Generalization of Perfectly Continuous, Regular Set-Connected and Clopen Functions

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 continuous, 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 almost clopen functions including the classes of perfectly continuous, regular set- connected and clopen functions. Furthermore, properties of almost clopen functions 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 including perfectly continuous, regular set-connected and clopen functions to obtain properties of this new class of functions and to investigate relationships 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 relationships 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 ).

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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 containing 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)

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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). Take V = f‡ (U). We have, by (4), V ∈ τco and x ∈ V , hence, f(V ) ⊂ G. (12) ⇒ (11). Let G be any regular open set of Y . Then, G ∈ υs. By (12), −1 −1 f (G) ∈ τco. Hence, f (G) is a union of clopen sets of (X, τ). (12) ⇔ (13). This is obvious.

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Definition 7. Let f :(X, τ) → (Y, υ) be a function. The functions fco : (X, τco) → (Y, υ), fs :(X, τ) → (Y, υs) and fco−s :(X, τco) → (Y, υs) associ- ated with f :(X, τ) → (Y, υ) are deﬁned as follows: fco(x) = fs(x) = fco−s(x) = f(x) for each x ∈ X. Corollary 8. Let f :(X, τ) → (Y, υ) be a function. Then the following statements are equivalent: (1) f is almost clopen; (2) fco is almost continuous; (3) fs is clopen; (4) fco−s is continuous. Proof. This is obvious from Theorem 6. Theorem 9. If f : X → Y is an almost clopen function and A is any subset of X, then the restriction f|A : A → Y is almost clopen. Proof. Let x ∈ A and let V be an open set in Y containing f(x). Since f is almost clopen, then there exists a clopen set U in X containing x such that f(U) ⊂ W = int cl (V ). We have U ⊂ f −1(W ). Then, U ∩ A ⊂ f −1(W ) −1 ∩ A = f|A (W ) and U ∩ A is clopen in A. Hence, f|A is almost clopen.

Definition 10. A cover Σ = {Uα : α ∈ I} of subsets of X is called a co-cover if Uα is clopen for each α ∈ I.

Theorem 11. Let f : X → Y be a function and Σ = {Uα : α ∈ I} be a co-cover of X. If for each α ∈ I, f| is almost clopen, then f : X → Y is Uα an almost clopen function. Let f| be an almost clopen function for each α ∈ I. Let V Proof. Uα be a regular open set in Y . Since f| is almost clopen for each α ∈ I, by Uα −1 −1 −1 Theorem 6, f| (V ) ⊂ int f| (V ) . We have f (V ) ∩ Uα Uα (τUα )co ( Uα ) −1 −1 ⊂ int f (V ) ∩ Uα = intτ f (V ) ∩ Uα. Since Σ = {Uα : α ∈ I} (τUα )co co −1 −1 is a co-cover of X, f (V ) ⊂ intτco f (V ) . Thus, by Theorem 6, f is almost clopen. Theorem 12. Let f : X → Y be a function and let g : X → X × Y be the graph function of f, deﬁned by g(x) = x, f(x) for every x ∈ X. If g is almost clopen, then f is almost clopen. Proof. Let G ∈ RO(Y ), then X × G = X × int cl (G) = int cl (X) × int cl (G) = int cl (X × G). Therefore, X × G ∈ RO(X × Y ). It follows −1 −1 from Theorem 6 that f (G) = g (X × G) ∈ τco. Thus f is almost clopen.

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Theorem 13. Let f : X → Y and g : Y → Z be functions. Then, the following properties hold: (1) If f is clopen and g is almost clopen, then g ◦ f : X → Z is almost clopen. (2) If f is clopen and g is almost continuous, then g ◦ f : X → Z is almost clopen. (3) If f is regular set-connected and g is almost clopen, then g ◦ f : X → Z is almost clopen. (4) If f is almost clopen and g is regular set-connected, then g ◦ f : X → Z is almost clopen. (5) If f is perfectly continuous and g is almost clopen, then g ◦ f : X → Z is almost clopen. (6) If f is almost clopen and g is perfectly continuous, then g ◦ f : X → Z is clopen.

