On Standard Concepts Using ii Open Sets

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ISSN Online: 2333-9721 ISSN Print: 2333-9705

DOI: 10.4236/oalib.1105604 Sep. 17, 2019 1 Open Access Library Journal

On Standard Concepts Using ii-Open Sets

Beyda S. Abdullah, Amir A. Mohammed

Department of Mathematics, College of Education for pure Sciences, University of Mosul, Mosul, Iraq

Abstract

Following Caldas in [1] we introduce and study topological properties of

ii-derived, ii-border, ii-frontier, and ii-exterior of a set using the concept of

ii-open sets. Moreover, we prove some further properties of the well-known notions of ii-closure and ii-interior. We also study a new decomposition of

ii-continuous functions. Finally, we introduce and study some of the separa-tion axioms specifically T0ii, T1ii.

Subject Areas

Mathematical Analysis

Keywords

α-Open Set, ii-Open Set, Separation Axioms

1. Introduction

The notion of α-open set was introduced by Njastad in [2]. Caldas in [1] intro-duced and studied topological properties of α-derived, α-border, α-frontier,

α-exterior of a set by using the concept of α-open sets. In this paper, we intro-duce and study the same above concepts by using ii-open sets. A subset A of X is called ii-open set [3] if there exists an open set G in the topology τ _of_X_{, such} that: G≠φ,X _, A CL A G⊆

(



)

and Int A

( )

=G _{, the complement of an}

ii-open set is an ii-closed set. We denote the family of ii-open sets in

(

X,

τ

)

by

ii

τ

. It is shown in [4] that each of

_{τ τ}

_⊂ ii_and

_τ

ii_{is a topology on}_X_{. This}

property allows us to prove similar properties of α-open set. Also, we define

ii-continuous functions and we study the relation between this type of function and continuous, semi-continuous, α-continuous and i-continuous functions. Finally, we introduce a new type of separation axioms namely T0ii, T1ii. We

prove similar properties and characterizations of T0 and T1.

How to cite this paper: Abdullah, B.S. and Mohammed, A.A. (2019) On Standard Concepts Using ii-Open Sets. Open Access Library Journal, 6: e5604.

https://doi.org/10.4236/oalib.1105604

Received: July 15, 2019 Accepted: September 14, 2019 Published: September 17, 2019

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

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DOI: 10.4236/oalib.1105604 2 Open Access Library Journal

2. Preliminaries

Throughout this paper,

(

X,τ

)

_and

(

Y,σ

)

_(simply_X_and_Y_{) always mean} to-pological spaces. For a subset A of a space X, Cl (A) and Int (A) denote the clo-sure of A and the interior of A respectively. We recall the following definitions, which are useful in the sequel.

Definition 2.1. A subset A of a space X is called 1) Semi-open set [5] if A CL Int A⊆

(

( )

)

_. 2) α-open set [2] if A Int CL Int A⊆

(

( )

)

_.

3) i-open set [3] if there exist an open set G in the topology τ _of_X_{, such that} i)

G

≠

φ

,

X

ii) A CL A G⊆

(



)

The complement of an i-open set is an i-closed set.

4) ii-open set [4] if there exist an open set G in the topology τ_of_X_{, such} that

i)

G

≠

φ

,

X

ii) A CL A G⊆

(



)

iii) Int A

( )

=G

The complement of an ii-open set is an ii-closed set.

5) int-open set [4] if there exist an open set G in the topology τ _of_X_and

,

G

≠

φ

X

_{such that}Int A

_{( )}

=G_{. The complement of}_int-_{open set is}_int-_closed set.

6) αo (X), So (X), io (X), iio (X), into (X) are family of α-open, semi-open,

i-open, ii-open, int-open sets respectively.

7)

_τ

i_,

_τ

ii_{denote the family of all}_i_{-open sets and}_ii_{-open sets respectively.}

Definition 2.2. [3] A topological space X is called

1) T0i if a, b are to distinct points in X, there exist an i-open set U such that

either a U∈ and b U∉ , or b U∈ and a U∉ .

