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ZANCO Journal of Pure and Applied Sciences The official scientific journal of Salahaddin University-Erbil ZJPAS (2017), 29 (1); 6-10 http://dx.doi.org/10.21271/ZJPAS.29.1.2 - Open Sets in Topological Spaces Sami A. Hussein Department of Mathematics, College of Basic Education, Salahaddin University,Erbil Kurdistan Region, Iraq ARTICLE INFO ABSTRACT Article History: Received: 29/04/2016 Accepted: 19/09/2016 Published: 17/ 4/2017 Keywords: The main aim of this paper is to introduce a new class of set called - open set. some general properties of it are investigated. The relationships between this class of set with other related classes of sets are studied. supra-topology nearly open sets operators *Corresponding Author: Sami A.Hussein [email protected] 1. INTRODUCTION By a space or we mean a topological space on which no separation axiom is assumed unless explicitly stated. In [Levine, N.1963] defined semi-open, which is a subset of a space satisfies ,or equivalently if there exists an open set such hat . [Njasted, O.,1965] defined a subset of a space to be open if , which plays an important role in the field of general topology. [Mashhour et al., 1982] defined a subset of a space to be pre-open if , [ Abd-El-Monsef et al.,1983] defined a subset of a space to be -open if , while in 1986 Andrijvic D. independently defined semipre- open set which is equivalent to -open set. A subset of a space to be b-open if , The family of all semiopen(resp.,preopen, open, -open and bopen)sets are denoted by [Khalaf, A.B. and Ameen, Z.A. ,2010] introduced and investigated a new way to define a new class of set called sc open set. and used it in the way to study some properties of continuity, Based on sc open set, pc-open, bc-open and -open sets were defined in [Ameen, Z.A., 2011, Ibrahim, H.Z., 2013, Mizyed, A Y., 2015] As a generalization of sc-open,pc-open,bcopen and -open sets. In this work, we 7 Hussein, S. /ZJPAS: 2017, 29(1): 6-10 introduce and study a new class of set, called - open sets, which is a subset of a space for which if for each , there exists a closed set such that . exists an open set such that 3. 2. Preliminaries We recall the following definitions and results which will be used in the sequel. Definition 2.1:[ Mashhour A.S. et al.,1983] Let be a topological space and let be a collection of subsets of such that . Then is called supra-topology associated with , if satisfies the following conditions 1) . 2) If for each arbitrary set, then Definition 3.1: A subset of a space called -open if for each there exists a closed set such that is , . From now on, by or simply , we mean a family of all -open subsets of a space is an The relationship between -open set and some other sets is given in the following results. Definition 2.2: A subset of a space is called sc-open [Khalaf, A.B. and Ameen, Z.A. ,2010] (resp., pc-open [Ameen, Z.A,2011], Bcopen [Ibrahim, H.Z.,2013], and -open [Mizyed, A Y.,2015]) if for each Theorem 3.2: A subset of a space is open if and only if is -open and it is a union of closed sets of . there exists a closed set , where . In this section we introduce our main definition, which is -open set, with some of its properties. such that Definition 2.3: [Velicko N., 1968]A subset of a space is called -open (resp., - – open) set, if for each , there exists an open( resp., –open) set such that . where is a supra topology. Definition 2.4: [Ahmed N.K,1990] A space is said to be -regular if for every and every -closed set such that , there exists disjoint open sets and such that and . Theorem 2.5: [Ahmed N.K,1990]A space is -regular if and only if for every and every -open set such that , there Proof: Assume that . Then , and for each there exists a closed set such that , thus , which implies that . Conversely suppose that and , where is a closed set which containing , then for each ,implies that , for some closed set and . Hence . Remark 3.3: It is clear that from Definition 3.1, -open set implies -open set, but the converse may not be true in general, as shown in the following example Example 3.4: then , then . Let , if we , take and but 8 Hussein, S. /ZJPAS: 2017, 29(1): 6-10 Remark 3.5: From the fact that union of arbitrary collection of closed sets may not be closed in general, therefore -open sets may not be closed sets, as shown in the following example Proof: Let be an open