International Journal of Analysis and Applications Volume 18, Number 6 (2020), 1029-1036 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-1029 Received March 31st, 2020; accepted June 1st, 2020; published October 13th, 2020. 2010 Mathematics Subject Classification. 54A05. Key words and phrases. topological spaces; semi generalized open sets; semi generalized closed sets. Β©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 1029 SEMI GENERALIZED OPEN SETS AND GENERALIZED SEMI CLOSED SETS IN TOPOLOGICAL SPACES ABDELGABAR ADAM HASSAN1,2,* 1Department of Mathematics, College of Science and Arts in Tabrjal, Jouf University, Kingdom of Saudi Arabia 2Department of Mathematics, University of Nyala, Nyala, Sudan *Corresponding author: aahassan@ju.edu.sa ABSTRACT. In this paper we introduce a new class of semi generalized open sets, generalized semi closed sets in topological spaces, and studied some of its basic properties. Moreover we define approximately semi generalized open sets and approximately generalized semi closed sets in topological spaces. Further we obtained some properties of closure, semi generalized open sets and generalized semi closed sets in topological spaces. 1. INTRODUCTION The study of generalized closed sets in topological space was initiated by Levine in [23]. Biswas [18], Njasted[15], Mashhour[12], Robert[19], Bhattacharya [13], Arya and Nour [12 ]. introduced and investigated semi closed, -open and -closed, pre-open, semi*-open, sg-closed, gs- closed, gp-closed, g-closed, g* closed, s*g-closed, w-closed, g*-closed respectively. Topology is an important and interesting area of mathematics, the study of which will not only introduce to new concepts and theorems but also put into context old ones like continuous functions [1]. However, to say just this is to understate the significance of topology. It is so fundamental that its influence is evident in almost every other branch of mathematics [3]. Topological notions like compactness, connectedness and denseness areas basic to mathematicians of today as sets and functions were https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-1029 Int. J. Anal. Appl. 18 (6) (2020) 1030 to those of last century. Topology has several different branches, genera l topology, algebraic topology, differential topology and topological algebra, the first, general topology, being the door to the study of the others [3,5]. The aim of this paper is to introduce the concept of semi generalized open Sets and generalized semi closed Sets in topological spaces, and provide Semi generalized open Sets and generalized semi closed Sets in topological spaces. 