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Generalized Form of Closure Set for Topological Space

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Generalized Form of Closure Set for Topological Space European Journal of Molecular & Clinical Medicine ISSN 2515-8260 Volume 07, Issue 07, 2020 Generalized Form Of Closure Set For Topological Space S. Mishra1 1School Chemical Engineering and Physical Sciences Lovely Professional University, Punjab Abstract: We are going to define the new general form of closure set for any subset of the topological space and study their properties. Furthermore, we also establish relationship of new defined form with existing form. Introduction For the first time, in 1970 N. Levine studied about new general form of closed set called 푔 − closed set in [1]. As we know that topology on a set is defined in term of open set and complementary of each open set is closed set. Several properties of topological space studied in term of closed set like continuity, conecctededness, compactness, countability, separation axiom. Levine and many reserachers studied several properties of topological space in term of this newly defined generalized form of closed set and found very singnficant reuslts. In continuoution of such study, researchers also intiated to defined other general forms of closed sets which help to understand many more properties of topological space in [2-16]. The reder can study some important generalization of closed sets in refer items. In 2012, S.Mishra and V.Joshi et.al. defined new general form of closed set as generalized pre-regular weakly closed set (in short 푔푝푟푤- closed set) and study thier properties. We are going to define new general form of closure set as called 푔푝푟푤-closure set (shortly dentoed by 훽퐶 −closure set) which is actually based of 푔푝푟푤-closed set. Furthermore, we investigate properties of 훽퐶 −closure set and also find the relationship of this newly defined closure set with other existing defined closure set. First, we look at definitions and results for better understanding about the main results. These are the notation and symbols used throughout the paper. Here 퐶(퐴) and 퐼(퐴) are notation of closure and interior of any subset 퐴 of space 푋 respectively. The space means a topological space (푋, 휏) (or 푋 ) otherwise we will mention. The 푔푝푟푤 − set means generalized pre-regular weakly set. We use symbol 훽 instead of 푔푝푟푤, for example 푔푝푟푤- closure set means 훽 −closure set. If 퐴 ⊂ 푋, then 퐴 is i. 푔 −closed set ⟺ 퐶(퐴) ⊆ 퐵, when 퐴 ⊆ 퐵, here 퐵 is open in 푋. ii. 푟-open (or 푟-closed) if 퐴 = 퐼(퐶(퐴)) or 퐴 = 퐶(퐼(퐴)). iii. pre-open (or pre-closed) if 퐴 ⊆ 퐼(퐶(퐴)) or 퐶(퐼(퐴)) ⊆ 퐴. iv. regular semiopen if ∃ 푟 −open set 푈 s.t. 