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Jordan Journal of Mathematics and Statistics (JJMS) 12(3), 2019, Pp 409 - 429

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Jordan Journal of Mathematics and Statistics (JJMS) 12(3), 2019, Pp 409 - 429 Jordan Journal of Mathematics and Statistics (JJMS) 12(3), 2019, pp 409 - 429 Sα-CONNECTEDNESS IN TOPOLOGICAL SPACES B. K. TYAGI(1), MANOJ BHARDWAJ(2) AND SUMIT SINGH(3) Abstract. In this paper, connectedness of a class of Sα-open sets in a topo- logical space X is introduced. The connectedness of this class on X, called Sα- connectedness, turns out to be equivalent to connectedness of X when X is locally indiscrete or with finite α-topology. The Sα-continuous and Sα-irresolute map- pings are defined and their relationship with other mappings such as continuous mappings and semi-continuous mappings are discussed. An intermediate value the- orem is obtained. The hyperconnected spaces constitute a subclass of the class of Sα-connected spaces. 1. Introduction The study of connectedness via various generalized open sets is not a new idea in topological spaces. Njastad [10] introduced the α-open sets and investigated the topological structure on the class of these sets; the α-open sets form a topology. The classes of semi-open sets [7] , β-open sets [1] , αβ-open sets [14], Pβ-open sets [13] and Sα-open sets [17] were introduced. The classes of β-open, α-open and semi-open sets contain the class of open sets. Based on these classes, the concepts of β-connectedness [9, 12, 4], α-connectedness [4], semi-connectedness [11] and αβ-connectedness [14] were 2010 Mathematics Subject Classification. 54A05, 54D05. Key words and phrases. α-connected, hyperconnected, semi-connected, Sα-connected. Copyright c Deanship of Research and Graduate Studies, Yarmouk University, Irbid, Jordan. The second and third authors acknowledge the fellowship grant of University Grant Commission, India. Received: Jun. 21, 2018 Accepted: Feb. 21, 2019 . 409 410 B. K. TYAGI, MANOJ BHARDWAJ AND SUMIT SINGH introduced, respectively. It is already known that connectedness of a topological space X is equivalent to α-connectedness of X. Here a connectedness based on the class of Sα-open sets called Sα-connectedness is introduced in a topological space X, which turns out to be stronger than the connectedness of the original topology, but it turns out to be equivalent to connectedness in case of finite α-topologies on X. The class of Sα-connected spaces contains the class of hyperconnected spaces and also contains the class of hyperconnected modulo an ideal spaces [15]. It is shown that in the class of locally indiscrete spaces, the classes of open sets, Sα-open sets and semi-open sets coincide. Through this, it is shown that R is not Sα-connected. One might be interesting to study the connectedness of smaller class than the class of semi-open sets under which R is not connected. It motivates to study Sα-connectedness. This paper is organised as follows. In Section-2, the basic properties of Sα-open sets are developed, Sα-closure and its properties are obtained and the notion of a Sα-continuous function is defined. In Section-3, the notion of Sα-connectedness is in- troduced and its relationship with various other weaker and stronger forms of connect- edness is investigated. It is shown that in a locally indiscrete space Sα-connectedness is equivalent to connectedness. Several characterizations of Sα-connected spaces are obtained. It is shown that the class of hyperconnected spaces is a subclass of Sα- connected spaces. Section-4 contains the properties of Sα-connected sets. In Section- 5, we introduced the notion of Sα-irresolute function and studied the behaviour of Sα-connected spaces with respect to several type of mappings. Section-6 covers the concept of Sα-component. It is shown that Sα-component of a space containing a point is contained in the component containing that point. Sα-CONNECTEDNESS IN TOPOLOGICAL SPACES 411 2. Preliminaries Let (X; τ) or X be a topological space or a space. We will denote by Cl(A) and Int(A) the closure of A and the interior of A, for a subset A of X, respectively. Definition 2.1. A subset A of a topological space X is said