2. PRELIMINARIES Definition 2.1. Let 𝑋 be a non – empty sets. A collection of subsets of 𝑋 is said to be a topology on 𝑋 if (i) 𝑋 and the empty set, βˆ…, belong to Ο„, (ii) The union of any (finite or infinite) number of sets in Ο„ belong to Ο„, (iii) The intersection of any two sets in Ο„ belongs to Ο„. The pair (𝑋, 𝜏) is called a topological space. Definition 2.2 Let 𝑋 be a non – empty sets and let Ο„ be the collection of all subsets of 𝑋. Then Ο„ is called discrete topology on the set 𝑋. The topological space (𝑋, 𝜏) is called a discrete space. Observe that the set 𝑋 in Definition 1.2. can be any non – empty set. So, there is an infinite number of discrete spaces – one for each𝑋. Definition 2.3. Let 𝑋 be any non – empty set and 𝜏 = {𝑋, βˆ…}. Then Ο„ is called indiscrete topology and (𝑋, 𝜏) is said to be indiscrete space. Definition 2.4. Let (𝑋, 𝜏) be a topological space. A subset 𝐴 of 𝑋 is said to be generalized closed if 𝑐𝑙(𝐴) βŠ† ⋃ when ever 𝐴 βŠ† ⋃ and ⋃ is an open in (𝑋, 𝜏). Definition 2.5. Let (𝑋, 𝜏) be a topological space and 𝐴 βŠ† 𝑋. The generalized closure of 𝐴, denote by π‘π‘™βˆ—(𝐴) and is defined by the intersection of all 𝑔 closed sets containing 𝐴 an generalized interior of 𝐴, denoted by π‘–π‘›π‘‘βˆ—(𝐴) and is defined by union of all 𝑔- open sets contained in 𝐴. Definition 2.6. A subset 𝐴 of a topological space (𝑋, 𝜏) is said to be (i) a semi – open set [9] if 𝐴 βŠ† 𝑐𝑙(𝑖𝑛𝑑(𝐴)) and a semi closed if 𝑖𝑛𝑑(𝑐𝑙(𝐴)) βŠ† 𝐴, (ii) a preopen set [11] if 𝐴 βŠ† 𝑖𝑛𝑑(𝑐𝑙(𝐴)) and a pre closed if 𝑐𝑙(𝑖𝑛𝑑(𝐴)) βŠ† 𝐴, (iii) an ∝ - open sets [12] if 𝐴 βŠ† 𝑖𝑛𝑑 (𝑐𝑙(𝑖𝑛𝑑(𝐴))) and ∝ - closed sets if 𝑐𝑙 (𝑖𝑛𝑑(𝑐𝑙(𝐴))) βŠ† 𝐴, (iv) A regular set [13] if 𝑖𝑛𝑑(𝑐𝑙(𝐴)) = 𝐴 and a regular closed set if 𝑐𝑙(𝑖𝑛𝑑(𝐴)) = 𝐴, Int. J. Anal. Appl. 18 (6) (2020) 1031 (v) A 𝑄 – set [10] if 𝑐𝑙(𝑖𝑛𝑑(𝐴)) = 𝑖𝑛𝑑(𝑐𝑙(𝐴)). The intersection of all semi – closed (resp. to pre closed, ∝ - closed) sets containing a subset 𝐴 of (𝑋, 𝜏) is called semi – closure [14] (resp. pre closure, ∝ - closed, 𝑐𝑙𝛼 (𝐴). The semi – interior of 𝐴 is the largest semi – open set contained in 𝐴 and denoted by 𝑆 βˆ’ 𝑖𝑛𝑑(𝐴). 3. OPEN SETS AND CLOSED SETS IN TOPOLOGICAL Definition 3.1. Let (𝑋, 𝜏) be any topological space. Then the members of 𝜏 are said to be open sets. Proposition 3.2. If (𝑋, 𝜏) is any topological space, then (i) 𝑋 and βˆ… are open sets, (ii) the union of any (finite or infinite) number of open sets is an open set, (iii) the intersection of any finite number of open sets is an open set. Proof. Clearly (i) and (ii) are trivial consequences of Definition 2.1. and Definition 2.1.(i) and (ii). The condition (iii) follows from Definition 2.1. Definition 3.3. Let (𝑋, 𝜏) be a topological space. A subset 𝑆 of 𝑋 is said to be a closed set in (𝑋, 𝜏) if its complement in 𝑋, namely 𝑋\𝑆, is open in (𝑋, 𝜏). Proposition 3.4. If (𝑋, 𝜏) is any topological space, then (i) βˆ… and 𝑋 are closed sets, (ii) the intersection of any (finite or infinite) number of closed sets is a closed set, (iii) the union of any finite number of closed sets is a closed set. Proof. (i) follows immediately Proposition 2.2. and Definition 2.3., as the complement of 𝑋 is βˆ… and the complement of βˆ… is 𝑋. To prove that (iii) is true, let 𝑆1, 𝑆2, … , 𝑆𝑛 be closed sets. We are required to prove that 𝑆1 βˆͺ 𝑆2 βˆͺ … βˆͺ 𝑆𝑛is a closed