SCJMSU JOURNAL

(1)

1 นิสิตปริญญาโท, ²ผู้ช่วยศาสตราจารย์, สาขาคณิตศาสตร์ คณะวิทยาศาสตร์ศาสตร์ มหาวิทยาลัยมหาสารคาม อาเภอกันทรวิชัย จังหวัดมหาสารคาม 44150

1 Master's Degree student, ² Assist. Prof., Department of Mathematics, Faculty of Science, Mahasarakham University, Kantharawichai District Maha Sarakham 44150, Thailand,

* Corresponding author ; Department of Mathematics, Faculty of Science, Mahasarakham University, Kantharawichai District Maha Sarakham 44150, Thailand, [email protected], [email protected]

สมบัติบางประการของปริภูมิใหญ่สุดย่อยแบบ sg ในปริภูมิโครงสร้างเล็กสุด Some properties of sg-submaximal in minimal structure space

ณัฐวรรณ พิมพิลา

¹

, ดรุณี บุญชารี

²

Nuthawan Pimpila

¹

, Daruni Boonchari

²

Received: 21 June 2018 ; Revised : 21 August 2018 ; Accepted: 24 August 2018 บทคัดย่อ

ในบทความนี้เราศึกษาคุณสมบัติของ เซตเปิด เซตปิด ตัวด�าเนินการส่วนปิดคลุม ตัวด�าเนินการภายใน บนปริภูมิโครงสร้างเล็ก สุด เราได้ให้ลักษณะของปริภูมิใหญ่สุดย่อยแบบ โดยใช้เซตปิดและเซตเปิดชนิดต่างๆ

ค�าส�าคัญ : หนาแน่นแบบ , ไม่หนาแน่นแบบ , ปริภูมิใหญ่สุดย่อยแบบ

Abstract

In this paper, we study the properties of open sets, closed sets, closure operator and interior operator on minimal structure space. We will provide characterization of -submaximal space by using various kinds of generalized closed sets and open sets.

Keywords : -dense , -codense, -Submaximal

Introduction

In the literature

¹

, Maki introduced the notion of minimal structure. Also Popa and Noiri

²

, introduced the notion of -open sets, -closed sets and then characterized those sets using -closure and -interior operators, respectively. After that Popa and Noiri

²

and Cao et al.

³

, defined some new types of open sets and closed sets in topological space and obtained some results in topological space. Late

⁴

, Rosas introduced some new types of open set and closed set in minimal structure. The concept of relationships of generalized closed sets and some new character-izations of sg-submaximal were introduced by Gansteer

⁵

. In this paper, we study the minimal structure space and properties of open set, closed set, closure and interior in this space, including the relationship between every type of closed set. We provide the characterization of -submaximal space.

Preliminaries

First, we recall some concepts and definitions which are useful in the results.

Definition 2.1

¹

Let be a non-empty set and the power set of A subfamily of is called

a (briefly ) on if

contains and The pair is called an

Each member of is said to be -open set and the complement of -open set is said to be -closed set.

Definition 2.2

²

Let be a non-empty set and be an on For a subset of the -closure of denoted by and the -interior of denoted by are defined as follows:

(1)

2. Preliminaries

First, we recall some concepts and definitions which are useful in the results.

Definition 2.1¹ Let ^X be a non-empty set and

 

P X the power set of ^X^. A subfamily mX of

 

P X is called a minimal structure (briefly

m structure ) on ^X if mX contains ^ and .

X The pair



X m, _X



is called an ^{m space}^ ^. Each member of mX is said to be mX-open set and the complement of mX-open set is said to be mX-closed set.

Definition 2.2² Let ^X be a non-empty set and

mX be an m structure on ^X^. For a subset

A of ^X^, the mX-closure of ^A denoted by

 

mCl A and the mX-interior of ^A denoted by

 

,

mInt A are defined as follows:

(1) mCl A

  

 B X X B m :   _X and

A B



,

(2) mInt A

  

 B X B m :  _Xand B A



.

Lemma 2.3² Let ^X be a non-empty set and

mX be an m structure on ^X^. For ^{A B X}^, ^ the following statements hold:

(1) mInt A

 

A. If A m X, then

mInt A A

 

 .

(2) A mCl A

 

. If X A m  X, then

mCl A A

 

 .

(3) If ^{A B}^ ^, then mInt A

 

mInt B

 

and

mCl A

 

mCl B

 

.

(4) mInt A B







mInt A mInt B

 



 

and

mCl A mCl B

 



 

mCl A B







.

(5) mInt mInt A

   

mInt A

 

and

mCl mCl A

   

mCl A

 

.