set in Then , and for any is closed subset of , such that which implies that . Example 3.6: Consider the usual space for R and let , then , but it is not closed set. Proposition 3.12: If a space every open set is -open set. Proposition 3.7: The union of Arbitrary collection of -open sets is -open set. Proof: Let be a collection of open sets for each , where is an arbitrary set. Then, we have , and for any there exists such that , for some . Since a closed , then , is regular, then Proof: Let be an open subset of a space , in the case where , then , otherwise, let , since is a regular space, then there exists an open set such that . Thus we have that and . By Theorem 3.1, . Proposition 3.13: For a , then there exists set such that Hence , . Remark 3.8: The intersection of two -open sets may not be -open set in general, as shown in the following example. Example . 3.9: Let . If , take we then and , Proof: From Remark 3.3, we have that , it is suffices to show that . Let , if , then , otherwise, let , since is , then is a closed set for each , and . Therefore . From Theorem 2.5. Remark 3.3, we have the following result. then but Proposition 3.14: For a . Note that from Proposition 3.7 and Remark 3.8, the family of all is a supra-topology for X. Proposition 3.15: Every space is -open set. Remark 3.10: and are incomparable in general. In Example 3.9, , but , and but . Proof: Let be a -open subset of a space , if , then , otherwise, let , then there exists an open set such that . Proposition 3.11: For a open set is -open set. So follows that every -open subset of a . It . From the fact 9 Hussein, S. /ZJPAS: 2017, 29(1): 6-10 , and by Theorem 3.2, . . The class of all is denoted by . space -sets of a Remark 3.16: The converse of Proposition 3.15 may not be true in general, as shown in the following example. Proposition 4.2: For any space Example 3.17: Let , and be a co-finite topology for . Since is T1 –space, and is an open set, then by Proposition 3.11, , but is not -open set. Proof: Clearly are in Let Then . This implies that By , for each . we have Thus , therefore 4.3: For any Proof: Let we have take . Then for all , , . In particular, if we . , then Remark 4.4: The converse of Proposition 4.3 may not be true in general, let we take the space as in Example 3.9, we see that but . Proposition 4.5: A subset of a space closed in if and only if Proof: A subset only if , open If we take B in , is - in . is closed in is an open in if of a space is , for each space . closed for each Definition 4.1: A subset called -set if Then . Proposition 4. The Topology generated by sets for each . . for all Proposition 3.7, . , The topological properties of the above concepts are coincides with those supratopology. . Let ( Proof: Follows from Theorem 3.2 and fact that union of finite closed sets is closed set. 1. A subset N of a space , is said to be -neighborhood of x, if there exists an -open set G such that . 2. A point , is called -interior point of , if there exists an -open set G such that . 3. A point , is called -limit point of , if for each -open set G containing x such that . 4. A point , is called -closure point of , if for each -open set G containing x such that . and . , where Definition 3.19: For a subset , of a space and , then is a topology for . Proposition 3.18: For a finite space , every -open set is -clopen set. For now, we define some topological concepts in the following definition , ,if and , if and only for each if and only if is - closed set in . , then for any - closed set -closed for each in . 10 Hussein, S. /ZJPAS: 2017, 29(1): 6-10 REFERENCES ABD El-MONSEF M.E, El-DEEB S.N and MAHMOUD R.A(1983), On β-Open Sets and βContinuous Mapping, Bull. Fac. Sci. Assiut Univ, 12 (1), (77-90). AHMED N.K(1990), On some types of Separation Axioms, M.Sc. Thesis, Salahaddin Univ. AMEEN Z.A.(2011), pc-Open sets and pc-Continuity in topological spaces, J. of Advanced Research in pure Mathematics Vol.3,(1),(123-134). ANDRIJEVIC D. (1986)., Semi-Preopen Sets, Mat. Vesnik , 38, (24-32). ANDRIJEVIC D. 1996), On b-open sets, Mat.Vesnik,48,(59-64). IBRAHIM H.Z.(2013), Bc-open sets in topological spaces, J. of Advanced in pure Mathematics Vol.3,(34-40). KHALAF A.B. and AMEEN Z.A (2010), sc-open Sets and sc-continuity in Topological Spaces, J. of Advanced Research in Pure Mathematics, 2 (3), 87101. LEVINE N. 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