CUBO A Mathematical Journal
Vol.15, No¯ 02, (65–69). June 2013

A Note on Modifications of rg-Closed Sets in Topological
Spaces

Takashi Noiri

2949-1 Shiokita-Cho, Hinagu,

Yatsushiro-Shi, Kumamoto-Ken,

869-5142 Japan.

t.noiri@nifty.com

Valeriu Popa

Department of Mathematics,

University of Bacǎu,,

600 115 Bacǎu,, Romania,

vpopa@ub.ro

ABSTRACT

We point out that a certain modification of regular generalized closed sets due to

Palaniappan and Rao [15] means nothing to the family of semi-open sets.

RESUMEN

Destacamos que una modificación de conjuntos cerrados regulares generalizados debido

a Palaniappan and Rao [15] no tiene importancia para la familia de conjuntos semi-

abiertos.

Keywords and Phrases: g-closed, rg-closed.

2010 AMS Mathematics Subject Classification: 54A05.



66 Takashi Noiri and Valeriu Popa CUBO
15, 2 (2013)

1 Introduction

In 1970, Levine [11] introduced the notion of generalized closed (briefly g-closed) sets in topological

spaces and showed that compactness, locally compactness, countably compactness, paracompact-

ness, and normality etc are all g-closed hereditary. And also he introduced a separation axiom

called T1/2 between T1 and T0. Since then, many modifications of g-closed sets are introduced and

investigated. Among them, Dontchev and Ganster [5] introduced the notion of T3/4-spaces which

are situated between T1 and T1/2 and showed that the digital line or the Khalimsky line [9] (Z, κ)

lies between T1 and T3/4.

As a modification of g-closed sets, regular generalized closed sets are introduced and investi-

gated by Palaniappan and Rao [15]. As the further modification of g-closed sets, Gnanambal [7]

introduced the notion of generalized preregular closed sets. The purpose of this note is to present

some remarks concerning modifications of regular generalized closed sets.

2 Preliminaries

Let (X, τ) be a topological space and A a subset of X. The closure of A and the interior of A are

denoted by Cl(A) and Int(A), respectively. We recall some generalized open sets in topological

spaces.

Definition 2.1. Let (X, τ) be a topological space. A subset A of X is said to be

(1) α-open [14] if A ⊂ Int(Cl(Int(A))),

(2) semi-open [10] if A ⊂ Cl(Int(A)),

(3) preopen [12] if A ⊂ Int(Cl(A)),

(4) semi-preopen [2] or β-open [1] if A ⊂ Cl(Int(Cl(A))),

(5) b-open [3] if A ⊂ Int(Cl(A)) ∪ Cl(Int(A)),

(6) regular open if A = Int(Cl(A)).

The family of all α-open (resp. semi-open, preopen, semi-preopen, b-open, regular open) sets

in (X, τ) is denoted by τα (resp. SO(X), PO(X), SPO(X), BO(X), RO(X, τ)).

For generalizations of open sets defined above, the following relations are well known:

DIAGRAM I

open ⇒ α-open ⇒ preopen

⇓ ⇓

semi-open ⇒ b-open ⇒ semi-preopen

Definition 2.2. Let (X, τ) be a topological space. A subset A of X is said to be α-closed [13] (resp.

semi-closed [4], preclosed [12], semi-preclosed [2], b-closed [3]) if the complement of A is α-open

(resp. semi-open, preopen, semi-preopen, b-open).



CUBO
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A Note on Modifications of rg-Closed Sets in Topological Spaces 67

Definition 2.3. Let (X, τ) be a topological space and A a subset of X. The intersection of all

α-closed (resp. semi-closed, preclosed, semi-preclosed, b-closed) sets of X containing A is called

the α-closure [13] (resp. semi-closure [4], preclosure [6], semi-preclosure [2], b-closure [3]) of A and

is denoted by αCl(A) (resp. sCl(A), pCl(A), spCl(A), bCl(A)).

Definition 2.4. Let (X, τ) be a topological space. A subset A of X is said to be

(1) generalized closed (briefly g-closed) [11] if Cl(A) ⊂ U whenever A ⊂ U and U ∈ τ,

(2) regular generalized closed (briefly rg-closed) [15] if Cl(A) ⊂ U whenever A ⊂ U and

U ∈ RO(X, τ),

(3) generalized preregular closed (briefly gpr-closed) [7] if pCl(A) ⊂ U whenever A ⊂ U and

U ∈ RO(X, τ).

For generalizations of closed sets defined above, the following relations are well known:

DIAGRAM II

closed ⇒ g-closed ⇒ rg-closed ⇒ gpr-closed

3 Modifications of rg-closed sets

First we shall define a modification of rg-closed sets.

Definition 3.1. Let (X, τ) be a topological space. A subset A of X is said to be regular generalized

α-closed (briefly rgα-closed) if αCl(A) ⊂ U whenever A ⊂ U and U ∈ RO(X, τ).

Lemma 3.2. If A is a subset of (X, τ), then τα-Int(τα-Cl(A)) = Int(Cl(A)).

Proof. This is shown in Corollary 2.4 of [8].

Lemma 3.3. Let V be a subset of a topological space (X, τ). Then V ∈ RO(X, τ) if and only if

V ∈ RO(X, τα).

Proof. This is an immediate consequence of Lemma 3.2.

Theorem 3.4. A subset A of a topological space (X, τ) is rgα-closed in (X, τ) if and only if A is

rg-closed in the topological space (X, τα).

Proof. Necessity. Suppose that A is rgα-closed in (X, τ). Let A ⊂ V and V ∈ RO(X, τα).

By Lemma 3.3, V ∈ RO(X, τ) and we have τα-Cl(A) = αCl(A) ⊂ V. Therefore, A is rg-closed in

(X, τα).

Sufficiency. Suppose that A is rg-closed in (X, τα). Let A ⊂ V and V ∈ RO(X, τ). By Lemma

3.3, V ∈ RO(X, τα) and hence αCl(A) = τα-Cl(A) ⊂ V. Therefore, A is rgα-closed in (X, τ).



68 Takashi Noiri and Valeriu Popa CUBO
15, 2 (2013)

Remark 3.5. It turns out that, by Therem 3.4, we can not obtain the essential notion even if we

replace Cl(A) in Definition 2.4(2) with αCl(A).

Next, we try to replace Cl(A) in Definition 2.4(2) with sCl(A).

Lemma 3.6. Let (X, τ) be a topological space. If A ⊂ V and V ∈ RO(X, τ), then sCl(A) ⊂ V.

Proof. Let A ⊂ V and V ∈ RO(X, τ). Then we have sCl(A) ⊂ sCl(V) = V ∪ Int(Cl(V)) = V

and hence sCl(A) ⊂ V.

Remark 3.7. (1) Lemma 3.6 shows that in case SO(X, τ) the condition ”sCl(A) ⊂ V whenever

A ⊂ V and V ∈ RO(X, τ)” does not define a subset like regualr generalized semi-closed sets.

(2) By Diagram I, SO(X) ⊂ BO(X) ⊂ SPO(X) and hence spCl(A) ⊂ bCl(A) ⊂ sCl(A) for any

subset A of X. Therefore, we can not obtain any notions even if we replace Cl(A) in Definitiion

2.4(2) with sCl(A), bCl(A) or spCl(A).

Received: October 2010. Accepted: September 2012.

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A Note on Modifications of rg-Closed Sets in Topological Spaces 69

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