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On Ω-Connectedness and Ω-Continuity in the Product Space Mhelmar A EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 3, 2018, 834-843 ISSN 1307-5543 { www.ejpam.com Published by New York Business Global On !-Connectedness and !-Continuity in the Product Space Mhelmar A. Labendia1,∗, Jan Alejandro C. Sasam1 1 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. In this paper, the concepts of !-open and !-closed functions between topological spaces will be introduced and characterized. Moreover, related concepts such as !-connectedness and !- continuity from an arbitarary topological space into the product space will also be characterized. 2010 Mathematics Subject Classifications: 54A05 Key Words and Phrases: !-open, !-closed, !-connected, !-continuous 1. Introduction One of the attempts to substitute numerous concepts in topology with concepts pos- sessing either weaker or stronger properties was done by N. Levine [4]. He introduced the concepts of semi-open, semi-closed set and semi-continuity of a function, which gener- ated new results, some of which are generalization of existing ones. After this noteworthy work of Levine several mathematicians became attracted in presenting other topological concepts which can substitute the concepts of open sets. In [5], Velicko introduced the concepts of θ-continuity between topological spaces and subsequently defined the concepts of θ-closure and θ-interior of a subset of topological space. In [1], Al-Hawary characterized θ-continuity and the other well-known variations of continuity such as strong continuity, semi-continuity and closure-continuity. Let (X; T) be a topological space and A ⊆ X. The θ-closure and θ-interior of A are, respectively, denoted and defined by Cls(A) = fx 2 X : Cl(U) \ A 6= ? for every open set U containing xg and Ints(A) = fx 2 X : Cl(U) ⊆ A for some open set U containing xg; ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3293 Email addresses: [email protected] (M. Labendia), [email protected] (J. A. Sasam) http://www.ejpam.com 834 c 2018 EJPAM All rights reserved. M. Labendia, J. A. Sasam / Eur. J. Pure Appl. Math, 11 (3) (2018), 834-843 835 where Cl(U) is the closure of U in X. A subset A of X is θ-closed if Cls(A) = A and θ-open if Ints(A) = A. Equivalently, A is θ-open if and only if XnA is θ-closed. In [2], Hdeib introduced the concepts of !-open and !-closed sets and !-closed map- pings on a topological space. He showed that !-closed mappings are strictly weaker than closed mappings and also showed that the Lindel¨ofproperty is preserved by counter im- ages of !-closed mappings with Lindel¨ofcounter image of points. In 2010, Ekici et al. [3] introduced the concepts of !θ-open and !θ-closed sets on a topological space. They showed that the family of all !θ-open sets in a topological space X forms a topology on X. They also introduced the notions of !θ-interior and !θ-closure of a subset of a topological space. A point x of a topological space X is called a condensation point of A ⊆ X if for each open set G containing x, G \ A is uncountable. A subset B of X is !-closed if it contains all of its condensation points. The complement of B is !-open. Equivalently, a subset U of X is !-open (resp., !θ-open [3]) if and only if for each x 2 U, there exists an open set O containing x such that OnU (resp., OnInts(U)) is countable. A subset B of X is !θ-closed [3] if its complement XnB is !θ-open. The !-closure (resp., !θ-closure [3]) and !-interior (resp., !θ-interior [3]) of A ⊆ X are, respectively, denoted and defined by Cl!(A) = \fF : F is an !-closed set containing Ag (resp., Cl!θ (A) = \fF : F is an !θ-closed set containing Ag) and Int!(A) = [fG : G is an !-open set contained in Ag (resp., Int!θ (A) = [fG : G is an !θ-open set contained in Ag): It is worth noting that A ⊆ Cl!(A) (resp., A ⊆ Cl!θ (A) [3]) and Int!(A) ⊆ A (resp., Int!θ (A) ⊆ A [3]). Let T! (resp., T!θ ) be the family of all !-open (resp., !θ-open) subsets of a topological space X. Since T! (resp., T!θ ) is a topology on X, for any set A ⊆ X, Int!(A) (resp., Int!θ (A)) is !-open (resp., !θ-open) and the largest !-open (resp., !θ-open set) contained in A. Moreover, for any set A ⊆ X, Cl!