Between Open Sets and Semi-Open Sets

Transcription

1 Univ. Sci. 23 (1): 9-20, doi: /Javeriana.SC23-1.bosa Bogotá original article Between Open Sets and Semi-Open Sets Samer Al Ghour 1, *, Kafa Mansur 1 Edited by Juan Carlos Salcedo-Reyes (salcedo.juan@javeriana.edu.co) 1. Jordan University of Science and Technology, Faculty of Science and Arts, Department of Mathematics and Statistics 22110, Irbid, Jordan, * algore@just.edu.jo Received: Accepted: Published on line: Citation: Al Ghour S, Mansur K Between Open Sets and Semi-Open Sets, Universitas Scientiarum, 23 (1): 9-20, doi: /Javeriana.SC23-1.bosa Funding: N.A. Electronic supplementary material: N.A. Abstract We introduce and investigate ω s -open sets as a new class of sets which lies strictly between open sets and semi-open sets. Then we use ω s -open sets to introduce ω s -continuous functions as a new class of functions between continuous functions and semi-continuous functions. We give several results and examples regarding our new concepts. In particular, we obtain some characterizations of ω s -continuous functions. Keywords: Semi-open set; ω-open set; Semi-continuous function Introduction Let (X,τ) be a topological space and A X. We will denote the complement of A in X, the closure of A, the interior of A, the exterior of A, and the relative topology on A, by X A, A, Int(A), Ext(A), and τ A, respectively. In 1963, Levine [7] defined semi-open sets as a class of sets containing the open sets as follows: A is semi-open if there exists an open set U such that U A U, this is equivalent to say that A Int(A). Using semi-open sets he also generalized continuity by semi-continuity as follows: A function f : (X,τ 1 ) (Y,τ 2 ) is semi-continuous if for all V τ 2, the preimage f 1 (V ) SO(X,τ 1 ). The complement of a semi-open set is called semi-closed [5]. A point x X is called a condensation point [6] of A if for every U τ with x U, the set U A is uncountable. Hdeib [6] defined ω-closed sets and ω-open sets as follows: A is called ω-closed if it contains all its condensation points. The complement of an ω-closed set is called ω-open. The collection of all ω-open sets of a topological space (X,τ) will be denoted by τ ω. In [1], the author proved that (X,τ ω ) is a topological space and τ τ ω. Moreover, it was observed that A is ω-open if and only if for every x in A there is an open set U and a countable subset C such that x U C A. The ω-closure of A in (X,τ), denoted by A ω, is the smallest ω-closed set in (X,τ) that contains A (cf. [1]). The ω-interior of Universitas Scientiarum, Journal of the Faculty of Sciences, Pontificia Universidad Javeriana, is licensed under the Creative Commons Attribution 4.0 International Public License

2 10 Open Sets A in (X,τ), denoted by (A), is the largest ω-open set in (X,τ) contained in A. The ω-exterior of A in (X,τ), denoted by Ext ω (A), is defined to be (X A). It is clear that the ω-closure (resp. ω-interior) of A in (X,τ) equals the closure (resp. interior) of A in (X,τ ω ). In 2002, Al-Zoubi and Al-Nashef [2] used ω-open sets to define semi ω-open sets as a weaker form of semi-open sets as follows: Ais semi ω-open if there exists an ω-open set U such that U A U. The collection of all semi ω-open sets of a topological space (X,τ) will be denoted by SωO Al-Zoubi [4] used semi ω-open sets to introduce semi ω-continuous functions as a weaker form of ω-continuous functions as follows: A function f : (X,τ 1 ) (Y,τ 2 ) is semi ω-continuous [4] if for all V τ 2, the preimage f 1 (V ) SωO(X,τ 1 ). This paper is devoted to define ω s -opennes as a property of sets that is strictly weaker than openness and stronger than semi-openness as follows: A is ω s -open if there exists an open set U such that U A U ω. We investigate this class of sets, and use it to study a new property of functions strictly between continuity and semi-continuity, and another new property of functions strictly between slight continuity and slight semi-continuity. Throughout this paper,,, and c, will denote the set of real numbers, the set of natural numbers, the set of rational numbers, and the set of irrational numbers, respectively. For any non-empty set X we denote by τ disc the discrete topology on X. Finally, by τ u we mean the usual topology on. The following sequence of theorems will be useful in the sequel: Theorem 1.1 ([3]). Let (X, τ) be a topological space and A X. Then (a) If A is non-empty, then (τ A = (τ ω ) A. (b) (τ ω = τ ω. Theorem 1.2 ([2]). Let (X, τ) be a topological space. Then (a) SO(X,τ) SωO(X,τ), and SO(X,τ) SωO(X,τ) in general. (b) τ ω SωO(X,τ), and τ ω SωO(X,τ) in general. Theorem 1.3 ([1]). Let (X, τ) be a topological space. Then (a) If (X,τ) is anti-locally countable, then A ω = A for all A τ ω, and (A) = Int(A) for all ω-closed set A in (b) If (X,τ) is locally countable, then τ ω is the discrete topology. ω s -Open sets Definition 2.1. Let A of be a subset of a topological space Then A is called ω s -open of (X,τ), if there exists U τ such that U A U ω and A