푈 ⊂ 퐴 ⊂ 퐶(푈). v. 훽 −closed if 푝퐶(퐴) ⊂ 푈, if 퐴 lies in regular semi open set 푈. vi. 푟푤 − closed if 퐶(퐴) ⊆ 푈 if 퐴 lies in (or equal) to regular semi open set 푈. Main Result Now we define new general form of closure set as 훽 −closure set for any subset of space 푋 and study their properties. Furthermore, we also find the relationship with other general form 3653 European Journal of Molecular & Clinical Medicine ISSN 2515-8260 Volume 07, Issue 07, 2020 of closed sets. The 휷 − closure of any subset 푈 of space 푋 is denoted by 훽퐶(푈) and it the intersection of all 훽 − closed set whcih containing 푈 i.e. 훽퐶(푈) = ⋂{퐹 ∶ 푈 ⊂ 퐹, where 퐹 is 훽 − closed}. Theorem 1 For a space 푋, 훽퐶(푋) = 푋 and 훽퐶(휙) = 휙. Proof. Due to defintion of 훽-closure, 푋 is the only 훽-closed lies in . Hence, 훽퐶(푋) is equal to intersection of all the 훽-closed set lies in 푋 i.e. =∩ {푋} = 푋. That is 훽퐶(푋) = 푋. Also due to the definition of 훽-closure, 훽퐶(휙) is equal to the intersection of all the β-closed sets containing 휙 that is 휙 ∩ any gprw-closed set lies in 휙 is only ϕ. Hence 훽퐶(휙) = 휙. Theorem 2 For subset 퐴 of space 푋, 퐴 ⊂ 훽퐶(퐴). Proof. By the definition of 훽-closure that is the intersection of all 훽-closed sets lies in 퐴, so it is quite obvious that 퐴 in contained in 훽퐶(퐴). Theorem 3 For two subsets 퐴 and 퐵 of space 푋, where 퐵 is 훽-closed set s.t. 퐴 ⊂ 퐵, then 훽퐶(퐴) ⊂ 퐵. Proof. As given that 퐵 is 훽 -closed set lies in 퐴 훽퐶(퐴) is the intersection of all 훽 −closed sets lies in 퐴. So, 훽퐶(퐴) is lies in all 훽 −closed set lies in 퐴. Therefore, in particular 훽퐶(퐴) ⊂ 퐵. Theorem 4 For two subsets 퐴 and 퐵 of space 푋 if 퐴 ⊂ 퐵, then 훽퐶(퐴) ⊂ 훽퐶(퐵). Proof. As given that 퐴 and 퐵 be subsets of 푋 and 퐴 ⊂ 퐵 so, 훽퐶(퐵) = ⋂ { 퐹: 퐵 ⊂ 퐹 ∈ 퐺푃푅푊퐶(푋)}, now when 퐵 ⊂ 퐹 ∈ 퐺푃푅푊퐶(푋), then 훽퐶(퐵) ⊂ 퐹. As, ⊂ 퐵, 퐴 ⊂ 퐵 ⊂ 퐹 ∈ 퐺푃푅푊퐶(푋), also we have 훽퐶(퐴) ⊂ 퐹. Therefore 훽퐶(퐴) ⊂ ⋂ {퐹: 퐵 ⊂ 퐹 ∈ 퐺푃푅푊퐶(푋)} = 푔푝푟푤 − 푐푙(퐵). Here 퐺푃푅푊퐶(푋) means the collection of all 훽 − closed set for space 푋. Theorem 5 For any subset 퐴 of space 푥, 훽퐶(퐴) = 훽퐶(훽퐶(퐴)) Proof. As given that 퐴 be a subset of 푋 and we have 훽퐶(퐴) = ⋂ {퐹: 퐴 ⊂ 퐹 ∈ 퐺푃푅푊퐶(푋)}. When 퐴 ⊂ 퐹 ∈ 퐺푃푅푊퐶(푋). Then 퐶(퐴) ⊂ 퐹. Since 퐹 is 훽-closed set containing 훽퐶(퐴), by standard result 훽퐶(훽퐶(퐴)) ⊂ 퐹. Now 훽퐶(훽퐶(퐴)) ⊂ ⋂{퐹: 퐴 ⊂ 퐹 ∈ 퐺푃푅푊퐶(푋)} = 훽퐶(퐴). Hence 훽퐶(훽퐶(퐴)) = 훽퐶(퐴). Theorem 6 For a 훽 −closed subset 퐴 of 푋, 훽퐶(퐴) = 퐴. Proof. As given that 퐴 be a 훽 −-closed subset of 푋 and since 퐴 ⊂ 훽퐶(퐴). Also 퐴 ⊂ 퐴 and 퐴 is 훽-closed. By standard result 훽퐶(퐴) ⊂ 퐴. Hence 훽퐶(퐴) = 퐴. Conversely of the above result is false, for example. Example 7 If 푋 = {1,2,3,4} be a space where topology is 휏 = {휙, 푋, {1}, {2}, {1,2}, {1,2,3}}. The collection of all 