set. It suffices to show, by Definition 2.3., that 𝑋\(𝑆1 βˆͺ 𝑆2 βˆͺ … βˆͺ 𝑆𝑛 ) is an open set. As 𝑆1, 𝑆2, … , 𝑆𝑛 are closed sets, their complement 𝑋\𝑆1, 𝑋\𝑆2, … , 𝑋\𝑆𝑛 are open sets. But 𝑋\(𝑆1 βˆͺ 𝑆2 βˆͺ … βˆͺ 𝑆𝑛) = (𝑋\𝑆1) ∩ (𝑋\𝑆2) ∩ … ∩ (𝑋\𝑆𝑛 ) (1) As the right hand side of (1) is a finite intersection of open sets, it is an open set. So that the left hand side of (1) is an open set. Hence 𝑆1 βˆͺ 𝑆2 βˆͺ … βˆͺ 𝑆𝑛 is a closed set, as required. So (iii) is true. Int. J. Anal. Appl. 18 (6) (2020) 1032 The proof of (ii) is similar to that of (iii). Example 3.5. On any set X there is the trivial topology {βˆ…, X}. There is also the discrete topology whereas any subset of X is open. Thus, on a set there can be many topologies. Example 3.6. (Euclidean topology). In ℝ𝑛 = {(π‘₯1, π‘₯2, … , π‘₯𝑛 )|π‘₯𝑖 ∈ ℝ}, the Euclidean norm of a point π‘₯ = (π‘₯1, π‘₯2, … , π‘₯𝑛 ) is β€–π‘₯β€– = [βˆ‘ π‘₯𝑖 2𝑛 𝑖=1 ] 1 2⁄ . The topology generated by this norm is called the Euclidean topology of ℝ𝑛. 4. SEMI GENERALIZED * b OPEN SETS In this part, we introduce semi generalized* b open sets in topological spaces by using the notion of semi generalized b- open sets, and study some of their properties. Definition 4.1. A subset 𝐴 of a topological space (𝑋, 𝜏), is called semi generalized * b open set if 𝐴𝑐 is semi generalized – b closed in 𝑋. We denote the family of all semi generalized – b open sets in 𝑋 by 𝑠𝑔 𝑏 βˆ’ 𝑂(𝑋). Theorem 4.2. If 𝐴 and 𝐡 are 𝑠𝑔 𝑏 – open sets in a space 𝑋. Then 𝐴 ∩ 𝐡 is also 𝑠𝑔 𝑏 – open set in 𝑋. Proof. If 𝐴 and 𝐡 are 𝑠𝑔 𝑏 – open sets in a space 𝑋. Then 𝐴𝑐 and 𝐡𝑐 are 𝑠𝑔 𝑏 – closed sets in a space 𝑋. Therefore 𝐴𝑐 βˆͺ 𝐡𝑐 is also 𝑠𝑔 𝑏 – closed set in 𝑋. (i.e.) 𝐴𝑐 βˆͺ 𝐡𝑐 = (𝐴 ∩ 𝐡)𝑐 is a 𝑠𝑔 𝑏 – closed set in 𝑋. Therefore 𝐴 ∩ 𝐡 𝑠𝑔 𝑏 – open set in 𝑋. 5. SEMI GENERALIZED * b CLOSED SETS In this part, we introduce semi generalized* b – closed set and investigate some of its properties. Definition 5.1. (Closed set). Let (𝑋, 𝜏) topological space. the set 𝐴 βŠ† 𝑋 is called closed set if and only if 𝐴𝐢 be open set. i.e. 𝐴 closed set 𝐴 ⇔ 𝐴𝐢 open set 𝐴 open set 𝐴 ⇔ 𝐴𝐢 closed set And denote that by 𝔍𝑋 or 𝔍. Example 5.2. Let the set 𝑋 = {π‘Ž, 𝑏, 𝑐, 𝑑, 𝑒}. Define topology 𝜏 into 𝑋 by the flowing: 𝜏 = {𝑋, βˆ…, {π‘Ž}, {𝑐, 𝑑}, {π‘Ž, 𝑐, 𝑑}, {𝑏, 𝑐, 𝑑, 𝑒}} Then 𝔍𝑋 = {𝑋, βˆ…, {𝑏, 𝑐, 𝑑, 𝑒}, {π‘Ž, 𝑏, 𝑒}, {𝑏, 𝑒}, {π‘Ž}}. Int. J. Anal. Appl. 18 (6) (2020) 1033 Example 5.3. Let(𝑋, 𝐷) Discrete topological space. We find that each subset from (𝑋, 𝐷) is open and closed at the same time because: 𝑃(𝑋) = 𝐷, βˆ€ 𝐴 βŠ† 𝑋 ⇔ 𝐴 ∈ 𝐷 ⇔ 𝐴𝐢 closed Definition 5.4. A subset 𝐴 of a topological space (𝑋, 𝜏), is called semi generalized * b closed set if 𝑐𝑙(𝐴) βŠ‚ βˆͺ whenever 𝐴 βŠ‚ βˆͺ and βˆͺ is semi generalized – b open in 𝑋. Theorem 5.5. Every closed set is 𝑠𝑔 𝑏 – closed. Proof. Let 𝐴 be any closed set in 𝑋 such that 𝐴 βŠ‚ βˆͺ, where βˆͺ is 𝑠𝑔 open. Since 𝑏 𝑐𝑙(𝐴) βŠ‚ 𝑐𝑙(𝐴) = 𝐴. Therefore 𝑏 𝑐𝑙(𝐴) βŠ‚ βˆͺ. Hence 𝐴 is 𝑠𝑔 𝑏- closed set in 𝑋. Is Example 5.6. Let 𝑋 = {π‘Ž, 𝑏, 𝑐} with 