3 Al Ghour & Mansur 11 is called ω s -closed if X A is ω s -open. The family of all ω s -open subsets of (X,τ) will be denoted by ω s Theorem 2.2. Let (X,τ) be a topological space. SO Then τ ω s (X,τ) Proof. Let A τ. Take U = A. Then U τ and U A U ω. This shows that A ω s It follows that τ ω s Let A ω s Then there exists U τ such that U A U ω, but U ω U. Thus A SO This shows that ω s (X,τ) SO In the following example we will see that, in general, neither of the two inclusions in Theorem 2.2 are equalities: Example 2.3. Consider (,τ), where τ = {,,, c, c }. It is not difficult to check that ω =, =, and c ω =. Thus SO(X,τ) ω s (X,τ) and ω s (X,τ) τ. Theorem 2.4. Let (X,τ) be a topological space. Then (a) If (X,τ) is anti-locally countable, then ω s (X,τ) = SO (b) If (X,τ) is locally countable, then τ = ω s Proof. (a) By Theorem 2.2 it is sufficient to show that SO(X,τ) ω s Let A SO Then there exists U τ such that U A U. Since (X,τ) is anti-locally countable, then by Theorem 1.3 (a), U = U ω. It follows that A ω s (b) By Theorem 2.2 it is sufficient to show that ω s (X,τ) τ. Let us take A ω s Then there exists U τ such that U A U ω. Since (X,τ) is locally countable, then by Theorem 1.3 (b), U ω = U. It follows that A = U and hence A τ. The following example shows that ω-open sets and ω s -open sets are independent: Example 2.5. Consider (,τ) where τ = {,,[0, )}. It is not difficult to check that [0, =. Thus [ 1, ) ω s (X,τ) τ ω and (0, ) τ ω ω s Theorem 2.6. A subset A of a topological space (X,τ) is ω s -open if and only if A Int(A. Proof. Necessity. Let A be ω s -open. Then there exists some U τ such that U A U ω. Since U A, then U = Int(U ) Int(A) and so U ω Int(A. Therefore, A Int(A.

4 12 Open Sets Sufficiency. Suppose that A Int(A. Take U = Int(A). Then U τ with U A U ω. It follows that A is ω s -open. Theorem 2.7. Arbitrary unions of ω s -open sets in a topological space is ω s -open. Proof. Let (X,τ) be a topological space and let {A α : α } ω s For each α, there exists U α τ such that U α A α U α ω. So, we have U α α τ, with ω ω U α A α U α U α. α α It follows that A α α ω s α Corollary 2.8. If {C α : α } is a collection of ω s -closed subsets of a topological space (X,τ), then {C α : α } is ω s -closed. The following example shows that the intersection of two ω s -open sets need not to be ω s -open in general: Example 2.9. Consider (,τ u ). Let A = [0,1], B = [1,2]. By Theorem 1.3 (a), (0,1 = (0,1) = A, and (1,2 = (1,2) = B. Thus A, B ω s (X,τ), but A B = {1} / ω s Theorem For any topological space, the intersection of two ω s -open sets where one of them is open is also ω s -open. Proof. Let (X,τ) be a topological space, A τ and B ω s Choose a set U τ such that U B U ω. Now we have A U τ, and then A U A B A U ω A U ω. This shows that, A B ω s Corollary For any topological space, the union of two ω s -closed sets where one of them is closed is also ω s -closed. Theorem Let (X,τ) be a topological space, B a non-empty subset of X and A B. Then (a) If A ω s (X,τ), then A ω s (B,τ B ). α (b) If B τ and A ω s (B,τ B ), then A ω s Proof. (a) Let A ω s Then there is U τ such that U A U ω. Then U = U B A U ω B. Note that U ω B is the closure of U in (τ ω ) B and by Theorem 1.1 (a), it is the closure of U in (τ B. This shows that A ω s (B,τ B ). (b) Let B τ and A ω s (B,τ B ). Since A ω s (B,τ B ), there is V τ B such that V A H where H is the closure of V in (B,(τ B ). Since B τ, then V τ. Also, V A H V ω. Therefore, A ω s