Proof. (1) Let V be any open set in Z containing (g ◦ f)(x). Since g is almost clopen, there exists a clopen set U in Y containing f(x) such that g(U) ⊂ int cl (V ). Since f is clopen, there exists a clopen set G in X containing x such that f(G) ⊂ U. This shows that (g ◦ f)(G) ⊂ int cl (V ). Therefore, g ◦ f is almost clopen. (2) It can be obtained similarly. (3) Let V be any open set in Z containing (g ◦ f)(x). Since g is almost clopen, there exists a clopen set U in Y containing f(x) such that g(U) ⊂ int cl (V ). Since f is regular set-connected, f −1(U) is clopen in X. Take G = f −1(U). Then f(G) ⊂ U. This shows that (g ◦ f)(G) ⊂ int cl (V ). Therefore, g ◦ f is almost clopen. (4), (5) and (6) are obvious similarly.

Lemma 14 (Singal and Singal [12]). Let {Xα : α ∈ I} be a family of Y spaces and let X = Xα be the product space. If x = (xα) ∈ X and V is α∈I a regular open subset of X containing x, then there exists a basic regular Y Y open set Vα such that x ∈ Vα ⊂ V where Vα is regular open in Xα for α∈I α∈I each α ∈ I and Vα = Xα for all α ∈ I except for a ﬁnite number of indices αi, i = 1, 2, 3, . . . , n.

Theorem 15. Let {fα : Xα → Yα | α ∈ I} be a family of almost clopen Y Y functions. Then the product function f : Xα → Yα deﬁned by f (xα) α∈I α∈I = fα(xα) is almost clopen.

Acta Mathematica Hungarica 107, 2005 PERFECTLY CONTINUOUS, REGULAR SET-CONNECTED AND CLOPEN FUNCTIONS 199 Y Y Proof. Let (xα) ∈ Xα and U be a regular open subset of Yα con- α∈I α∈I Y taining f (xα) . Then by Lemma 14, there exists a basic open set Gα α∈I where Gα = Yα for α 6= αi, i = 1, 2, 3, . . . , n and Gα is a regular open subset Y of Yα for α = α1, α2, α3, . . . , αn such that f (xα) ∈ Gα ⊂ U. We have α∈I fαi (xαi ) ∈ Gαi = int cl (Gαi ) for each i. Now since for i = 1, 2, 3, . . . , n, fα is almost clopen, then for i = 1, 2, 3, . . . , n, there exists a clopen set Vα i i containing xαi such that fαi (Vαi ) ⊂ int cl (Gαi ) . Y Let V = Vα where Vα = Xα for α 6= αi, i = 1, 2, 3, . . . , n. Then α∈I Y (xα) ∈ V and f(V ) ⊂ U where V is clopen set in Xα. This shows that f α∈I is almost clopen.

4. Separation axioms

In this section, we investigate the relationships between almost clopen functions and separation axioms.

Definition 16. A space X is said to be r-T1 if for each pair of distinct points x and y of X, there exist regular open sets U and V containing x and y respectively such that y 6∈ U and x 6∈ V .

Definition 17. A space X is said to be clopen T1 if for each pair of distinct points x and y of X, there exist clopen sets U and V containing x and y respectively such that y 6∈ U and x 6∈ V . Remark 18. The following implications hold for a topological space X: (1) r-T1 ⇒ T1, (2) clopen T1 ⇒ T1. None of these implications is reversible. Example 19. Let R be the real numbers with the ﬁnite complements topology τ. Then (R, τ) is T1 but not r-T1 and not clopen T1. Theorem 20. If f : X → Y is an almost clopen injection and Y is r-T1, then X is clopen T1.

Proof. Suppose that Y is r-T1. For any distinct points x and y in X, there exist V , W ∈ RO(Y ) such that f(x) ∈ V , f(y) 6∈ V , f(x) 6∈ W and f(y) ∈ W . Since f is almost clopen, by Theorem 6, there exist clopen sets

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G and H in X containing x and y respectively such that f(G) ⊂ V and f(H) ⊂ W . Thus, we obtain y 6∈ G and x 6∈ H. This shows that X is clopen T1. Definition 21. A space X is said to be clopen T2 (clopen Hausdorﬀ) if for each pair of distinct points x and y in X, there exist disjoint clopen sets U and V in X such that x ∈ U and y ∈ V .