𝜏 = {𝑋, βˆ…, {𝑏}, {π‘Ž, 𝑏}}. The set {π‘Ž, 𝑏} is 𝑠𝑔 𝑏- closed set but not a closed set. Theorem 5.7. Every semi closed set is 𝑠𝑔 𝑏- closed set. Proof. Let 𝐴 be any closed set in 𝑋 and βˆͺ be any 𝑠𝑔 open set containing 𝐴. Since 𝐴 is semi closed set, 𝑏 𝑐𝑙(𝐴) βŠ‚ 𝑆 𝑐𝑙(𝐴) βŠ‚ βˆͺ. Therefore 𝑏𝑐𝑙(𝐴) βŠ‚ βˆͺ. Hence 𝐴 is 𝑠𝑔 𝑏- closed set. Corollary 5.8. Let 𝑋 = {π‘Ž1, π‘Ž2, π‘Ž3} with 𝜏 = {𝑋, βˆ…, {π‘Ž1, π‘Ž2}}. The set {π‘Ž1, π‘Ž2} is 𝑠𝑔 𝑏- closed set but not a semi closed set. Theorem 5.9. Every ∝- closed set is 𝑠𝑔 𝑏- closed set. Proof. Let 𝐴 be any ∝-closed set in, and 𝑋 and βˆͺ be any 𝑠𝑔 set containing 𝐴. Since 𝐴 is ∝- closed 𝑏 𝑐𝑙(𝐴) βŠ‚βˆ 𝑐𝑙(𝐴) βŠ‚ βˆͺ. Therefore 𝑏𝑐𝑙(𝐴) βŠ‚ βˆͺ. Hence 𝐴 is 𝑠𝑔 𝑏- closed set. Theorem 5.10. Every 𝑠𝑔 𝑏- closed set is 𝑔 𝑏- closed set. Proof. Let 𝐴 be any 𝑠𝑔 𝑏-closed set in, such that βˆͺ be any open set containing 𝐴. Since every open set is 𝑠𝑔 open, we have 𝑏 𝑐𝑙(𝐴).. Hence 𝐴 is 𝑔 𝑏- closed set. 6. THE CLOSURE AND INTERIOR Definition 6.1. Let 𝑋 be a topological space and 𝐴 βŠ† 𝑋 a subset. The closure of 𝐴 denoteοΏ½Μ…οΏ½ is the intersection of all the closed subsets of 𝑋 that contain 𝐴. The interior of 𝐴 is the union of all the open subsets of 𝑋 that are contained in A. Example 6.2. Let 𝑋 = ℝ and A = [a, b) with a < b. The closure of 𝐴 is the closed interval [a, b], and the interior of 𝐴 is the open interval (a, b). The closure cannot be smaller, since [a, b) is not closed, and the interior cannot be larger, since[a, b) is not open. Theorem 6.3. Let 𝑋 be a topological space, π‘Œ βŠ‚ 𝑋 a subspace, and 𝐴 βŠ‚ π‘Œ a subset. Let οΏ½Μ…οΏ½ denote the closure of 𝐴 in 𝑋. Then the closure of 𝐴 in π‘Œ equals Y ∩ AΜ… Int. J. Anal. Appl. 18 (6) (2020) 1034 Proof. Let 𝐡 denote the closure of 𝐴 in π‘Œ. To see that B βŠ‚ Y ∩ AΜ…, note that οΏ½Μ…οΏ½ is closed in 𝑋, so Y ∩ AΜ… is closed in π‘Œ and contains 𝐴. Hence it contains the closure 𝐡 of A in π‘Œ. To prove the opposite inclusion, note that 𝐡 is closed in π‘Œ, hence has the form B = Y ∩ C for some C that is closed in 𝑋. Then A βŠ‚ B βŠ‚ C, so C is closed in 𝑋 and contains 𝐴. Hence AΜ… βŠ‚ C and Y ∩ AΜ… βŠ‚ Y ∩ C = B. Example 6.4. Topologist ’s sine curve. The closure in the Euclidean plane of the graph of the function 𝑦 = 𝑠𝑖𝑛 1 π‘₯ , π‘₯ > 0 is often called the Topologist’s sine curve. Figure 1: Topologist’s sine curve. Denote 𝐴 = {𝑠𝑖𝑛 1 π‘₯ |π‘₯ > 0} and 𝐡 = {0} Γ— [βˆ’1,1]. Then the Topologist’s sine curve is 𝑋 = 𝐴 βˆͺ 𝐡. 7. CONCLUSION The aim of this paper is to introduce the concepts of semi generalized open Sets and generalized semi closed Sets in topological spaces sets and we study some of their properties. Furthermore, we discuss the conditions which are added to these concepts in order to coincide with the concept of semi-closed. Int. J. Anal. Appl. 18 (6) (2020) 1035 Data Availability: No data were used to support this study. Acknowledgment: I’m forever indebted my family for their endless patience and encouragement, also I want to recognize and express my thank to anyone helped me. Conflicts of Interest: The author declares that there are no conflicts of interest regarding the publication of this paper. References [1] J. Dugundji, Topology, Allyn and Bacon, Inc., Boston. 