5 Al Ghour & Mansur 13 Theorem Let (X,τ) be a topological space. Let A ω s (X,τ) and suppose that A B A ω, then B ω s Proof. Since A ω s (X,τ), there exists U τ such that U A U ω. Since A U ω, then A ω U ω. Since B A ω, then B U ω. Therefore, we have U τ and U A B U ω. This shows that B ω s Theorem For any topological space (X,τ) we have that SO(X,τ ω ) = ω s (X,τ ω ). Proof. By Theorem 2.2, we have ω s (X,τ ω ) SO(X,τ ω ). Conversely, let A SO(X,τ ω ), then there exists U τ ω such that U A H, where H is the closure of U in (X,τ ω ). By Theorem 1.1 (b), we have (τ ω = τ ω and so H = U ω. It follows that A ω s (X,τ ω ). Theorem For any topological space (X,τ) we have the relation τ = {Int(A) : A ω s (X,τ)}. Proof. It follows because from Theorem 2.2 we have τ ω s Theorem A subset C of a topological space (X,τ) is ω s -closed if and only if (C ) C. Proof. Necessity. Suppose that C is ω s -closed in Then X C ω s -closed and by Theorem 2.6, X C Int(X C. So is (C ) Ext ω (X C ) = Ext ω (Ext(C )) = X Ext(C = X Int(X C C. Sufficiency. Suppose that (C ) C. Then X C X (C ) = X Ext ω (X C ) = X Ext ω (Ext(C )) = (Ext(C ) = Int(X C. By Theorem 2.6 it follows that X C is ω s -open, and hence C is ω s -closed.

6 14 Open Sets Definition Let (X,τ) be a topological space and let A X. (a) The ω s -closure of A in (X,τ) is denoted by A ω s and defined as follows: A ω s = {C : C is ω s -closed in (X,τ) and A C }. (b) The ω s -interior of A in (X,τ) is denoted by s (A) and defined as follows: s (A) = {U : U is ω s -open in (X,τ) and U A}. Remark Let (X, τ) be a topological space and let A X. Then (a) A ω s is the smallest ω s -closed set in (X,τ) containing A. (b) A is ω s -closed in (X,τ) if and only if A = A ω s. (c) s (A) is the largest ω s -open set in (X,τ) contained in A. (d) A is ω s -open in (X,τ) if and only if A = s (A). (e) x A ω s if and only if for every B ω s (X,τ) with x B, A B. (f) s (X A) A ω s =. (g) X = s (X A) A ω s. (h) X A ω s = s (X A) and X s (X A) = A ω s. Theorem Let f : (X, τ) (Y, σ) be an open function such that f : (X,τ ω ) (Y,σ ω ) is continuous. Then for every A ω s (X,τ) we have f (A) ω s (Y,σ). Proof. Let A ω s (X,σ). Then there exists U τ such that U A U ω, and so f (U ) f (A) f (U ω ). Since f : (X,τ) (Y,σ) is open, then f (U ) σ. Since f : (X,τ ω ) (Y,σ ω ) is continuous, then f (U ω ) f (U. It follows that f (A) ω s (Y,σ). The condition open function cannot be dropped from Theorem 2.19 as shown by: Example Consider f : (,τ disc ) (,τ u ), where f (x) = 0 for all x. Then it is obvious that f : (,(τ disc ) (,(τ u ) is continuous. On the other hand, {0} ω s (,τ disc ) but f ({0}) = {0} / ω s (,τ u ).

