ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING
Peer Reviewed and Refereed Journal IMPACT FACTOR: 2.104 SSN NO. 2456-1037 (INTERNATIONAL JOURNAL) Vol. 04, Special Issue 02, 13th Conference (ICOSD) February 2019 Available Online: www.ajeee.co.in/index.php/AJEEE
1
THE NOTION OF G-SEMI-CLOSURE, G-PRE-CLOSURE IN GENERALIZED TOPOLOGICAL SPACES: A STUDY
Dr. H. K. Tripathi
Lecturer, Govt. Women’s Polytechnic College, Jabalpur-482001
Abstract - The notion of g-semi-closure, g-pre-closure of a set in generalized topological spaces and obtained their significant properties. Further we have introduced the notion of g-semi- interior, g-pre-interior, of a set in generalized topological spaces and obtained their significant properties.
Keywords: Generalized Topology.
1 INTRODUCTION
1.1 G-Semi-Closure and G-Semi-Interior In this section we have studied notions of g- semi-closure and g-semi-interior in generalized topological spaces and obtained their significant properties.
Definition 1.1: Let(𝑋, 𝜏𝑔) be a generalized topological space and 𝐴 ⊆ 𝑋.Then the g- semi-closure of A is defined as the intersection of all g-semi-closed sets in X containing A. The g-semi-closure of A is denoted by 𝑆𝑐𝑔 𝐴 .
Remark 1.1: We note that 𝑆𝑐𝑔 𝐴 is the smallest g-semi-closed set in 𝑋, 𝜏𝑔 containing A.
Proposition 1.1: Let 𝑋, 𝜏𝑔 be a generalized topological space and let 𝐴 ⊆ 𝑋. Then A is g- semi-closed set if and only if 𝑆𝑐𝑔 𝐴 = 𝐴.
Proof: Let A be a g-closed set in X. Then clearly the smallest g-semi-closed set containing A, is itself A. Hence 𝑆𝑐𝑔 𝐴 = 𝐴.
Conversely suppose 𝐴 ⊆ 𝑋 and 𝑆𝑐𝑔 𝐴 = 𝐴.
Since 𝑆𝑐𝑔 𝐴 is a g-semi-closed set in X, it follows that A is g-semi-closed set in X.
Proposition 1.2: Let 𝑋, 𝜏𝑔 be a generalized topological space and let A, B be subsets of X. Then following properties holds:
(i) 𝑆𝑐𝑔 𝜙 = 𝜙, 𝑆𝑐𝑔 𝑋 = 𝑋.
(ii) If 𝐴 ⊆ 𝐵then 𝑆𝑐𝑔 𝐴 ⊆ 𝑆𝑐𝑔 𝐵 .
(iii)𝑆𝑐𝑔 𝐴 ∪ 𝑆𝑐𝑔 𝐵 ⊆ 𝑆𝑐𝑔(𝐴 ∪ 𝐵).
(iv) 𝑆𝑐𝑔 𝐴 ∩ 𝐵 ⊆ 𝑆𝑐𝑔 𝐴 ∩ 𝑆𝑐𝑔 𝐵 . (v) 𝑆𝑐𝑔 𝑆𝑐𝑔 𝐴 = 𝑆𝑐𝑔 𝐴 .
Proof:
(i) Since 𝜙 and 𝑋 are g-closed sets, from Proposition, we have, 𝑆𝑐𝑔 𝜙 = 𝜙 and𝑆𝑐𝑔 𝑋 = 𝑋.
(ii) Suppose 𝐴 ⊆ 𝐵 in X. Since 𝐵 ⊆ 𝑆𝑐𝑔 𝐵 and 𝐴 ⊆ 𝐵, we have 𝐴 ⊆ 𝑆𝑐𝑔 𝐵 . Now 𝑆𝑐𝑔 𝐵 is a g-semi-closed set and 𝑆𝑐𝑔 𝐴 is the smallest g-semi-closed set containing 𝐴, we find that 𝑆𝑐𝑔 𝐴 ⊆ 𝑆𝑐𝑔 𝐵 .