2) T1i if

a b X

,

∈

and a b≠ , there exist i-open sets U, V containing a, b

respectively, such that b U∉ and a V∉ .

Definition 2.3. A function f X:

(

,

τ

) (

→ Y,

σ

)

_{is called}

1) Continuous [6], if _f−1

( )

_G _{is open in}

(

_X_,

_τ

)

_{for every open set}_G_of

(

Y,

σ

)

_.

2) α-continuous [6], if _f−1

( )

_G _is_α-_{open in}

(

_X_,

_τ

)

(

Y,σ

)

_.

3) Semi-Continuous [5], if _f−1

( )

_G _{is semi-open in}

(

_X_,

_τ

)

_{for every open} set G of

(

Y,

σ

)

_.

4) i-Continuous [3], if _f−1

( )

_G _is_i-_{open in}

(

_X_,

_τ

)

(

Y,

σ

)

_.

3. Applications of

ii

-Open Sets

Definition 3.1. Let A be a subset of a topological space

(

X,

τ

)

. A derived set of A denoted by D(A) is defined as follows:

( )

{

:

(

) { }

\ ,

}

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DOI: 10.4236/oalib.1105604 3 Open Access Library Journal point of A if it satisfies the following assertion:

(

_∀_G_∈_τii

)

(

_{x G}_∈ _⇒

(

_{G A}_

) { }

\ _x _≠_φ

)

_{. The set of all}_ii_{-limit points of}_A_{is called} the ii-derived set of A and is denoted by D Aii

( )

. Note that x X∈ is not

ii-limit point of A if and only if there exist an ii-open set G in X such that (x G∈ and

(

G A

) { }

\ x =

φ

).

Theorem 3.2. For subsets A, B of a space X, the following statements hold: 1) D Aii

( )

⊂D A

( )

2) If A B⊆ , then Dii

( )

A ⊆Dii

( )

B

3) Dii

( )

A Dii

( )

B ⊂Dii

(

AB

)

and Dii

(

A B

)

⊂Dii

( )

A Dii

( )

B

4) Dii

(

Dii

( )

A

)

\A⊂Dii

( )

A

5) Dii

(

ADii

( )

A

)

⊂ADii

( )

A

Proof. 1) Since every open set is ii-open [4], it follows that D Aii

( )

⊂D A

( )

.

2) Let x∈Dii

( )

A . Then G is ii-open set containing x such that

(

A G

) { }

\ x ≠

φ

(3.1) Since A B⊆ _{we get}

(

A G

) (

⊆ B G

)

, it implies that

(

A G

) { } (

\ X ⊆ B G

) { }

\ X ≠

φ

, from (3.1) we get

(

B G

) { }

\ X ≠

φ

. Hence, x∈Dii

( )

B . Therefore Dii

( )

A ⊆Dii

( )

B .

3) Since

A A B

⊆

_

_and

B A B

⊆

_

_{, from (2) we get} Dii

( )

A ⊆Dii

(

A B

)

,

( )

(

)

ii ii

D B ⊆D A B .

This implies to Dii

( )

A Dii

( )

B ⊂Dii

(

AB

)

.

We shall prove that Dii

(

A B

)

⊂Dii

( )

A Dii

( )

B . Since

A B A



⊆

,

A B B

_

⊆

_{, from (2) we get} Dii

(

A B

)

⊂Dii

( )

A and Dii

(

A B

)

⊂Dii

( )

B .

ThereforeDii

(

A B

)

⊂Dii

( )

A Dii

( )

B .

4) If x∈D Dii

(

ii

( )

A

)

\A and G is an ii-open set containing x, then

( ) { }

(

ii \

)

G D A X ≠φ . Let y∈G

(

Dii

( ) { }

A \ x

)

. Then, since y∈Dii

( )

A

and

y G

∈

_, G_

(

A Y\

_{{ }}

)

≠φ_{. Let}z G∈ _

(

A Y\

{ }

)

_{. Then,}z x≠ _for z A∈ and x A∉ _{. Hence,} G

(

A X\

{ }

)

≠φ. Therefore, x∈Dii

( )

A .