7 Al Ghour & Mansur 15 ω s -Continuous functions Definition 3.1. A function f : (X,τ) (Y,σ) is called ω s -continuous, if for each V σ, the preimage f 1 (V ) ω s (X,σ). Theorem 3.2. The notions of continuity satisfy that (a) Every continuous function is ω s (b) Every ω s -continuous function is semi Proof. Theorem 2.2. The following example will show that the converse of each of the two implications in Theorem 3.2 is not true in general: Example 3.3. Let f, g : (,τ) ({a, b},τ disc ), with τ as in Example 2.3 and a if x a if x c f (x) = and g(x) = b if x b if x. Since f 1 ({a}) = τ ω s (,τ) and f 1 ({b}) = ω s (,τ) τ, then f is ω s -continuous but not continuous. Also, Since g 1 ({a}) = c τ SO(X,τ) and g 1 ({b}) = SO(,τ) ω s (,τ), then f is semi-continuous but not ω s Theorem 3.4. Let f : (X,τ) (Y,σ) be a function. (a) If (X,τ) is locally countable, then f is continuous if and only if f is ω s (b) If (X,τ) is anti-locally countable, then f is ω s -continuous if and only if f is semi Proof. (a) It is a consequence of Theorems 2.4 (a) and 3.2 (a). (b) It is a consequence of Theorems 2.4 (b) and 3.2 (b). Theorem 3.5. A function f : (X,τ) (Y,σ) is ω s -continuous if and only if for every x X and every open set V containing f (x) there exists U ω s (X,τ) such that x U and f (U ) V. Proof. Necessity. Assume that f : (X,τ) (Y,σ) is ω s Let us take V σ with f (x) V. By ω s -continuity, f 1 (V ) ω s Set U = f 1 (V ). Then U ω s (X,τ) satisfies x U and f (U ) V. Sufficiency. Let V σ. For each x f 1 (V ) we have f (x) V, and thus there exists U x ω s (X,τ) such that x U x, f (U x ) V, and x U x f 1 (V ). Thus f 1 (V ) = {U x : x f 1 (V )}. Therefore, by Theorem 2.7, it follows f 1 (V ) ω s This shows that f is ω s

8 16 Open Sets Theorem 3.6. Let f : (X,τ) (Y,σ) be a function. conditions are equivalent: Then the following (a) The function f is ω s (b) Inverse images of all members of a base for σ are in ω s (c) Inverse images of all closed subsets of (Y,σ) are ω s -closed in (d) For every A X we have f (A ω s ) f (A). (e) For every B Y we have f 1 (B s f 1 (B). (f ) For every B Y we have f 1 (Int(B)) s ( f 1 (B)). Proof. (a) = (b). Obvious. (b) = (c). Suppose is a base for σ such that f 1 (B) ω s (X,τ) for every B. Let C be a non-empty closed subset of (Y,σ). Then Y C τ { }. Choose such that Y C = {B : B }. Then X f 1 (C ) = f 1 (Y C ) = f 1 {B : B } = { f 1 (B) : B }. By assumption f 1 (B) ω s (X,τ) for every B, then by Theorem 2.7 we have X f 1 (C ) ω s (X,τ), and hence f 1 (C ) is ω s -closed in (c) = (d). Let A X. Then f (A) is closed in (Y,σ), and by (c) f 1 ( f (A)) is ω s -closed in Since A f 1 ( f (A)) f 1 ( f (A)), and f 1 ( f (A)) is ω s -closed in (X,τ), then A ω s f 1 ( f (A)), and thus f (A ω s ) f ( f 1 ( f (A))) f (A). (d) = (e). Let B Y. Then f 1 (B) X, and by (d) f ( f 1 (B s ) f ( f 1 (B)) B. Therefore, f 1 (B s f 1 (B). (e) = (f). Let B Y. Then by (e), f 1 (Y B s f 1 (Y B). Also by Theorem 2.19 (h), X X f 1 (B s = s ( f 1 (B)). Thus, f 1 (Int(B)) = f 1 (Y Y B) = X f 1 (Y B) X f 1 (Y B s = X X f 1 (B s = s ( f 1 (B)). Lemma 3.7. Let (X, τ) be a topological space and let A X. Then A ω s = A (A).