(iii) Since 𝐴 ⊆ 𝐴 ∪ 𝐵, 𝐵 ⊆ 𝐴 ∪ 𝐵 from (ii) we have 𝑆𝑐𝑔 𝐴 ⊆ 𝑆𝑐𝑔(𝐴 ∪ 𝐵) and 𝑆𝑐𝑔 𝐵 ⊆ 𝑆𝑐𝑔 𝐴 ∪ 𝐵 . This implies 𝑆𝑐𝑔 𝐴 ∪ 𝑆𝑐𝑔 𝐵 ⊆ 𝑆𝑐𝑔(𝐴 ∪ 𝐵).
(iv) Since 𝐴 ∩ 𝐵 ⊆ 𝐴 and 𝐴 ∩ 𝐵 ⊆ 𝐵 from (ii) we have 𝑆𝑐𝑔(𝐴 ∩ 𝐵) ⊆ 𝑆𝑐𝑔(𝐴)and 𝑆𝑐𝑔 𝐴 ∩ 𝐵 ⊆ 𝑆𝑐𝑔 𝐵 .This
implies𝑆𝑐𝑔(𝐴 ∩ 𝐵) ⊆ 𝑆𝑐𝑔(𝐴) ∩ 𝑆𝑐𝑔 𝐵 . (v) Since 𝑆𝑐𝑔(𝐴) is a g-semi-closed set in
X, it follows that 𝑆𝑐𝑔 𝑆𝑐𝑔 𝐴 = 𝑆𝑐𝑔 𝐴 .
Proposition 1.3: Let 𝑋, 𝜏𝑔 be a generalized topological space and let 𝐴∝ ∝∈Λbe a family of subsets of X. Then
(i) ∝∈Λ𝑆𝑐𝑔(𝐴∝) ⊆ 𝑆𝑐𝑔 ( ∝∈Λ𝐴∝).
(ii) 𝑆𝑐𝑔 ∝∈Λ𝐴∝ ⊆ ∝∈Λ 𝑆𝑐𝑔 (𝐴∝).
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2 Proof: Similar to proof of Proposition 1.3 (iii) and (iv).
Definition 1.2: Let 𝑋, 𝜏𝑔 be a generalized topological space and let 𝐴 ⊆ 𝑋. Then the g- semi-interior of A is defined as the union of all g-semi-open sets in X contained in A.
The g-semi-interior of A is denoted by 𝑆𝑖𝑔 𝐴 .
Remark 1.2: We note that 𝑆𝑖𝑔 𝐴 is the largest g-semi-open set in 𝑋, 𝜏𝑔 contained in A.
Proposition 1.4: Let 𝑋, 𝜏𝑔 be a generalized topological space and let 𝐴 ⊆ 𝑋. Then A is g- semi-open if and only if 𝑆𝑖𝑔 𝐴 = 𝐴.
Proof: Let A be a g-open set in X. Then clearly the largest g-semi-open set contained in A is itself A. Hence 𝑆𝑖𝑔 𝐴 = 𝐴.
Conversely suppose 𝐴 ⊆ 𝑋 and 𝑆𝑖𝑔 𝐴 = 𝐴.
Since 𝑆𝑖𝑔 𝐴 is a g-semi-open set in X, it follows that A is a g-semi-open set in X.
Proposition 1.5: Let 𝑋, 𝜏𝑔 be a generalized topological space and let A, B be subsets of X. Then following properties holds:
(i) 𝑆𝑖𝑔 𝜙 = 𝜙, 𝑆𝑖𝑔 𝑋 = 𝑋.
(ii) If 𝐴 ⊆ 𝐵 then 𝑆𝑖𝑔 𝐴 ⊆ 𝑆𝑖𝑔 𝐵 . (iii)𝑆𝑖𝑔 𝐴 ∪ 𝑆𝑖𝑔 𝐵 ⊆ 𝑆𝑖𝑔(𝐴 ∪ 𝐵).