(6) ^X^mCl A

 

mInt X A







and

XmInt A

 

mCl X A







.

(7) mCl

 

  , mCl X

 

X,

mInt

 

  and mInt X

 

X.

Definition 2.4^6,7 Let



X m, X



be an ^{m space}^ and^{A X}^ ^. Then ^A is called:

(1) mX-semi open set if A mCl mInt A

   

^,

(2) mX-pre open set if A mInt mCl A

   

^,

(3) mX-b open set if A mInt mCl A

   

 ^{mCl mInt A}

   

^,

(4) mX-^open set if

A mInt mCl mInt A

     

^,

(5) mX-regular open set ifA mInt mCl A

   

^.

The complement of an mX-semi open (resp. mX

-preopen, mX-b open, mX-^open, mX-regular open) set is called an mX-semi closed (resp.

mX-pre closed, mX-b closed, mX-^closed,

mX-regular closed) set. The collection of all mX- semi open (resp. mX-preopen, mX-b open, mX

-^open, mX-regular open) sets of ^Xis denoted by m SO XX

  

resp.m PO XX

 

,m BO XX

 

,

X

 

m O X , m RO XX

  

^.



X m, X



be an ^{m space}^ and ^{A X}^ ^. Then ^A is called:

(1) sCl A

  

 B X B : is a mX-semi closed set and A B



,

(2) pCl A

  

 B X B : is a mX-pre closed set and A B



,

(3) bCl A

  

 B X B : is a mX-b closed set

and A B



.



X m, _X



be an ^{m space}^ and ^{A X}^ ^. Then^A is called:

(1) mX gbclosed if bCl A U

 

 whenever ^{A U}^ and U m X^,

(2) mX sgclosed if sCl A U

 

 whenever ^{A U}^ and U m SO X _X

 

,

(3) mX gsclosed if sCl A U

 

(4) mX gpclosed if pCl A U

 

 whenever ^{A U}^ and U m X^.

The complement of an mX gbclosed (resp.

mX sgclosed, mXgsclosed, mX gpclosed) set is called an mX gbopen (resp.

and (2)

2. Preliminaries

 

X The pair



X m, X



 

,

(1) mCl A

  

 B X X B m :   _X and

A B



,

(2) mInt A

  

 B X B m :  _Xand B A



.

(1) mInt A

 

mInt A A

 

 .

(2) A mCl A

 

mCl A A

 

 .

 

mInt B

 

and

mCl A

 

mCl B

 

.

(4) mInt A B







mInt A mInt B

 



 

and

mCl A mCl B

 



 

mCl A B







.

(5) mInt mInt A

   

mInt A

 

and

mCl mCl A

   

mCl A

 

.

(6) ^X^mCl A

 

mInt X A







and

XmInt A

 

 mCl X A







.

(7) mCl

 

  , mCl X

 

X,

mInt

 

X.



X m, X



   

^,

   

^,

   

 ^{mCl mInt A}

   

^,

     

^,

   

^.

  

resp.m PO XX

 

,m BO XX

 

,

X

 

  

^.



X m, _X



(1) sCl A

  



,

(2) pCl A

  



,

(3) bCl A

  

and A B



.



X m, _X



(1) mXgbclosed if bCl A U

 

,

 

(4) mXgpclosed if pCl A U

 

and

Lemma 2.3

²

Let be a non-empty set and

be an on . For

2. Preliminaries

Definition 2.1¹Let ^X be a non-empty set and

 

X The pair



X m, X



 

,

(1) mCl A

  

 B X X B m :   X and

A B



,

(2) mInt A

  

 B X B m :  Xand B A



.

(1) mInt A

 

mInt A A

 

 .

(2) A mCl A

 

mCl A A

 

 .

 

mInt B

 

and

mCl A

 

mCl B

 

.

(4) mInt A B







mInt A mInt B

 



 

and

mCl A mCl B

 



 

mCl A B







.

(5) mInt mInt A

   

mInt A

 

and

mCl mCl A

   

mCl A

 

.

(6) ^X^mCl A

 

mInt X A







and

XmInt A

 

 mCl X A







.

(7) mCl

 

  , mCl X

 

X,

mInt

 

X.



X m, X



   

^,

   

^,

   

 ^{mCl mInt A}

   

^,

     

^,

   

^.

mX-regular closed) set. The collection of all mX- semi open (resp. mX-preopen, mX-b open,mX

  

resp.m PO XX

 

,m BO XX

 

,

X

 

  

^.



X m, _X



(1) sCl A

  



,

(2) pCl A

  



,

(3) bCl A

  

and A B



.