5) Let x∈Dii

(

ADii

( )

A

)

. If x A∈ , the result is obvious. So, let,

( )

(

)

\

ii ii

x∈D AD A A then for ii-open set G containing x,

( ) { }

(

)

(

G AD_ii A \ x

)

≠φ. Thus, G_

(

A X\

{ }

)

≠φ_or

( ) { }

(

ii \

)

G D A x ≠φ.

Now, it follows similarly from (4) that G_

(

A X\

{ }

)

≠φ_{. Hence,} x∈D_ii

( )

A . Therefore, in any case, Dii

(

ADii

( )

A

)

⊂ADii

( )

A .

In general, the converse of (1) may not true and the equality does not hold in (3) of theorem 3.2.

Example 3.3. Let X =

{

a b c, ,

}

and τ=

{

φ, ,X b

{ }

}

_{. Thus,}

( )

{

, ,

{ } { } { }

, , , ,

}

iio x = φ X b a b b c _{. Take the following:}

1) A=

{ }

c . Then, D A

( ) { }

= a b, and D Aii

( )

=

φ

. Hence, D A

( )

⊄Dii

( )

A ;

2) C=

{ }

a b, _and E=

{ }

c _{. Then} D cii

( ) { }

= a c, and D Eii

( )

=

φ

. Hence

(

)

( )

ii C E ii C ii

D  ≠D D E .

Theorem 3.4. For any subset A of a space X, CLii

( )

A =ADii

( )

A .

Proof. Since Dii

( )

A ⊂CLii

( )

A , ADii

( )

A ⊂CL Aii

( )

. On the other hand,

(4)

G containing x intersects A at a point distinct from x; so x∈Dii

( )

A . Thus,

( )

ii A A ii

CL ⊂ D A , which completes the proof.

Definition 3.5. A point x X∈ is said to be ii-interior point of A if there ex-ist an ii-open set G containing x such that G⊂A. The set of all ii-interior points of A is said to be ii-interior of A and denoted by Int Aii

( )

.

Theorem 3.6. For subset A, B of a space X, the following statements are true: 1) Int Aii

( )

is the union of all ii-open subset of A

2) A is ii-open if and only if A=Intii

( )

A

3) Int Intii

(

ii

( )

A

)

=Int Aii

( )

4) I tnii

( )

A A \= D X Aii

(

\

)

5) X \Intii

( )

A =CL Xii

(

\A

)

6) X \CLii

( )

A =Intii

(

X A\

)

7) If A B⊆ _{, then} Int_ii

( )

A ⊆Int_ii

( )

B 8) Intii

( )

A Intii

( )

B ⊆Intii

(

AB

)

9) Intii

( )

A Intii

( )

B ⊇Intii

(

AB

)

Proof. 1) Let

{

G iiii\ ∈ ∧

}

be a collection of all ii-open subsets of A. If

( )

ii

x∈Int A _{, then there exist}

j

∈ ∧

_{such that} x∈G_j⊆A . Hence

ii ii G

x∈

_

_∈∧ , and so Intii

( )

A ⊆



ii∈∧Gii. On the other hand, if y∈



ii∈∧Gii,

then y∈Gk ⊆A for somek∈ ∧. Thus y∈Intii

( )

A , and



ii∈∧Gii ⊆Int Aii

( )

.

Accordingly,



ii∈∧Gii ⊆Int Aii

( )

.

2) Straightforward.

3) It follows from (1) and (2).

4) If x∈A D\ ii

(

X A\

)

, then x∉D Xii

(

\A

)

and so there exist an ii-open set

G containing x such that G

(

X A\

)

=

φ

. Thus, x G∈ ⊂A and hence

( )

ii

x∈Int A . This shows that A\Dii

(

X A\

)

⊂Int Aii

( )

. Now let x∈Intii

( )

A .

Since

( )

ii

ii A

Int ∈τ and Intii

( ) (

A  X \A

)

=

φ

. We havex∉D Xii

(

\A

)

.

There-fore, Intii

( )

A =A\Dii

(

X \A

)

.