(iv) 𝑆𝑖𝑔 𝐴 ∩ 𝐵 ⊆ 𝑆𝑖𝑔 𝐴 ∩ 𝑆𝑖𝑔 𝐵 . (v) 𝑆𝑖𝑔 𝑆𝑖𝑔 𝐴 = 𝑆𝑖𝑔 𝐴 .
Proof:
(i) Since 𝜙 and X are g-semi-open sets, from Proposition, we have, 𝑆𝑖𝑔 𝜙 = 𝜙 and 𝑆𝑖𝑔 𝑋 = 𝑋.
(ii) Suppose 𝐴 ⊆ 𝐵 in X. Since 𝑆𝑖𝑔 𝐴 ⊆ 𝐴and 𝐴 ⊆ 𝐵, we have 𝑆𝑖𝑔 𝐴 ⊆ 𝐵. Now 𝑆𝑖𝑔 𝐴 is a g-semi-open set and 𝑆 𝑖𝑔 𝐵 is the largest g-semi-open set contained in B, we find that 𝑆𝑖𝑔 𝐴 ⊆ 𝑆𝑖𝑔 𝐵 .
(iii) Since 𝐴 ⊆ 𝐴 ∪ 𝐵, 𝐵 ⊆ 𝐴 ∪ 𝐵 from (ii) we have 𝑆𝑖𝑔 𝐴 ⊆ 𝑆𝑖𝑔 𝐴 ∪ 𝐵 and 𝑆𝑖𝑔 𝐵 ⊆
𝑆𝑖𝑔 𝐴 ∪ 𝐵 . This implies 𝑆𝑖𝑔 𝐴 ∪ 𝑆𝑖𝑔 𝐵 ⊆ 𝑆𝑖𝑔 𝐴 ∪ 𝐵 .
(iv) Since 𝐴 ∩ 𝐵 ⊆ 𝐴 and 𝐴 ∩ 𝐵 ⊆ 𝐵, from (ii) we have 𝑆𝑖𝑔 𝐴 ∩ 𝐵 ⊆ 𝑆𝑖𝑔 𝐴 and 𝑆𝑖𝑔 𝐴 ∩ 𝐵 ⊆ 𝑆𝑖𝑔 𝐵 . This implies 𝑆𝑖𝑔 𝐴 ∩ 𝐵 ⊆ 𝑆𝑖𝑔 𝐴 ∩ 𝑆𝑖𝑔 𝐵 .
(v) Since 𝑆𝑖𝑔 𝐴 is a g-semi-open set in X, it follows that 𝑆𝑖𝑔 𝑆𝑖𝑔 𝐴 = 𝑆𝑖𝑔 𝐴 .
Proposition 1.6: Let 𝑋, 𝜏𝑔 be a generalized topological space and let 𝐴∝ ∝∈Λ be a family of subsets of X. Then
(i) ∝∈Λ𝑆𝑖𝑔(𝐴∝) ⊆ 𝑆𝑖𝑔 ( ∝∈Λ𝐴∝).
(ii) 𝑆𝑖𝑔 ∝∈Λ𝐴∝ ⊆ ∝∈Λ𝑆𝑖𝑔 (𝐴∝).
Proof: Similar to proof of Proposition 1.5 (iii) and (iv).
Proposition 1.7: Let 𝑋, 𝜏𝑔 be a generalized topological space and let 𝐴 ⊆ 𝑋. Then
(i) 𝑆𝑖𝑔 𝑋 − 𝐴 = 𝑋 − 𝑆𝑐𝑔 𝐴 . (ii) 𝑆𝑐𝑔 𝑋 − 𝐴 = 𝑋 − 𝑆𝑖𝑔 𝐴 .
Proof:
(i) We have𝑋 − 𝑆𝑐𝑔 𝐴 = ∝∈Λ { 𝐹∝: 𝐹∝is a g- semi-colsed set in 𝑋 and 𝐴 ⊆ 𝐹∝}
= ∝∈Λ {𝑋 −𝐹∝: 𝑋 − 𝐹∝ is ag-semi-open set in X and 𝑋 − 𝐹∝ ⊆ 𝑋 − 𝐴 }.
=𝑆𝑖𝑔 𝑋 − 𝐴 .