Theorem 22. If f : X → Y is an almost clopen injection and Y is T2, then X is clopen T2. Proof. For any pair of distinct points x and y in X, there exist disjoint open sets U and V in Y such that f(x) ∈ U and f(y) ∈ V . Since f is almost clopen, there exist clopen sets G and H in X containing x and y, respectively, such that f(G) ⊂ int cl (U) and f(H) ⊂ int cl (V ). Since int cl (U) ∩ int cl (V ) = ∅, then G ∩ H = ∅. This shows that X is clopen T2. Theorem 23. If f, g : X → Y are almost clopen functions and Y is Hausdorff, then E = x ∈ X : f(x) = g(x) is closed in X. Proof. If x ∈ X\E, then it follows that f(x) 6= g(x). Since Y is Haus- dorff, there exist open sets U and W containing f(x) and g(x), respectively, such that V ∩ W = ∅. Since f and g are almost clopen, there exist clopen sets U and G in X with x ∈ U and x ∈ G such that f(U) ⊂ int cl (V ) and g(G) ⊂ int cl (W ). Set O = U ∩ G. Then O is open. Therefore f(O) ∩ g(O) = ∅ and it follows that x 6∈ cl (E). This shows that E is closed in X. Theorem 24. If f : X → Y is an almost clopen function and Y is Haus- dorff, then E = (x, y) ∈ X × X : f(x) = f(y) is closed in X × X. Proof. Let (x, y) ∈ (X × X)\E. It follows that f(x) 6= f(y). Since Y is Hausdorff, there exist open sets V and W containing f(x) and f(y), respectively, such that V ∩ W = ∅. Since f is almost clopen, there exist clopen sets U and G in X containing x and y, respectively, such that f(U) ⊂ int cl (V ) and f(G) ⊂ int cl (W ). Hence, (U × G) ∩ E = ∅. We have U × G is open in X × X containing (x, y). This means that E is closed in X × X. Definition 25. A space is called almost regular if for each regular closed set F and each point x 6∈ F , there exist disjoint open sets U and V such that F ⊂ U and x ∈ V (Singal and Arya [13]). Definition 26. A space is said to be mildly normal if for every pair of disjoint regular closed subsets F1 and F2 of X, there exist disjoint open sets U and V such that F1 ⊂ U and F2 ⊂ V (Singal and Singal [15]). Theorem 27. If f is an almost clopen injective open function from a regular space X onto a space Y , then Y is almost regular.

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Proof. Let F be a regular closed set in Y and y 6∈ F . Take y = f(x). Since f is almost clopen, by Theorem 6, f −1(F ) is an intersection of clopen −1 \ \ sets Gα of X for α ∈ I. Then f (F ) = Gα. Take G = Gα. We have α∈I α∈I x 6∈ G. Since X is regular, there exist disjoint open sets U and V such that G ⊂ U and x ∈ V . We obtain that F = f(G) ⊂ f(U) and y = f(x) ∈ f(V ) such that f(U) and f(V ) are disjoint open sets. This shows that Y is almost regular. Theorem 28. If f is an almost clopen injective open function from a normal space X onto a space Y , then Y is mildly normal.

Proof. Let F1 and F2 be disjoint regular closed subsets of Y . Since f −1 is almost clopen, by Theorem 6, f (F1) is an intersection of clopen sets −1 Uα of X for α ∈ I and f (F2) is a intersection of clopen sets Vβ of X for −1 \ −1 \ \ β ∈ J. Then f (F1) = Uα and f (F2) = Vβ. Take U = Uα and α∈I β∈J α∈I \ V = Vβ. We have U ∩ V = ∅. Since X is normal, there exist disjoint β∈J open sets A and B such that U ⊂ A and V ⊂ B. We obtain that F1 = f(U) ⊂ f(A) and F2 = f(V ) ⊂ f(B) such that f(A) and f(B) are disjoint open sets. Thus Y is mildly normal.

5. Connectedness

In this section we study the relationships between almost clopen functions and connectedness. Definition 29. A space X is said to be almost connected if X cannot be written as a disjoint union of two non-empty regular open sets. Theorem 30. If f : X → Y is an almost clopen surjective function and X is a connected space, then Y is an almost connected space. Proof. Suppose that Y is not an almost connected space. Then Y can be written as Y = U ∪ V such that U and V are disjoint non-empty regular open sets. Since f is almost clopen, then f −1(U) and f −1(V ) are unions of clopen sets of X. We have X = f −1(U) ∪ f −1(V ) such that f −1(U) and f −1(V ) are disjoint. This shows that X is not connected. This contradicts that Y is not almost connected as assumed. Hence, Y is almost connected. Definition 31. A topological space X is called hyperconnected if every non-empty open set is dense (Steen and Seebach [17]).

