Research Paper
Mathematics
E-ISSN No : 2454-9916 | Volume : 4 | Issue : 5 | May 2018
rgrα - CLOSED SETS IN TOPOLOGICAL SPACES
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V. Jeyanthi | M. Anupriya 1
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Department of Mathematics, Sree Narayana Guru College, Coimbatore, India.
ABSTRACT Here in this paper, a new class of closed set called rgrα-closed set is defined and obtained few of its characteristics. KEY WORDS: rgrα-closed set, regular α-open set. 1. INTRODUCTION: N.Levine[11] introduced generalized closed sets in general topology as a generalization of closed sets and a class of topological spaces called T1/2 -spaces. Regular open and regular α-open sets have been introduced and investigated by Stone [22], A. Vadivel and K. Vairamanickam [23] respectively. Many researchers like Balachandran, Sundaram and Maki [4], Bhattacharyya and Lahiri [5], Arockiarani [2], Dunham [8], Gnanambal [9], Malghan [14], Palaniapan and Rao [19], Park [20], Arya and Gupta [3] and Devi [7], wali et al [24] have worked on generalized closed sets, their generalizations and related concepts in general topology. In this paper, we define and study the properties of regular generalized regular α-closed sets (briefly rgrα-closed). Moreover, in this paper we define rgrα-open sets and obtained some of its basic properties as results. 2. PRELIMINARIES: Throughout this paper (X, τ) and (Y,σ) represent non empty topological spaces on which no separation axioms are assumed unless otherwise mentioned. For a subset A of a space (X, τ), cl(A) and int(A) denote the closure of A and the interior of A respectively. (X, τ) will be replaced by X if there is no chance of confusion.
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a pre-generalized regular α-closed set (briefly, pgrα-closed)[6] if pcl(A) ⊂ U whenever A⊂ U and U is regular α-open in X.
Definition 2.3: Regular closure(α-closure) of A is defined as the intersection of all regular closed sets(α-closed sets) containing A. It is denoted by rcl(A)(resp. αcl(A)). 3: rgrα -CLOSED SETS We introduce the following definition: Definition 3.1: A subset A of a space (X,τ) is called regular α-open set (briefly, rα-open) if there is a regular open set U such that U⊂A⊂αcl(U). The family of all regular α-open sets of (X,τ) is denoted by RαO(X). Definition 3.2: A subset A of a space (X,τ) is called a regular generalized regular α-closed set (briefly, rgrα-closed) if rcl(A)⊂U whenever A⊂U and U is regular α-open. We denote the set of all rgrα-closed sets in (X,τ) by RGRαC(X). Proposition 3.3: Every regular closed set is rgrα-closed.
Let us recall the following definition which we shall require later. Definition 2.1: A subset A of (X, τ) is called
Proof: Let A be a regular-closed set and A⊂U and U is regular α-open. Then
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rcl(A)=A. Therefore, rcl(A)⊂U whenever A⊂U and U is regular α-open. Hence A is rgrα-closed.
a pre-open set [19] if A⊂int(cl(A)) and a pre-closed set if cl(int(A))⊂ A
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a semi-open set [12] if A⊂cl(int(A)) and a semi-closed set if int(cl(A))⊂A
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an α-open set [7] if A⊂int(cl(int(A))) and an α-closed set if cl(int(cl(A)))⊂ A
Remark 3.4: Converse of the above proposition need not be true as in the following example.
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a semi-preopen [1] if A⊂cl(int(cl(A))) and a semi-pre closed set if int
Example 3.5: Let X = {a,b,c,d}and τ = {ϕ,{a},{b},{a,b},{b,d},{a,b,d},X}. Let A= {c}.Then A is rgrα-closed but not regular closed set.
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(cl(int(A)))⊂ A a regular open set [19] if A=int(cl(A)) and a regular closed set if A=cl(int(A)) A subset A of (X, τ) is called clopen if it is both open and closed in (X, τ).
Proof: Similar to that of the proposition 3.3.
Definition 2.2: A subset A of a space (X, τ) is called 1. 2.
a weakly closed set (briefly, w-closed) [17] if cl(A) ⊂ U whenever A⊂U and U is semi open in X. a generalized closed set (briefly, g-closed)[11] if cl(A) ⊂U whenever A⊂U and U is open in X.
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a regular generalized closed set (briefly, rg-closed)[2] if
Proposition 3.6: Every gr-closed set is rgrα-closed set.
cl(A) ⊂ U
Remark 3.7: Converse of the above proposition need not be true as seen in the following example. Example 3.8:Let X= {a,b,c,d}, τ ={ ϕ,{a},{b},{a,b},{b,d},{a,b,d}, X}. Let A ={a,b}.Then A is rgrα-closed , but not gr-closed. Proposition 3.9: Every rgrα-closed set is rg-closed.
whenever A⊂U and U is regular open in X. a generalized regular set [19](briefly, gr-closed) if rcl(A) ⊂ U whenever
Proof: Let A be a rgrα-closed set in X and let A⊂U, U is regular open. Since A is
A⊂U and U is open in X.
rgrα-closed, rcl(A)⊂U. Since every regular closed set is closed, cl(A)⊂rcl(A).
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a generalized pre regular closed set (briefly, gpr-closed)[9] if pcl(A) ⊂ U
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whenever A⊂U and U is regular open in X. a regular weakly generalized closed set (briefly, rwg-closed)[17] if cl
That is cl(A)⊂rcl(A)⊂U. This implies, cl(A) ⊂U whenever A⊂U & U is regular open. Hence A is rg-closed.
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(int(A)) ⊂ U whenever A⊂U and U is regular open in X. 7.
a regular generalized α- closed set (briefly, rgα-closed)[23] if αcl(A) ⊂ U
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whenever A⊂U and U is regular α-open in X. a weakly generalized regular α-closed set (briefly, wgrα-closed)[10] if
Remark 3.10: Converse of the above proposition need not be true as seen in the following example. Example 3.11:Let X= {a,b,c,d}, τ ={ ϕ,{a},{b},{a,b},{b,d},{a,b,d}, X}. Let A = {c}.Then A is rg-closed, but not rgrα-closed. Proposition 3.12: Every rgrα-closed set is gpr-closed.
cl(int(A)) ⊂ U whenever A⊂U and U is regular α-open in X. Copyright© 2018, IERJ. This open-access article is published under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License which permits Share (copy and redistribute the material in any medium or format) and Adapt (remix, transform, and build upon the material) under the Attribution-NonCommercial terms.
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