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Journal of Babylon University/Pure and Applied Sciences/ No.(2)/ Vol.(24): 2016 ON -Syndetic Sets Aqeel ketab Mezaal Al-khafaji Department of Computer Sciences, College of Sciences for women Babylon University, [email protected] Abstract In this paper, we introduce and investigate the notion of -syndetic sets, using the notion of -open sets which was introduce by Njastad. And the work of W. Gottschalk , and S. Al-Kutaibi . Keywords: -open, − , −closure, − , − ﺍﻟﺨﻼﺼﺔ ﻓﻲ ﻫﺫﺍ ﺍﻟﺒﺤﺙ ﻗﺩﻤﻨﺎ ﻤﻔﻬﻭﻡ ﺍﻟﻤﺠﻤﻭﻋﺎﺕ ﺍﻟﺭﺍﺒﻁﺔ ﻤﻥ ﺍﻟﻨﻭﻉ ﺍﻟﻔﺎ ﻤﻊ ﻋﺩﺩ ﻤﻥ ﺍﻟﻨﻅﺭﻴﺎﺕ ﺒﺎﺴﺘﺨﺩﺍﻡ ﺘﻌﺭﻴﻑ ﺍﻟﻤﺠﻤﻭﻋﺎﺕ ﺍﻟﻤﻔﺘﻭﺤﺔ .ﻤﻥ ﺍﻟﻨﻭﻉ ﺍﻟﻔﺎ ﺍﻟﺘﻲ ﻗﺩﻤﻬﺎ ﻨﺠﺎﺴﺘﺩ ﻭﻤﻌﺘﻤﺩﻴﻥ ﺒﺫﻟﻙ ﻋﻠﻰ ﺍﻟﻤﻔﺎﻫﻴﻡ ﺍﻟﺘﻲ ﻗﺩﻤﻬﺎ ﻜل ﻤﻥ ﺠﻭﺘﺴﺠﻭﻙ ﻭﺍﻟﻜﺘﺒﻲ ﻓﻲ ﻫﺫﺍ ﺍﻟﻤﻭﻀﻭﻉ ﺍﻟﻤﺠﻤﻭﻋﺔ، ﺍﻻﻨﻐﻼﻕ ﻤﻥ ﺍﻟﻨﻭﻉ ﺍﻟﻔﺎ،ﺍﻟﻤﺠﻤﻭﻋﺔ ﺍﻟﻤﻐﻠﻘﺔ ﻤﻥ ﺍﻟﻨﻭﻉ ﺍﻟﻔﺎ، ﺍﻟﻤﺠﻤﻭﻋﺔ ﺍﻟﻤﻔﺘﻭﺤﺔ ﻤﻥ ﺍﻟﻨﻭﻉ ﺍﻟﻔﺎ:ﺍﻟﻜﻠﻤﺎﺕ ﺍﻻﻓﺘﺘﺎﺤﻴﺔ . ﺍﻟﻤﺠﻤﻭﻋﺔ ﺍﻟﻤﺘﺭﺍﺒﻁﺔ ﻤﻥ ﺍﻟﻨﻭﻉ ﺍﻟﻔﺎ،ﺍﻟﻤﺭﺼﻭﺼﺔ ﻤﻥ ﺍﻟﻨﻭﻉ ﺍﻟﻔﺎ 1.Introduction Gottschalk in [Gottschalk, 1955] introduced the notions of left (right) syndetic set in topological group. He defined a subset of topological group , is called left )= (right) syndetic if there exists a compact subset of , such that = ( . In 1998 Al-kutaibi [Al-kutaibi, 1998] introduced paper semi-syndetic and feebly syndetic subsets. The aim of this paper is to introduce and investigate a new class of syndetic called − which is a subset of topological group , is called left (right) )= − if there exists a − set of , such that = ( . And we obtain several Theorem abut − . 2. Preliminaries In this section we recall some of the basic Definitions. Throughout this paper will denote a topological space. If A is a subset of the space ( ), − ( ) denote the closure, interior and the −closure of A , ( ), respectively. A subset of is said to be − set [Njastad, 1965] if A ⊂ Int(C1(Int(A))). The complement of a − set is said to be − . Let ⊆ and ℋ be a family of − subsets of , then ℋ is called a −open cover of , if ⊆ ⋃ ∈ℋ , A is said to be − if and only if any − cover has a finite sub cover. Clear that, every open set is − and then every − set is compact also, the union of two − subsets of is − . A topological group is a set which carries group stricture and a topology and satisfied the two axioms: (i) The map ( , ) ⟶ of × into is continuous.(That is, the operation of is continuous). (ii) The map ⟶ (The inversion map) of into is continuous.[Higgins, 1979] 3. The main results 3.1 Theorem: If is − set in a topological group 329 , then is − . Journal of Babylon University/Pure and Applied Sciences/ No.(2)/ Vol.(24): 2016 Proof: Let : ⟶ be the invers map, that is, ( ) = for all in , let ℋ be a )=( − cover of , then (ℋ ) is − cover of ( ) = ,but is − , which implies (ℋ ) has a finite sub cover ℋ ∗ , then (ℋ ∗ ) covers ( ) = . Hence, is − set. 3.2 Definition: Let be a subset of a topological group , then is called left (right) − if there exists a − subset of such that = ( = ). 