훽-closed sets is {휙, 푋, {3}, {4}, {1,2}, {3,4}, {1,2,3}, {2,3,4}, {1,3,4}, {1,2,4}}. If we take 퐴 = {1} then 푔푝푟푤 − 푐푙({1}) = {1}. But 퐴 = {1} is not 푔푝푟푤-closed, since collections of all pre-closed and regular semi open sets is {휙, 푋, {3}, {4}, {3,4}, {2,3,4}, {1,3,4}}, and {휙, 푋, {1}, {2}, {2,3}, {1,3}, {2,4}, {1,4}, {2,3,4}, {1,3,4}}. Now we have regular semi open set {1} containing 퐴, but 푝푐푙(퐴 = {1}) = {1,3,4} is not contained in {1}. Theorem 8 For two subsets 퐴 and 퐵 of space 푋, 훽퐶(퐴 ∩ 퐵) ⊂ 훽퐶(퐴) ∩ 훽퐶(퐵). Proof: As given that 퐴 and 퐵 are two subsets of space 푋, as we know that 퐴 ∩ 퐵 ⊂ 퐴 and 퐴 ∩ 퐵 ⊂ 퐵 this shows that 훽퐶(퐴 ∩ 퐵) ⊂ 훽퐶(퐴) and 훽퐶(퐴 ∩ 퐵) ⊂ 훽퐶(퐵) . Hence 훽퐶(퐴 ∩ 퐵) ⊂ 훽퐶(퐴) ∩ 훽퐶(퐵). Theorem 9 If 푃퐶(푋) is closed under finite union where 푃퐶(푋) be the family of all pre closed set in 푋, then the family of all generalized pre regular weakly closed in 푋 denoted as 퐺푃푅푊퐶(푋) is closed under finite union. Proof. 푃퐶(푋) is given to be closed under finite union, and let 퐴 and 퐵 belong to 퐺푃푅푊퐶(푋) and 퐴 ∪ 퐵 ⊂ 푈, here 푈 is regular semi open in 푋, so 퐴 ⊂ 푈 and 퐵 ⊂ 푈. 3654 European Journal of Molecular & Clinical Medicine ISSN 2515-8260 Volume 07, Issue 07, 2020 Therefore 푝푐푙(퐴) ⊂ 푈 and 푝푐푙(퐵) ⊂ 푈 , this shows that 푝푐푙(퐴) ∪ 푝푐푙(퐵) ⊂ 푈 . By hypothesis 푝푐푙(퐴 ∪ 퐵) ⊂ 푈. That means 퐴 ∪ 퐵 is also 훽-closed set, so we can conclude that 퐺푃푅푊퐶(푋) is closed under finite union. Theorem 10 For two subsets 퐴 and 퐵 of space 푋, 훽퐶(퐴 ∪ 퐵) = 훽퐶(퐴) ∪ 훽퐶(퐵). Proof. As given that 퐴 and 퐵 are two subsets of space 푋, clearly 퐴 ⊂ 퐴 ∪ 퐵 and also 퐵 ⊂ 퐴 ∪ 퐵. So we have 훽퐶(퐴) ⊂ 훽퐶(퐴 ∪ 퐵) and 훽퐶(퐵) ⊂ 훽퐶(퐴 ∪ 퐵). Hence we got 훽퐶(퐴) ∪ 훽퐶(퐵) ⊂ 훽퐶(퐴 ∪ 퐵). Now we shall prove the reverse inequality so in that context let 푥 ∈ 훽퐶(퐴 ∪ 퐵) and suppose that 푥 does not belong to 훽퐶(퐴) ∪ 훽퐶(퐵), then there exist 훽 −closed set 퐴1 and 퐵1 with 퐴 ⊂ 퐴1, 퐵 ⊂ 퐵1 and 푥 ∈ 퐴1 ∪ 퐵1 so we have 퐴 ∪ 퐵 ⊂ 퐴1 ∪ 퐵1 and since the space is given to be closed under finite union therefore 퐴1 ∪ 퐵1 is 푔푝푟푤-closed set by 푥 ∈ 퐴1 ∪ 퐵1. Thus 푥 doses not belong to 훽퐶(퐴 ∪ 퐵). Which is a contradiction to 푥 ∈ 훽퐶(퐴 ∪ 퐵). So we got 훽퐶(퐴 ∪ 퐵) ⊂ 훽퐶(퐴) ∪ 훽퐶(퐵). Hence 훽퐶(퐴 ∪ 퐵) = 훽퐶(퐴) ∪ 훽퐶(퐵). Theorem 11 For subset 퐴 of space 푋, 훽퐶(퐴) ⊂ 퐶(퐴). Proof. Since 퐴 ⊂ 푋, so 퐶(퐴) = ⋂{퐹 ⊂ 푋: 퐴 ⊂ 퐹 ∈ 퐶(푋)} where 퐶(푋) be the family all closed sets in 푋. If 퐴 ⊂ 퐹 ∈ 퐶(푋) then 퐴 ⊂ 퐹 ∈ 퐺푃푅푊퐶(푋). Beacuse all closed set is 훽 − closed set. That is 훽퐶(퐴) ⊂ 퐹 . So, 훽퐶(퐴) ⊂ ⋂{퐹 ⊂ 푋: 퐴 ⊂ 퐹 ∈ 퐶(푋)} = 푐푙(퐴) . Therefore, 훽퐶(퐴) ⊂ 퐶(퐴). Remark 12 Reverse inequality does not hold in the above theorem, for example. Example 13 The set 푋 = {1,2,3,4} space with 휏 = {휙, 푋, {1}, {2}, {1,2}, {1,2,3}}.
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