1966. [2] S.A. Morris, Topology without tears, Unisia, 2001. [3] S. Sekar, B. Jothilakshmi, On semi generalized star b – closed set in topological spaces, Int. J. Pure Appl. Math. 113 (2017), 93-102. [4] S. Lipschutz, Schaum’s outline of theory and problems of general topology, McGraw-Hill, New York. 1965. [5] D. Lyappan, N. Nagaveni, On semi generalized b-continuous maps, semi generalized b-closed maps in topological space, Int. J. Math. Anal. 6 (2012), 1251-1264. [6] S. S. Benchalli, P. G. Patil, P. M. Nalwad, Generalized ωα-Closed Sets in Topological Spaces, J. New Results Sci. 7 (2014), 7–19. [7] N. Biswas, On Characterization of semi - continuous functions, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fsi. Mat. Natur. 48 (1970), 399–402. [8] P. Bhattacharyya, B. K. Lahiri, Semi-generalized closed sets in topology, Ind. J. Math. 29 (1987), 375- 382. [9] N. Biswas, On characterizations of semi-continuous functions, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 8 (1970), 399–402. [10] S.G. Crossly, S.K. Hildebrand, Semi- closure, Texas J. Sci. 22(1971), 99-112. [11] S.G. Crossly, S.K. Hildebrand, Semi- topological properties, Fund. Math. 74 (1974), 233-254. [12] S.P. Arya, T. Nour, Characterizations of S- normal spaces, Indian J. Pure. Appl. Math. 21 (8) (1990), 717- 719. [13] P. Bhattacharya, B. K. Lahiri, Semi- generalized closed sets in Topology, Indian J. Math. 29 (1987), 376- 382. [14] R. Devi, Studies on generalizations of closed maps and homeomorphisms in topological spaces, Ph.D. Thesis, Bharathiar University, Coimbatore (1994). Int. J. Anal. Appl. 18 (6) (2020) 1036 [15] N. Levine, On the commutivity of the closure and interior operator in topological spaces, Amer. Math. Mon. 68 (1961), 474-477. [16] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer Math. Mon. 70 (1963), 36- 41. [17] N. Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo, 19 (1970), 89–96. [18] N. Biswas, On characterizations of semi- continuous functions, Atti. Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. 48 (1970), 399-402. [19] S. S. Benchalli, P. G. Patil, T. D. Rayanagaudar, ωα-Closed Sets is Topological Spaces, Glob. J. Appl. Math. Math. Sci. 2 (2009), 53-63. [20] D. Andrijivic Semi, pre-open sets, Mat, 1986, Vesnic, pp. 24–32. [21] J. Weidmann, Spectral Theory of Ordinary Differential Operators, Springer-Verlag Berlin Heidelberg New York, 1987. [22] P. Bhattacharya, B.K. Lahiri, Semi-generalized closed sets in topology, Indian J. Math. 29 (1987), 376- 382. [23] N. Levine, Generalized closed sets in topology, Rand. Circ. Mat. Palermo, 19 (1970), 89- 96. [24] Y. Gnanambal, On Generalized Pre-Regular Closed Sets in Topological Spaces, Indian J. Pure Appl. Math. 28(1997), 351–360.