(ii) From (i), we have 𝑋 − 𝑆𝑐𝑔 𝑋 − 𝐴 = 𝑆𝑖𝑔(𝑋 − 𝑋 − 𝐴 )=𝑆𝑖𝑔 𝐴 . Hence 𝑋 − 𝑆𝑖𝑔 𝐴 = 𝑆𝑐𝑔 𝑋 − 𝐴 .
2 G-PRE-CLOSURE AND G-PRE- INTERIOR:
In this section we have studied notions of g- pre-closure and g-pre-interior in generalized topological spaces and obtained their significant properties.
Definition 2.1: Let (𝑋, 𝜏𝑔) be a generalized topological space and let𝐴 ⊆ 𝑋.Then the g- pre-closure of A is defined as the intersection of all g-pre-closed sets in X
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3 containing A. The g-pre-closure of A is denoted by 𝑃𝑐𝑔 𝐴 .
Remark 2.1: We note that 𝑃𝑐𝑔 𝐴 is the smallest g-pre-closed set in 𝑋, 𝜏𝑔 containing A.
Proposition 2.1: Let 𝑋, 𝜏𝑔 be a generalized topological space and let 𝐴 ⊆ 𝑋. Then A is g- pre-closed set if and only if 𝑃𝑐𝑔 𝐴 = 𝐴.
Proof: Let A be a g-closed set in X. Then clearly the smallest g-pre-closed set containing A, is itself A. Hence 𝑃𝑐𝑔 𝐴 = 𝐴.
Conversely suppose 𝐴 ⊆ 𝑋 and𝑃𝑐𝑔 𝐴 = 𝐴.
Since 𝑃𝑐𝑔 𝐴 is a g-pre-closed set in X, it follows that A is g-pre-closed set in X.
Proposition 2.2: Let 𝑋, 𝜏𝑔 be a generalized topological space and let A, B be subsets of X. Then following properties holds:
(i) 𝑃𝑐𝑔 𝜙 = 𝜙, 𝑃𝑐𝑔 𝑋 = 𝑋.
(ii) If 𝐴 ⊆ 𝐵then 𝑃𝑐𝑔 𝐴 ⊆ 𝑃𝑐𝑔 𝐵 . (iii) 𝑃𝑐𝑔 𝐴 ∪ 𝑃𝑐𝑔 𝐵 ⊆ 𝑃𝑐𝑔(𝐴 ∪ 𝐵).
(iv) 𝑃𝑐𝑔 𝐴 ∩ 𝐵 ⊆ 𝑃𝑐𝑔 𝐴 ∩ 𝑃𝑐𝑔 𝐵 . (v) 𝑃𝑐𝑔 𝑃𝑐𝑔 𝐴 = 𝑃𝑐𝑔 𝐴 .
Proof:
(i) Since 𝜙 and 𝑋 are g-closed sets, from Proposition 4.3.1, we have, 𝑃𝑐𝑔 𝜙 = 𝜙 and 𝑃𝑐𝑔 𝑋 = 𝑋.
(ii) Suppose 𝐴 ⊆ 𝐵 in X. Since 𝐵 ⊆ 𝑃𝑐𝑔 𝐵 and 𝐴 ⊆ 𝐵, we have 𝐴 ⊆ 𝑃𝑐𝑔 𝐵 . Now 𝑃𝑐𝑔 𝐵 is a g-pre-closed set and 𝑃𝑐𝑔 𝐴 is the smallest g-pre-closed set containing 𝐴, we find that 𝑃𝑐𝑔 𝐴 ⊆ 𝑃𝑐𝑔 𝐵 .
(iii) Since 𝐴 ⊆ 𝐴 ∪ 𝐵, 𝐵 ⊆ 𝐴 ∪ 𝐵 from (ii) we have 𝑃𝑐𝑔 𝐴 ⊆ 𝑃𝑐𝑔(𝐴 ∪ 𝐵) and 𝑃𝑐𝑔 𝐵 ⊆ 𝑃𝑐𝑔 𝐴 ∪ 𝐵 . This implies𝑃𝑐𝑔 𝐴 ∪ 𝑃𝑐𝑔 𝐵 ⊆ 𝑃𝑐𝑔(𝐴 ∪ 𝐵).