9 Al Ghour & Mansur 17 Proof. Since A ω s is ω s -closed, then by Theorem 2.15 ((A ω s )) = (A ω s ) A ω s. Therefore, (A) ((A ω s )) A ω s, and hence A (A) A ω s. To see that A ω s = A (A), it is sufficient to show that A (A) is ω s -closed. Since (A) A, then (A) A. Therefore, (A (A)) = (A (A)) = (A) A (A), and by Theorem 2.15 it follows that A (A) is ω s -closed. Theorem 3.8. Let f : (X,τ) (Y,σ) be a function. statements are equivalent: Then the following (a) f is ω s (b) For every A X we have f ( (A)) f (A). (c) For every B Y we have ( f 1 (B)) f 1 (B). Proof. (a) = (b). Suppose that f is ω s Let A X. Then by Theorem 3.6 (d), f (A ω s ) f (A). Therefore, by Lemma 3.7 we have f ( (A)) f (A ω s ) f (A). (b) = (a). We will apply Theorem 3.6 (d). Let A X. Then by (b), we have f ( (A)) f (A). Also, we have f (A) f (A) always. Therefore, by Lemma 3.7 we have f (A ω s ) = f (A (A)) = f (A) f ( (A)) f (A). (a) = (c). Suppose that f is ω s Let B Y. Then by Theorem 3.6 (e), f 1 (B s f 1 (B). Therefore, by Lemma 3.7 we have ( f 1 (B)) f 1 (B s f 1 (B). (c) = (a). We will apply Theorem 3.6 (e). Let B Y. Then by (c), we have ( f 1 (B)) f 1 (B). Also, we have f 1 (B) f 1 (B) always. Therefore, by Lemma 3.7 we have f 1 (B s = f 1 (B) ( f 1 (B)) f 1 (B).

10 18 Open Sets Theorem 3.9. If f : (X,τ) (Y,σ) is ω s -continuous and g : (Y,σ) (Z,λ) is continuous, then g f : (X,τ) (Z,λ) is a ω s Proof. Let V λ. Since g is continuous, then g 1 (V ) σ. Since f is ω s -continuous, then (g f ) 1 (V ) = f 1 (g 1 (V )) ω s In general, the composition of two ω s -continuous functions does not need to be ω s -continuous as the following example clarifies: Example Let f, g : (,τ u ) (,τ u ), where Then f (x) = x if x 1, and g(x) = 0 if x > 1 (g f )(x) = 0 if x 1 3 if x = 1. 0 if x < 1 3 if x 1. Since f and g are obviously semi-continuous and (,τ u ) is anti-locally countable, then by Theorem 3.4 (b) f and g are ω s On the other hand, since (2, ) τ u but (g f ) 1 (2, ) = {1} / ω s (,τ u ), then g f is not ω s Theorem Let { f α : (X,τ) (Y α,σ α )} α be a family of functions. If the function f : (X,τ) ( α Y α,τ prod ) defined by f (x) = ( f α (x)) α is ω s -continuous, then f α is ω s -continuous, for every α. Proof. Suppose that f is ω s -continuous and let β. Then f β = π β f where π β : ( α Y α,τ prod ) (Y β,σ β ) is the projection function on Y β. Since π β is continuous, then by Theorem 3.9, f β is ω s The following example will show that the converse of Theorem 3.11 is not true in general: Example Define f, g : (,τ u ) (,τ u ), and h : (,τ u ) (,τ prod ) by f (x) = 2 if x 0 2 if x > 0, g(x) = 2 if x < 0, and h(x) = ( f (x), g(x)). 2 if x 0 Since f and g are obviously semi-continuous, and (,τ u ) is anti-locally countable, then by Theorem 3.4 (b) f and g are ω s On the other hand, since (0, ) (,0) τ prod but h 1 (0, ) (,0) = {0} / ω s (,τ u ), then h is not ω s

