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December
2015
PURE AND APPLIED
MATHEMATICS LETTERS
An International Journal
A note on “Regular generalized b-closed Set” *Int.
Journal of Math. Anal. 7(13) (2013) 613 – 624]
by
Shuker Mahmood Al-Salem
HCTM Technical Campus
P A M L December
2015
Edited by Prof. V. K. Kaushik
Pure and Applied Mathematics Letters, Volume 2015, pages 55-57
Contents list available at HCTM TECHNICAL CAMPUS
PURE AND APPLIED MATHEMATICS LETTERS
ISSN NO 2349-4956 (journal home page: www.pamletters.org)
A note on “Regular generalized b-closed Set” *Int. Journal of Math. Anal. 7(13) (2013)
613 – 624]
Shuker Mahmood Al-Salem
Department of Mathematics, College of Science, University of Basra, Iraq
ARTICLE
INFORMATION
Manuscript No: 10010872014
th
Received: 10 August 2014
th
Revised: 26 November 2014
rd
Accepted: 24 December 2014
Communicated by Prof. V. K. Kaushik
ABSTRACT
It is pointed out that the assertion of Mariappa and Sekar [K. Mariappa and S. Sekar, On Regular
generalized b -closed Set, Int. Journal of Math. Anal. 7(13) (2013) 613 – 624], every rgb -closed
set is gb -closed set, every rgb -closed set is gsp -closed set, every rgb -closed set is gb closed set, rgb -closed set and rg -closed set are independent to each other, are incorrect by a
counterexample. Moreover, we show that all of these examples [(2.4), (2.6), (2.8), (2.9. b)] in [6],
are incorrect. Also, in this paper we prove that every gb -closed set is rgb -closed set, every gsp
-closed set is rgb -closed set, and every gb -closed set is rgb -closed set. Finally, we proved
that every rg -closed set is rgb -closed set that means rgb -closed set and rg -closed set are not
independent to each other.
Mathematical Classification:
54A10, 54C8, 54C10, 54D10, 54A05.
Keywords:
rgb - closed set, gαb -closed set, gb -closed set.
1. Introduction
In recent years a number of generalizations of b -closed sets have been introduced. This type of sets was discussed by Ekici and
Caldas [3] under the name of b -closed sets. The class of b -closed sets is contained in the class of semi-preclosed sets and contains
all semi-closed sets and preclosed sets. Later in 2007 M. Ganster and M. Steiner [5] gave a new type of generalized closed set in
topological space called generalized b -closed sets and study some of its fundamental properties. In 2013 K. Mariappa and S. Sekar [6]
introduced a new class of sets called regular generalized b -closed sets in topological spaces and made a theoretical study of the
regular generalized b -closed sets in more detail. In this paper, we give the related concepts and the assertion in [6], then we verify that
the assertion in [6], every rgb -closed set is gb -closed set, every rgb -closed set is gsp -closed set, every rgb -closed set is gb closed set, rgb -closed set and rg -closed set are independent to each other, are incorrect by a counterexample. Moreover, we show
that all of these examples [(2.4), (2.6), (2.8), (2.9. b)] in [6], are incorrect. Also, in this paper we prove that every gb -closed set is rgb
-closed set, every gsp -closed set is rgb -closed set, and every gb -closed set is rgb -closed set. Finally, we proved that every rg closed set is rgb -closed set that means rgb -closed set and rg -closed set are not independent to each other.
2. Preliminaries
Definition 2.1 A subset A of a space X is said to be: (1) a semi-open set [4] if A  cl (int(A)) and a semi-closed set if int(cl ( A))  A ; (2) a
pre-open set [8] if A  int(cl ( A)) and a pre-closed set if cl (int( A))  A ; (3) an  -open set [9] if A  int(cl (int( A))) and an  -closed set if
cl (int(cl ( A)))  A ; (4) a semi-preopen set [2] if A  cl (int(cl ( A))) and a semi-preclosed set if int(cl (int( A)))  A ; (5) a regular open set if
A  int(cl ( A)) and a regular closed set if A  cl (int(A)) ; (6) a regular generalized closed set (briefly rg -closed set) [10] if cl ( A)  U
whenever A  U and U is regular open in X ; (7) a generalized semi-pre closed set (briefly gsp -closed) [7] if spcl ( A)  U whenever
A  U and U is open in X ; (8) a generalized b - closed set (briefly gb - closed) [11] if bcl ( A)  U whenever A  U and U is  -open in
X and (9) a generalized b -closed set (briefly gb -closed) [1] if bcl ( A)  U whenever A  U and U is open in X .
______________________________________
“Corresponding Author: Shuker Mahmood Al-Salem, Email: [email protected]”
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Definition 2.2 [6] A subset A of a topological space ( X , ) , is called regular generalized b - closed set (briefly rgb -closed set) if bcl ( A)  U
whenever A  U and U is regular open in X .
Theorem 2.1 [6] Every rgb -closed set is gb -closed set.