(iv) Since 𝐴 ∩ 𝐵 ⊆ 𝐴 and 𝐴 ∩ 𝐵 ⊆ 𝐵 from (ii) we have 𝑃𝑐𝑔 𝐴 ∩ 𝐵 ⊆ 𝑃𝑐𝑔 𝐴 and 𝑃𝑐𝑔 𝐴 ∩ 𝐵 ⊆ 𝑃𝑐𝑔 𝐵 . This implies 𝑃𝑐𝑔 𝐴 ∩ 𝐵 ⊆ 𝑃𝑐𝑔 𝐴 ∩ 𝑃𝑐𝑔 𝐵 .
(v) Since 𝑃𝑐𝑔(𝐴) is a g-pre-closed set in X, it follows that 𝑃𝑐𝑔 𝑃𝑐𝑔 𝐴 = 𝑃𝑐𝑔 𝐴 .
Proposition 2.3: Let 𝑋, 𝜏𝑔 be a generalized topological space and let 𝐴∝ ∝∈Λbe a family of subsets of X. Then
(i) ∝∈Λ𝑃𝑐𝑔(𝐴∝) ⊆ 𝑃𝑐𝑔 ( ∝∈Λ𝐴∝).
(ii) 𝑃𝑐𝑔 ∝∈Λ𝐴∝ ⊆ ∝∈Λ 𝑃𝑐𝑔 (𝐴∝).
Definition 2.2: Let 𝑋, 𝜏𝑔 be a generalized topological space and let 𝐴 ⊆ 𝑋. Then the g- pre-interior of A is defined as the union of all g-pre-open sets in X contained in A. The g-pre-interior of A is denoted by 𝑃𝑖𝑔 𝐴 .
Remark 2.2: We note that 𝑃𝑖𝑔 𝐴 is the largest g-pre-open set in 𝑋, 𝜏𝑔 contained in A.
Proposition 2.4: Let 𝑋, 𝜏𝑔 be a generalized topological space and let𝐴 ⊆ 𝑋. Then A is g- pre-open if and only if 𝑃𝑖𝑔 𝐴 = 𝐴.
Proof: Let A be a g-open set in X. Then clearly the largest g-pre-open set contained in A is itself A. Hence 𝑃𝑖𝑔 𝐴 = 𝐴. Conversely suppose 𝐴 ⊆ 𝑋and 𝑃𝑖𝑔 𝐴 = 𝐴. Since 𝑃𝑖𝑔 𝐴 is a g-pre-open set in X, it follows that A is a g-pre-open set in X.
Proposition 2.5: Let 𝑋, 𝜏𝑔 be a generalized topological space and let A, B be subsets of X. Then following properties holds:
(i) 𝑃𝑖𝑔 𝜙 = 𝜙, 𝑃𝑖𝑔 𝑋 = 𝑋.
(ii) If 𝐴 ⊆ 𝐵 then 𝑃𝑖𝑔 𝐴 ⊆ 𝑃𝑖𝑔 𝐵 . (iii)𝑃𝑖𝑔 𝐴 ∪ 𝑃𝑖𝑔 𝐵 ⊆ 𝑃𝑖𝑔(𝐴 ∪ 𝐵).
(iv) 𝑃𝑖𝑔 𝐴 ∩ 𝐵 ⊆ 𝑃𝑖𝑔 𝐴 ∩ 𝑃𝑖𝑔 𝐵 . (v) 𝑃𝑖𝑔 𝑃𝑖𝑔 𝐴 = 𝑃𝑖𝑔 𝐴
Proof:
(i) Since 𝜙 and X are g-pre-open sets, from Proposition, we have, 𝑃𝑖𝑔 𝜙 = 𝜙 and 𝑃𝑖𝑔 𝑋 = 𝑋.