X m, _X



 

(2) mXsgclosed if sCl A U

 

 whenever ^{A U}^ and U m SO X X

 

,

(3) mXgsclosed if sCl A U

 

The complement of an mXgbclosed (resp.

mX sgclosed, mX gsclosed, mX gpclosed) set is called an mX gbopen (resp.

the following

(2)

statements hold:

(1) If , then

(2) If , then

(3) (4) (5) (6) (7)

Definition 2.4

^6,7

Let be an and Then is called:

(1) -semi open set if (2) -pre open set if (3) -b open set if

(4) - open set if (5) -regular open set if

The complement of an -semi open (resp. -preopen, -b open, - open, -regular open) set is called an -semi closed (resp. -pre closed, -b closed, - closed, -regular closed) set. The collection of all -semi open (resp. -preopen, -b open, - open,

-regular open) sets of is denoted by

Definition 2.5

^6,7

Let be an and Then is called:

(1) 2. Preliminaries

 

m structure ) on ^X if mX contains ^ and X. The pair



X m, X



 

,

(1) mCl A

  

 B X X B m :   X and

A B



,

(2) mInt A

  

 B X B m :  X and B A



.

(1) mInt A

 

mInt A A

 

 .

(2) A mCl A

 

mCl A A

 

 .

 

mInt B

 

and

mCl A

 

mCl B

 

.

(4) mInt A B







mInt A mInt B

 



 

and

mCl A mCl B

 



 

mCl A B







.

(5) mInt mInt A

   

mInt A

 

and

^{mCl mCl A}

   

^^{mCl A}

 

^.

(6) ^X^mCl A

 

mInt X A







and

XmInt A

 

 mCl X A







.

(7) mCl

 

  , mCl X

 

X,

mInt

 

X.



X m, _X



   

^,

   

^,

   

 ^{mCl mInt A}

   

^,

     

^,

   

^.

  

resp.m PO XX

 

,m BO XX

 

,

X

 

  

^.



X m, _X



(1) sCl A

  



,

(2) pCl A

  



,

(3) bCl A

  

and A B



.



X m, X



 

,

 

is a -semi closed set and

2. Preliminaries

 

X The pair



X m, X



 

,

(1) mCl A

  

 B X X B m :   _X and

A B



,

(2) mInt A

  

 B X B m :  _Xand B A



.

(1) mInt A

 

mInt A A

 

 .

(2) A mCl A

 

mCl A A

 

 .

 

mInt B

 

and

mCl A

 

mCl B

 

.

(4) mInt A B







mInt A mInt B

 



 

and

mCl A mCl B

 



 

mCl A B







.

(5) mInt mInt A

   

mInt A

 

and

mCl mCl A

   

mCl A

 

.

(6) ^X^^{mCl A}

 

^^{mInt X A}



^



_and

XmInt A

 

 mCl X A







.

(7) mCl

 

  , mCl X

 

X,

mInt

 

X.



X m, X



   

^,

   

^,

   

 ^{mCl mInt A}

   

^,

     

^,

   

^.

mX-regular closed) set. The collection of all mX- semi open (resp. mX-preopen, mX-b open,mX

  

resp.m PO XX

 

,m BO XX

 

,

X

 

  

^.



X m, _X



(1) sCl A

  



,

(2) pCl A

  



,

(3) bCl A

  

and A B



.



X m, _X



 

(2) mXsgclosed if sCl A U

 

,

(3) mXgsclosed if sCl A U

 

The complement of an mXgbclosed (resp.

(2) 2. Preliminaries

 

X The pair



X m, X



 

,

(1) mCl A

  

 B X X B m :   X and

A B



,

(2) mInt A

  

 B X B m :  Xand B A



.

(1) mInt A

 

mInt A A

 

 .

(2) A mCl A

 

mCl A A

 

 .

 

mInt B

 

and

mCl A

 

mCl B

 

.

(4) mInt A B







mInt A mInt B

 



 

and

mCl A mCl B

 



 

mCl A B







.

(5) mInt mInt A

   

mInt A

 

and

mCl mCl A

   

mCl A

 

.

(6) ^X^mCl A

 

mInt X A







and

XmInt A

 

mCl X A







.

(7) mCl

 

  , mCl X

 

X,

mInt

 

X.



X m, X



   

^,

   

^,

   

 ^{mCl mInt A}

   

^,

     

^,

   

^.

  

resp.m PO XX

 

,m BO XX

 

,

X

 

  

^.



X m, _X



(1) sCl A

  



,

(2) pCl A

  



,

(3) bCl A

  

and A B



.



X m, _X



 

,

 

is a -pre closed set and

2. Preliminaries

 

P X the power set of