Example 2.1 [6] (The converse of theorem (2.3) is not true) Let X  {a, b, c} with   {X ,  ,{c},{b},{b, c}} . The set {b} is gb -closed set but
not a rgb -closed set.
Theorem 2.2 [6] Every rgb -closed set is gsp -closed set.
Example 2.2 [6] (The converse of theorem (2.5) is not true) Let X  {a, b, c} with   { X ,  ,{a},{b},{a, b}} . The set {a} is gsp -closed set
but not a rgb -closed set.
Theorem 2.3 [6] Every rgb - closed set is gb - closed set.
Example 2.3 [6] (The converse of theorem (2.7) is not true) Let X  {a, b, c} with   { X ,  ,{a},{b},{a, b}} . The set {a} is gb -closed set
but not a rgb -closed set.
Remark 2.1 [6] rgb -closed set and rg -closed set are independent to each other as seen from the following examples.
Example 2.4 (a) [6] Let X  {a, b, c} with   { X ,  ,{a},{b},{a, b}} . The set {b} is rgb -closed set but not a rg -closed set.
Example 2.5 (b) [6] Let X  {a, b, c} with   {X ,  ,{a},{a, c},{a, b}} . The set {a, c} is rg - closed set but not a rgb - closed set.
3. A note on Regular generalized b-closed Set
In a recent paper [6], Mariappa and Sekar posed the following theorems, examples and remark [2.3-2.9, 2.9 (b)] concerning regular generalized
b -closed sets, in this paper we show that all of these are incorrect.
Remark 3.1 see examples [(2.4), (2.6), (2.8) and (2.9.b)] respectively, in example (2.4) we consider only two regular open sets X and {b} which
are containing {b} and such that bcl ({b})  X , bcl ({b})  {b} . Therefore {b} is a regular generalized b -closed set. In a similar way, we
consider {a} is a regular generalized b -closed set in examples (2.6), (2.8) Moreover, in example (2.9.b) we consider only one regular open set
X containing {a, c} such that bcl ({a, c})  X , where {a, c} is open set containing {a, c} but not regular, since int(cl ({a, c}))  X  {a, c} .
Therefore {a, c} is a regular generalized b -close set. ∎
Theorem 3.1 Every gb -closed set is rgb -closed set.
Proof Let A be gb -closed set in X and U be any regular open set containing A . However, every regular open set is open set and A  U ,
then bcl ( A)  U (since A is gb -closed set). Hence A is rgb -closed set. ∎
The converse of above theorem need not be true as seen from the following example.
Example 3.1 Let X  {a, b, c} with   {X ,  ,{b},{a},{a, c},{a, b}} . The set {a} is rgb -closed set but is not a gb -closed set, since {a} is
open set containing {a} but {a} not containing bcl ({a})  {a, c} .∎
Theorem 3.2 Every gsp -closed set is rgb -closed set.
Proof Let A be gsp -closed set in X and U be any regular open set containing A . Since every regular open sets are open sets, and A is gsp closed set. Then spcl (A)  U , thus A  int(cl (int( A)))  U , but int(cl (int( A)))  cl (int( A))  int(cl ( A))  int(cl ( A)) . Hence
spcl ( A)  bcl ( A)  scl ( A) . However, A  U and U is regular open set, then A  int(cl ( A))  scl ( A)  scl (U )  U  int(cl (U ))  U .
Therefore bcl ( A)  U . Hence A is rgb -closed set. ∎
The converse of above theorem need not be true as seen from the following example.
Example 3.2 Let X  {a, b, c} with   {X ,  ,{a},{b},{a, b},{a, c}} . The set {a} is rgb -closed set but not a gsp -closed set, since {a, b} is
open set containing {a} but {a, b} not containing spcl ({a})  {a, c} .∎
Theorem 3.3 Every gb -closed set is rgb -closed set.
Proof Let A be gb -closed set in X and U be any regular open set containing A . Since every regular open set is  -open set. Therefore, U
is  -open set containing A , thus bcl ( A)  U (since A is gb -closed set). Hence A is rgb -closed set. ∎
The converse of above theorem need not be true as seen from the following example.
Example 3.3 Let X  {a, b, c} with   {X ,  ,{a},{b},{a, b},{a, c}} . The set {a, b} is rgb -closed set but not a gb -closed set, since {a, b} is
 -open set containing {a, b} but {a, b} not containing bcl ({a, b})  X .∎
Theorem 3.4 Every rg - closed set is rgb -closed set.
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Proof Let A be rg -closed set in X and U be any regular open set containing A . However, bcl ( A)  cl ( A) and cl ( A)  U since A is rg closed set and A  U , U is regular open. Therefore bcl ( A)  U . Hence A is rgb -closed set. ∎
The converse of above theorem need not be true as seen from the following example.
Example 3.4 Let X  {a, b, c} with   { X ,  ,{a},{b},{a, b}} . The set {b} is rgb -closed set but not a rg -closed set, since {b} is regular
open set containing {b} but {b} not containing cl ({b})  {c, b} . ∎
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