(ii) Suppose 𝐴 ⊆ 𝐵 in X. Since𝑃𝑖𝑔(𝐴) ⊆ 𝐴and𝐴 ⊆ 𝐵, we have 𝑃𝑖𝑔 𝐴 ⊆ 𝐵. Now
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4 𝑃𝑖𝑔 𝐴 is a g-pre-open set and 𝑃𝑖𝑔 𝐵 is the largest g-pre-open set contained in B, we find that 𝑃𝑖𝑔 𝐴 ⊆ 𝑃𝑖𝑔 𝐵 .
(iii) Since 𝐴 ⊆ 𝐴 ∪ 𝐵, 𝐵 ⊆ 𝐴 ∪ 𝐵 from (ii) we have 𝑃𝑖𝑔 𝐴 ⊆ 𝑃𝑖𝑔 𝐴 ∪ 𝐵 and 𝑃𝑖𝑔 𝐵 ⊆ 𝑃𝑖𝑔 𝐴 ∪ 𝐵 . Thisimplies 𝑃𝑖𝑔 𝐴 ∪ 𝑃𝑖𝑔 𝐵 ⊆ 𝑃𝑖𝑔 𝐴 ∪ 𝐵 .
(iv) Since 𝐴 ∩ 𝐵 ⊆ 𝐴 and 𝐴 ∩ 𝐵 ⊆ 𝐵, from (ii) we have 𝑃𝑖𝑔 𝐴 ∩ 𝐵 ⊆ 𝑃𝑖𝑔 𝐴 and 𝑃𝑖𝑔 𝐴 ∩ 𝐵 ⊆ 𝑃𝑖𝑔 𝐵 . This implies 𝑃𝑖𝑔 𝐴 ∩ 𝐵 ⊆ 𝑃𝑖𝑔 𝐴 ∩ 𝑃𝑖𝑔 𝐵 .
(v) Since 𝑃𝑖𝑔 𝐴 is a g-pre-open set in X, it follows that 𝑃𝑖𝑔 𝑃𝑖𝑔 𝐴 = 𝑃𝑖𝑔 𝐴 .
Proposition 2.6: Let 𝑋, 𝜏𝑔 be a generalized topological space and let 𝐴∝ ∝∈Λ be a family of subsets of X. Then
(i) ∝∈Λ𝑃𝑖𝑔(𝐴∝) ⊆ 𝑃𝑖𝑔 ( ∝∈Λ𝐴∝).
(ii) 𝑃𝑖𝑔 ∝∈Λ𝐴∝ ⊆ ∝∈Λ𝑃𝑖𝑔 (𝐴∝).
Proposition 2.7: Let 𝑋, 𝜏𝑔 be a generalized topological space and let𝐴 ⊆ 𝑋. Then
(i) 𝑃𝑖𝑔 𝑋 − 𝐴 = 𝑋 − 𝑃𝑐𝑔 𝐴 . (ii) 𝑃𝑐𝑔 𝑋 − 𝐴 = 𝑋 − 𝑃𝑖𝑔 𝐴 .
Proof: (i) We have 𝑋 − 𝑃𝑐𝑔 𝐴 = 𝑋 − { 𝐹∝
∝∈Λ : 𝐹∝is a g-pre-closed setin 𝑋 and 𝐴 ⊆ 𝐹∝} = ∝∈Λ {𝑋 −𝐹∝: 𝑋 − 𝐹∝ is ag-pre-open set in X and 𝑋 − 𝐹∝ ⊆ 𝑋 − 𝐴 }. =𝑃𝑖𝑔 𝑋 − 𝐴 . (ii) From (i), we have 𝑋 − 𝑃𝑐𝑔 𝑋 − 𝐴 = 𝑃𝑖𝑔(𝑋 − 𝑋 − 𝐴 )=𝑃𝑖𝑔 𝐴 . Hence 𝑋 − 𝑃𝑖𝑔 𝐴 = 𝑃𝑐𝑔 𝑋 − 𝐴 .
3 CONCLUSION
Research conclude for the notion of g-semi- closure, g-pre-closure of a set in generalized topological spaces and obtained their significant properties. Further we have introduced the notion of g-semi-interior, g- pre-interior, of a set in generalized topological spaces and obtained their significant properties.
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