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Theorem 32. If X is connected and f : X → Y is almost clopen and surjective, then Y is hyperconnected. Proof. Assume that Y is not hyperconnected. Then there exists a non- empty open set V such that V is not dense in Y . Then there exist disjoint non-empty regular open subsets B1 and B2 in Y , namely int cl (V ) and −1 Y \ cl (V ). Since f is almost clopen and onto, by Theorem 6, A1 = f (B1) −1 and A2 = f (B2) are disjoint non-empty open subsets of X. By assump- tion, the connectedness of X implies that A1 and A2 must intersect. By contradiction, Y is hyperconnected.

6. Clopen almost closed graphs

In this section, we investigate the relationships between clopen almost closed graphs and almost clopen functions. 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 33. A graph G(f) of a function f : X → Y is said to be clopen almost closed if for each (x, y) ∈ (X × Y )\G(f), there exist a clopen set U of X and a V ∈ RO(Y ) such that (x, y) ∈ U × V and (U × V ) ∩ G(f) = ∅. Theorem 34. If f : X → Y is almost clopen and Y is Hausdorff, then G(f) is clopen almost closed in X × Y . Proof. Suppose that Y is Hausdorff. Let (x, y) ∈ (X × Y )\G(f), then f(x) 6= y. Since Y is Hausdorff, there exist open sets U and V containing f(x) and y, respectively, such that U ∩ V = ∅. Since f is almost clopen, there exists a clopen set A in X such that x ∈ A and f(A) ⊂ int cl (U). Therefore, we obtain y ∈ int cl (V ) ∈ RO(Y ) and f(A) ∩ int cl (V ) = ∅. This shows that G(f) is clopen almost closed. Theorem 35. Let f : X → Y have a clopen almost closed graph G(f). If f is injective, then X is clopen T1. Proof. Let x and y be any two distinct points of X. Then we have x, f(y) ∈ (X × Y )\G(f). By clopen almost closedness of the graph G(f), there exist a clopen set U of X and V ∈ RO(Y ) such that x, f(y) ∈ U × V and (U × V ) ∩ G(f) = ∅. Then we have f(U) ∩ V = ∅, hence U ∩ f −1(V ) = ∅. Therefore we have y 6∈ U. This implies that X is clopen T1. Theorem 36. Let f : X → Y have a clopen almost closed graph G(f). If f is a surjective open function, then Y is T2.

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Proof. Let y1 and y2 be any distinct points of Y . Since f is surjective, f(x) = y1 for some x ∈ X and (x, y2) ∈ (X × Y )\G(f). By clopen almost closedness of the graph G(f), there exist a clopen set U of X and V ∈ RO(Y ) such that (x, y2) ∈ U × V and (U × V ) ∩ G(f) = ∅. Then we have f(U) ∩ V = ∅. Since f is open, then f(U) is open such that f(x) = y1 ∈ f(U). This implies that Y is T2. Corollary 37. Let f : X → Y have a clopen almost closed graph G(f). If f is a surjective open function, then Y is T1. Definition 38. A subset A of a space X is N-closed if every cover of A by regular open sets of X has a ﬁnite subcover (Corhan [2]). Theorem 39. If a function f : X → Y has a clopen almost closed graph G(f), f −1(K) is closed in X for every subset K which is N-closed.

Proof. Let K be N-closed and x 6∈ f −1(K). For each y ∈ K, we have (x, y) ∈ (X × Y )\G(f) and there exist a clopen neighborhood Uy of x in X and Vy ∈ RO(Y ) such that y ∈ Vy and f(Uy) ∩ Vy = ∅. The family {Vy : y ∈ K} is a cover of K by regular open sets of Y and there exists [ a ﬁnite number of points, say, y1, y2, . . . , yn of K such that K ⊂ Vyi : \ i = 1, 2, . . . , n . Put U = Uyi : i = 1, 2, . . . , n , then U is an open neighborhood of x and f(U) ∩ K = ∅. This shows that f −1(K) is closed in X.

7. Compactness

In this section, we investigate the relationships between almost clopen functions and compactness. Definition 40. A space X said to be (1) mildly compact if every clopen cover of X has a finite subcover (Staum [16]). (2) mildly Lindelöfif every cover of X by clopen sets has a countable subcover (Staum [16]). (3) nearly compact if every regular open cover of X has a finite subcover (Singal, Singal and Mathur [14]). (4) nearly countably compact if every countable cover of X by regular open sets has a finite subcover (Ergun [4]). (5) nearly Lindelöfif every cover of X by regular open sets has a countable subcover.

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Theorem 41. Let f : X → Y be an almost clopen surjection. Then the following statements hold: (1) if X is mildly compact, then Y is nearly compact. (2) if X is mildly Lindel¨of,then Y is nearly Lindel¨of. (3) if X is compact, then Y is nearly countably compact.