11 Al Ghour & Mansur 19 Theorem Let { f α : (X,τ) (Y α,σ α )} α be a family of functions. If f α0 is ω s -continuous for some α 0, and if f α is continuous for all α {α 0 }, then the function f : (X,τ) ( α Y α,τ prod ) defined by f (x) = ( f α (x)) α is ω s Proof. We will apply statement (b) of Theorem 3.6. Let A be a basic open set of ( α Y α,τ prod ), without loss of generality we may assume that A = πα 1 0 (U α0 ) πα 1 1 (U α1 ) πα 1 n (U αn ), where U αi is a basic open set of Y αi for all i = 0,1,..., n. Then f 1 (A) = ((π α0 f ) 1 (U α0 )) ((π α1 f ) 1 (U α1 )) ((π αn f ) 1 (U αn )) = ( f 1 α 0 (U α0 )) ( f 1 (U α1 )) ( f 1 α n (U αn )). α 1 By assumption fα 1 0 (U α0 ) ω s (X,τ) and fα 1 i (U αi ) τ for all i = 0,1,..., n. Thus ( fα 1 1 (U α1 )) ( fα 1 n (U αn )) τ, and by Theorem 2.9, we have that f 1 (A) ω s It follows that f is ω s Corollary Let f : (X, τ) (Y, σ) be a function and denote by g : (X,τ) (X Y,τ prod ) the graph function of f given by g(x) = (x, f (x)), for every x X. Then g is ω s -continuous if and only if f is ω s Proof. Necessity. Suppose that g is ω s Then by Theorem 3.11, f is ω s Sufficiency. Suppose that f is ω s Note that h(x) = (I (x), f (x)) where I : (X,τ) (X,τ) is the identity functions. Since the function I is continuous, then by Theorem 3.13, g is ω s Conflic of interest The authors declare that they have no conflicts of interest to disclse. References [1] Al Ghour S. Certain covering properties related to paracompact-ness. Doctoral dissertation, University of Jordan, Jordan [2] Al-Zoubi K, Al-Nashef B. Semi ω-open subsets, Abhath Al-Yarmouk, 11: , [3] Al-Zoubi K, Al-Nashef B. The topology of ω-open subsets, Al-Manarah Journal, 9: , [4] Al-Zoubi K. Semi ω-continuous functions, Abhath Al-Yarmouk, 12: , [5] Crossley SG, Hildebrand SK. Semi-closure, Texas Journal of Sciences, 22:99-112, [6] Hdeib H. ω-closed mappings, Revista Colombiana de Mathematicas, 16:65-78, [7] Levine N. Semi-open sets and semi-continuity in topological spaces, The American Mathematical Monthly, 70:36-41, 1963.

12 20 Open Sets Intermedio entre conjuntos abiertos y semiabiertos Resumen. Introducimos e investigamos los conjuntos ω s -abiertos como una nueva clase de conjuntos que se ubica estrictamente entre los conjuntos abiertos y semi-abiertos. Usamos los conjuntos ω s -abiertos para introducir las funciones ω s -continuas como un nuevo tipo de funciones que se encuentran entre las funciones continuas y semicontinuas. Proporcionamos varios resultados y ejemplos relacionados con nuestros nuevos conceptos. En particular, obtenemos algunas caracterizaciones de las funciones ω s -continuas. Palabras clave: Conjunto semiabierto; Conjunto ω-abierto; Función semicontinua Intermédio entre conjuntos abertos e semiaberto Resumo. Introduzimos e investigamos os conjuntos ω s -abertos como uma nova classe de conjuntos que se localiza estritamente entre os conjuntos abertos e semiabertos. Usamos os conjuntos ω s -abertos para introduzir as funções ω s -contínuas como um novo tipo de função que se encontram entre as funções contínuas e semicontínuas. Proporcionamos vários resultados e exemplos relacionados com nossos novos conceitos. Particularmente, obtemos algumas caracterizações das funções ωs-contínuas. Palavras-chave: Conjunto semiaberto; Conjunto ω-aberto; Função semicontínua Samer Al Ghour He is a professor of mathematics at Jordan University of Science Technology and obtained his PhD degree in mathematics at University of Jordan. His research interests includes homogeneity and classes of sets in the structures: Topology, Generalized Topology, Bitopology and Fuzzy Topology. Kafa Mansur She is graduated with a master's degree in mathematics from Jordan University of Science Technology. She is working now a teacher at the Jordanian Ministry of Education.