in X containing x such that f(U) ⊂ int cl (V ). (2) ⇔ (3). Let F be any regular closed set in Y not containing f(x). Then, Y \F is a regular open set containing f(x). By (2), there exists a clopen set U in X containing x such that f(U) ⊂ Y \F . Hence, U ⊂ f −1(Y \F ) ⊂ X\f −1(F ) and then f −1(F ) ⊂ X\U. Take H = X\U. We obtain that H is a clopen set in X not containing x. The converse can be shown easily. (2) ⇔ (4). Let G be any regular open set in Y and let x ∈ f −1(G). Since f(x) ∈ G, by (2), there exists a clopen set Ux in X containing x such that −1 −1 [ Ux ⊂ f (G). Hence, we have f (G) = Ux. x∈f −1(G) Acta Mathematica Hungarica 107, 2005 196 E. EKICI The converse can be shown easily. (4) ⇔ (5). This can be obtained easily. (1) ⇔ (6). Let G be an open subset of Y and let x ∈ f −1(G). Since f(x) ∈ G, by (1), there exists a clopen set U in X containing x such that f(U) ⊂ int cl (V ). We have x ∈ U ⊂ f −1(int cl (V )). By the definition −1 of interior, x ∈ intτco (f int cl (V ) ). The converse can be shown by the definition of interior. (6) ⇔ (7). Obvious. (6) ⇔ (8). Let F be a closed subset of Y . Then Y \F is open. By (6), −1 −1 f (Y \F ) ⊂ intτco f (int cl (Y \F ) ) −1 −1 and X\f (F ) ⊂ intτco (f int cl (Y \F ) ). We have −1 −1 intτco (f int cl (Y \F ) ) = intτco f (int Y \ int (F ) ) −1 −1 = intτco f (Y \ cl int (F ) ) = intτco (X\f (cl int (F ) ) −1 = X\ clτco f (cl int (F ) ) . −1 −1 Hence, we obtain that clτco (f cl int (F ) ) ⊂ f (F ). The converse is similar. (8) ⇔ (9). Obvious. (1) ⇔ (10). Let G be any open set in Y containing f∗(x). By (1), there exists a clopen set U in X containing x such that f(U) = f∗(U) ⊂ int cl (G) . We have U ∈ τco. Hence f∗ is almost continuous. The converse can be obtained easily. (1) ⇔ (11). Let G ∈ υs with f∗∗(x) ∈ G. By the definition of υs, there exists a regular open subset V in (Y, υ) such that f∗∗(x) ∈ V ⊂ G. Since V ∈ υ, by (1), there exists a clopen set U in X containing x such that f(U) = f∗∗(U) ⊂ int cl (V ) = V ⊂ G. Now we prove the converse. Let G be any open set in Y containing f(x). Since int cl (G) ∈ υs and f∗∗(x) ∈ int cl (G) , by (11), there exists a clopen set U in X containing x such that f(U) = f∗∗(U) ⊂ int cl (G) . (4) ⇒ (12). Let G ∈ υs with f‡(x) ∈ G. By the definition of υs, there exists a regular open subset U in (Y, υ) such that f‡(x) ∈ U ⊂ G. Then, −1 −1 −1 x ∈ f‡ (U) ⊂ f (G).