Proof. We prove only (1), the proofs of (2) and (3) being entirely anal- ogous. Let {Vα : α ∈ I} be any regular open cover of Y . Since f is almost clopen, by Theorem 6, for each α ∈ I, there exists a family Ψ = Giα : Giα −1 [ [ is a clopen set, iα ∈ Iα such that f (Vα) = Giα . Take ∆ = Iα. iα∈Iα α∈I Then Φ = Giα : iα ∈ ∆ is a clopen cover of X and hence there exists a [ finite subset ∆0 of ∆ such that X = Giα : iα ∈ ∆0 . We have X = [ −1 −1 [ f (Vα): Giα ⊂ f (Vα), iα ∈ ∆0 . Therefore we obtain Y = Vα : −1 Giα ⊂ f (Vα), iα ∈ ∆0 and Y is nearly compact. Corollary 42. Let f : X → Y be an almost clopen surjection. Then the following statements hold: (1) if X is compact (nearly compact), then Y is nearly compact. (2) if X is Lindelöf(nearly Lindelöf), then Y is nearly Lindelöf.

8. Comparison and examples

In this section, we investigate relationships between almost clopenness and other related functions. Remark 43. The following diagram holds:

clopen ⇒ almost clopen ⇒ almost continuous ⇑ ⇑ perfectly continuous ⇒ regular set-connected

None of the implications is reversible for almost clopenness as shown by the following examples.

Example 44. 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 clopen.

Acta Mathematica Hungarica 107, 2005 PERFECTLY CONTINUOUS, REGULAR SET-CONNECTED AND CLOPEN FUNCTIONS 205

Example 45. Let R and Q be the real numbers and rational numbers, respectively. Let A = {x ∈ R : x is rational and 0 < x < 1}. We deﬁne two topologies on R as τ = {R, ∅, A, R\A} and υ = R, ∅, {0} . Let f :(R, τ) → (R, υ) be a function which is deﬁned by f(x) = 1 if x ∈ Q and f(x) = 0 if x 6∈ Q. Then f is almost clopen, but f is not clopen since for f(x) = 0 ∈ {0} ∈ υ (x 6∈ Q), there is no clopen set U containing x such that f(U) ⊂ {0}.

Example 46. 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, since (0, 1) is not clopen in X, f is not regular set- connected. However, f is almost clopen because (a, b] is both closed and open in X.

Theorem 47. If f : X → Y is almost clopen from a space X into a semi-regular space Y , then f is clopen.

Proof. Let x ∈ X and let U be an open set with f(x) ∈ U. Since Y is a semi-regular space, there exists an open set G in Y containing f(x) such that G ⊂ int cl (G) ⊂ U. By almost clopenness of f, there exists a clopen set V containing x such that f(V ) ⊂ int cl (G) ⊂ U. Thus f is clopen. Recall that a space is 0-dimensional if its topology has a base consisting of clopen sets. Theorem 48. If f : X → Y is almost continuous and X is a 0-dimensional space, then f is almost clopen.

Proof. Let x ∈ X and let V be a regular open subset of Y containing f(x). Since f is almost continuous, there exists an open set U in X containing x such that f(U) ⊂ V . Since X is 0-dimensional, there exists a clopen set G in X containing x such that G ⊂ U. Hence we obtain that f(G) ⊂ V and hence f is almost clopen. Definition 49. A space X is called Alexandroﬀ space if any intersection of open sets is open (Alexandroﬀ [1]).

Theorem 50. If f : X → Y is almost clopen and X is an Alexandroﬀ space, then f is regular set-connected.

Proof. Let V be a regular open subset of Y . By Theorem 6, f −1(V ) is a union of clopen subsets of X. Since X is Alexandroﬀ, f −1(V ) ∈ RO(X). Thus, f is regular set-connected.

Acknowledgements. I would like to express my sincere gratitude to the referees and to the editors.

Acta Mathematica Hungarica 107, 2005 206 E. EKICI: PERFECTLY CONTINUOUS, REGULAR SET-CONNECTED . . .

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(Received September 3, 2003; revised October 27, 2003)

DEPARTMENT OF MATHEMATICS CANAKKALE ONSEKIZ MART UNIVERSITY TERZIOGLU CAMPUS 17020 CANAKKALE TURKEY E-MAIL: [email protected]

Acta Mathematica Hungarica 107, 2005