(A) (resp., Cl!θ (A)) is !-closed (resp., !θ-closed) and the smallest !-closed (resp., !θ-closed) set containing A. A topological space X is said to be !-connected (resp., θ-connected, !θ-connected [3]) if X cannot be written as the union of two nonempty disjoint !-open (resp., θ-open, !θ- open) sets. Otherwise, X is !-disconnected (resp., θ-disconnected, !θ-disconnected [3]). A subset B of X is !-connected (resp., θ-connected, !θ-connected) if it is !-connected (resp., θ-connected, !θ-connected) as a subspace of X. Throughout the paper, related results of !θ-open, !θ-closed, !θ-closure, !θ-interior, and !θ-connectedness are due to [3]. A function f from a topological space X to another topological space Y is said to be (i) !-open (resp., θ-open, !θ-open) if f(G) is !-open (resp., θ-open, !θ-open) in Y for every open set G in X; (ii) !-closed (resp., θ-closed, !θ-closed) if f(G) is !-closed (resp., θ-closed, !θ-closed) in Y for every closed set G in X; M. Labendia, J. A. Sasam / Eur. J. Pure Appl. Math, 11 (3) (2018), 834-843 836 −1 (iii) !-continuous if f (V ) is !-open (resp., θ-open, !θ-open) in X for every open subset V of Y ; (iv) !-irresolute if for every x 2 X and every !θ-open set A containing f(x), there exists an !-open set U containing x such that f(U) ⊆ A. Let A be an indexing set and fYα : α 2 Ag be a family of topological spaces. For each α 2 A, let Tα be the topology on Yα. The Tychonoff topology on ΠfYα : α 2 Ag −1 is the topology generated by a subbase consisting of all sets pα (Uα), where the projec- tion map pα :ΠfYα : α 2 Ag ! Yα is defined by pα(hyβi) = yα, Uα ranges over all members of Tα, and α ranges over all elements of A. Corresponding to Uα ⊆ Yα, denote −1 pα (Uα) by hUαi. Similarly, for finitely many indices α1; α2; : : : ; αn; and sets Uα1 ⊆ Yα1 , Uα2 ⊆ Yα2 ;:::;Uαn ⊆ Yαn ; the subset −1 −1 −1 hUα1 i \ hUα2 i \ · · · \ hUαn i = pα1 (Uα1 ) \ pα2 (Uα2 ) \···\ pαn (Uαn ) is denoted by hUα1 ;Uα2 ;:::;Uαn i. We note that for each open set Uα subset of Yα, hUαi = −1 pα (Uα) = Uα × Πβ6=αYβ. Hence, a basis for the Tychonoff topology consists of sets of the form hBα1 ;Bα2 ; :::; Bαk i, where Bαi is open in Yαi for every i 2 K = f1; 2; :::; kg. Now, the projection map pα :ΠfYα : α 2 Ag ! Yα is defined by pα(hyβi) = yα for each α 2 A. It is known that every projection map is a continuous open surjection. Also, it is well known that a function f from an arbitrary space X into the Cartesian product Y of the family of spaces fYα : α 2 Ag with the Tychonoff topology is continuous if and only if each coordinate function pα ◦f is continuous, where pα is the α-th coordinate projection map. 2. !-Open and !-Closed Functions In this section, we investigate the connection of !-open (resp., !-closed) function to the other well-known functions such as open, θ-open, and !θ-open (resp., closed, θ-closed, !θ- closed) functions. We also give some characterizations of !-open and !-closed functions. Throughout, if no confusion arises, let X and Y be topological spaces. We shall be using the following lemma later. Lemma 1. Let f : X ! Y be a bijective function. Then f is !-open on X if and only if it is !-closed on X. Remark 1. [3, Remark 4] Let A ⊆ X. Then (i) If A is open, then A is !-open; (ii) If A is θ-open, then A is open; (iii) If A is θ-open, then A is !θ-open; and (iv) If A is !θ-open, then A is !-open. M. Labendia, J. A. Sasam / Eur. J. Pure Appl. Math, 11 (3) (2018), 834-843 837 It is shown in [3, p.295] that the implications above are not reversible. Remark 2. Let f : X ! Y be a function. Then (i) If f is open (resp., closed), then f is !-open (resp., !-closed). (ii) If f is !θ-open (resp., !θ-closed), then f is !-open (resp., !-closed). Remark 3. The converses of Remark 2 (i) and (ii) do not necessarily hold. (i): First, consider the topological spaces X = Z = Y = fa; b; c; dg with respective topologies TX = f?; X; fa; b; cgg, TZ = f?; Z; fa; bgg, and TY = f?; Y; fag; fdg; fa; dg; fa; bg; fa; b; dgg. Define f : X ! Y by f(a) = a, f(b) = b, f(c) = c, and f(d) = a. Define g : Z ! Y by g(a) = g(b) = g(c) = g(d) = d. Since Y is countable, f(A) is !-open in Y for every open set A of X.
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