to be (1) α-open [10] if A ⊆ Int(Cl(Int(A))) (2) β-open [1] if A ⊆ Cl(Int(Cl(A))) (3) semi-open [7] if A ⊆ Cl(Int(A))). The complement of a α-open (β-open, semi-open) set is said to be α-closed (resp. β-closed, semi-closed). Definition 2.2. (1) A semi-open subset A of a topological space X is said to be Sα-open [17] if for each x 2 A there exists a α-closed set F such that x 2 F ⊆ A. (2) A α-open subset A of a topological space X is said to be αβ-open [14] if for each x 2 A there exists a β-closed set F such that x 2 F ⊆ A. A subset B of topological space X is Sα-closed (αβ-closed) if X n A is Sα-open (resp., αβ-open) in X. The family of all α-open ( β-open, Sα-open, semi-open, α-closed, semi-closed, β- closed, Sα-closed) subsets of X is denoted by αO(X)(resp:; βO(X);SαO(X); SO(X), αC(X); SC(X); βC(X);SαC(X)). We have the following inclusions: τ ⊆ αO(X) ⊆ SO(X) ⊆ βO(X) and SαO(X) ⊆ SO(X) ⊆ βO(X). Definition 2.3. [17] A point x 2 X is said to be an Sα-interior point of A ⊆ X if there exists an Sα-open set U containing x such that x 2 U ⊆ A. The set of all Sα-interior points of A is said to be Sα-interior of A and it is denoted by SαInt(A). 412 B. K. TYAGI, MANOJ BHARDWAJ AND SUMIT SINGH Lemma 2.1. The interior of a nonempty semi-open and hence Sα-open set in a space is nonempty. Proof. Follows directly from the definition of a semi-open set. Definition 2.4. [17] Intersection of all Sα-closed sets containing F is called the Sα- closure of F and it is denoted by SαCl(F ). It may be noted that Sα-open sets are obtained from semi-open sets but this col- lection neither contains the collection of open sets nor it is contained in the collection of open sets. Thus, the study of Sα-open sets is meaningful. Example 2.1. Let X = fa; b; cg; τ = f;; fag; fbg; fa; bg;Xg. Then SO(X) = f;; fag; fbg; fa; bg; fa; cg; fb; cg;Xg and SαO(X) = f;; fa; cg; fb; cgXg. Here fag 2 τ but fag 2= SαO(X) and fb; cg 2 SαO(X) and fb; cg 2= τ Lemma 2.2. If A is dense in a space X, then SαCl(A) = X. Proof. Suppose that SαCl(A) = F ⊂ X. Then F is semi-closed since F is Sα-closed. Therefore, Int(Cl(A)) ⊆ Int(Cl(F )) ⊆ F . Theorem 2.1. The Sα-closure of a dense subset of a connected space is connected. Proof. Follows from Lemma 2.2. Definition 2.5. A topological space X is said to be (1) locally indiscrete [6] if every open subset of X is closed. (2) hyperconnected [6] if every nonempty open subset of X is dense in X. Theorem 2.2. If X is a hyperconnected space, then the Sα-closure of any non-empty open set is X. Proof. Follows from Lemma 2.2. Sα-CONNECTEDNESS IN TOPOLOGICAL SPACES 413 Definition 2.6. Let X and Y be two topological spaces. A function f from X to Y is Sα-continuous at a point x 2 X if for each open set V in Y containing f(x), there exists an Sα- open set U in X containing x such that f(U) ⊆ V . If f is Sα-continuous at every point x of X, then it is called Sα-continuous on X. Theorem 2.3. Let X and Y be two topological spaces. A function f from X to Y is Sα-continuous if and only if the inverse image of every open set in Y is Sα-open in X. −1 Proof. Let V be any open set in Y . If f (V ) = ;, then it is obviously Sα-open. If −1 −1 f (V ) 6= ;, then for any x 2 f (V ); f(x) 2 V . Since f is Sα-continuous, there −1 exists an Sα open set U in X containing x such that f(U) ⊆ V . Then f (V ), being the union of Sα-open sets, is Sα-open in X. The converse follows from the definition. Definition 2.7. Let X and Y be two topological spaces. A function f from X to Y is semi-continuous [7](α-continuous [5, 8], αβ-continuous [14]) at a point x 2 X if for each open set V in Y containing f(x), there exists a semi-open(resp. α-open, αβ-open) set U in X containing x such that f(U) ⊆ V . If f is semi-continuous (α- continuous, αβ-continuous) at every point x of X, then it is called semi-continuous (resp., α-continuous, αβ-continuous) on X. 3. Sα-Connected Space Definition 3.1. Two nonempty subsets A and B of a topological space X are said to be (1) Sα-separated if A \ SαCl(B) = ; = SαCl(A) \ B. (2) α-separated [4] if A \ αCl(B) = ; = αCl(A) \ B. (3) semi-separated [11] if A \ SCl(B) = ; = SCl(A) \ B. (4) β-separated [4, 9, 12] if A \ βCl(B) = ; = βCl(A) \ B.
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