3.3 Note: In the following results we will prove the case of left − and the case of right − will be similar. 3.4 Theorem: Let be a topological group and let ⊆ , then is left (right) − set in if and only if there exists a − subset of such that every left (right) translation of intersects . Proof: Sufficiency, suppose is a left − set then, there exists a − subset of such that = , let ∈ then, there exists ∈ , ∈ such that = which implies = and then ∈ but is − . Hence ∩ ≠ ∅, (i.e is the − set we need). Efficiency, let ∈ , there exists − subset of such that, ∩ ≠ ∅, for each in , there exists ∈ , ∈ such that = , so = , which implies = , and since is − then is a left − . 3.5 Theorem: Let be a subset of a topological group , then is left (right) − in , if and only if right (left) − . Proof: Let is a left − , then there exists − subset of such that = . Since = =( ) = and since is − , then is right − . 3.6 Theorem: Let be a topological group, let , be two subsets of such that ⊆ . If A is left (right) − set, then so is . Proof: Let be a left − set, then there exists a − subset of such that = , since ⊆ , then = , which implies that is left − . 3.7 Theorem: Let be a topological group, then 1- If is a left − set in ,then ( ) − ( ) are left 2- The union of any family of left (right) − sets is left (right) − . Proof: Directly from (Theorem 3.6). 330 Journal of Babylon University/Pure and Applied Sciences/ No.(2)/ Vol.(24): 2016 3.8 Theorem: Let be a topological group, let , are two left (right) − sub sets of , then ∩ is a left (right) − . Proof: Since both are left − , then there exist − sets such that = = and then ( ∩ )( ∪ ) = ( ∪ ) ∩ ( ∪ ) = ∩ = and since ( ∪ ) is − , then ( ∩ ) is left − . 3.9 Theorem: Let be a subset of topological group .If is a subgroup of or if is an abelian group, then is left − in if and only if is a right − in . Proof: Let is a left − subgroup of , then = , that means = and hence = , but is − , which implies is a right − in . 3.10 Theorem: Let be a − subgroup of a topological group quotient space ∕ is compact. , then the Proof: Let be a exists a − map. Clear that ( ⊂ , that is is continuous and is compact set. − subgroup of a topological group , then there set such that = . Let : ⟶ ∕ is the quotient ) ⊂ ⁄ . Let ∈ ∕ , then ⊂ which implies ∈ ( ) and then, ⁄ ⊂ ( ). Hence ⁄ = ( ). Since is − and hence is compact, then ( ) = ⁄ Refrences ALkutaibi,S.H., On semi-syndetic and feebly syndetic subsets, Journal of sciences. College of Education .Tikrit University, Vol 4, No.3, (1998).pp. Gottschalk, W.H., Hedlund , G.A., Topological dynamics, American Mathematical Society, (1955) Higgins, P.J., An introduction to topological groups, Cambridge University Press, (1979). Munkres , J.R., Topology, 2nd Ed. , PHI , (2000). Njåstad O., On some classes of nearly open sets, Pacific J. Math. 15 (1965), 961−970. 331