5) Using (4) and Theorem (3.4), we have

( )

(

)

(

)

(

)

(

)

\ ii \ \ ii \ \ ii \ ii \

X Int A =X A D X A = X A D X A =CL X A . 6) Using (4) and Theorem (3.4), we get.

(

\

) (

\

)

\

( )

\

(

( )

)

\

( )

ii X A X A ii A X A ii A ii

Int = D = _D = X CL A

7) Since A B⊆ _and Int A_ii

( )

⊆A_, Int B_ii

( )

⊆B_{, we get}

( )

ii ii

Int A ⊆Int B .

8) Since A⊆

(

A B

)

and B⊆

(

A B

)

, from (7) we get

( )

(

)

ii ii

Int A ⊆Int A B , Intii

( )

B ⊆Intii

(

A B

)

. Therefore

( )

(

)

ii ii ii

Int A Int B ⊆Int AB .

9) Since

A B A

_

⊆

_and

A B B

_

⊆

_{, from (7) we get}

(

)

( )

ii A B ii

Int  ⊆Int A , Intii

(

A B

)

⊆Int Bii

( )

. Therefore

(

)

( ) ( )

ii A B ii A B

Int  ⊆Int  .

Definition 3.7. bii

( )

A =A nt\I ii

( )

A is said to be the ii-border of A.

Theorem 3.8. For a subset A of a space X, the following statements hold: 1) b Aii

( )

⊂b A

( )

where b A

( )

denotes the border of A

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DOI: 10.4236/oalib.1105604 5 Open Access Library Journal 3) Intii

( )

A bii

( )

A =

φ

4) b Aii

( )

=

φ

if and only if A is ii-open set

5) b Int Aii

(

ii

( )

)

=φ

6) Int b Aii

(

ii

( )

)

=φ

7) b bii

(

ii

( )

A

)

=bii

( )

A

8) bii

( )

A =ACLii

(

X \A

)

9) bii

( )

A = ADii

(

X \A

)

Proof.

1) Since Int

( )

A ⊂Intii

( )

A , we have

( )

\

( )

\

( ) ( )

ii A A ii A A In

b = Int ⊆ t A =b A _. 2) and (3). Straightforward.

4) Since Int Aii

( )

⊆A, it follows from Theorem 3.6 (2). That A is ii-open ⇔

( )

ii

A=Int A ⇔ bii

( )

A =A nt\I ii

( )

A =

φ

.

5) Since Int Aii

( )

is ii-open, it follows from (4) that b Int Aii

(

ii

( )

)

=φ.

6) If x I∈ ntii

(

bii

( )

A

)

, then x∈bii

( )

A . On the other hand, sinceb Aii

( )

⊂A,

( )

(

)

( )

ii ii ii

x∈Int b A ⊂Int A _{. Hence,} x∈Int_ii

( )

A b_ii

( )

A . Which contradicts (3). Thus Int b Aii

(

ii

( )

)

=φ.

7) Using (6), we get b bii

(

ii

( )

A

)

=bii

( )

A \Int bii

(

ii

( )

A

)

=bii

( )

A .

8) Using Theorem 3.6 (6), we have

( )

\

( )

\

(

\

(

\

)

(

\

)

ii ii ii ii

b A = A Int A =A X CL X A =A_CL X A 9) Applying (8) and the Theorem (3.4), we have

( )

(

\

)

(

\

)

(

\

)

(

\

)

ii A A ii X A A X A ii ii

b = CL =  D X A =AD X A .

Example 3.9. Consider the topological space

(

X,

τ

)

_{given in Example (3.3).}

If A=

{ }

a b, , then b Aii

( )

=

φ

and b A

( ) { }

= a . Hence, b A

( )

⊄bii

( )

A , that is,

in general, the converse Theorem 3.9 (1) may not be true.

Definition 3.10. Frii

( )

A =CLii

( )

A \Intii

( )

A is said to be the ii-frontier of A.

Theorem 3.11. For a subset A of a space X, the following statements hold: 1) Frii

( )

A ⊂Fr A

( )

where Fr A

( )

denotes the frontier of A

2) CLii

( )

A =Intii

( )

A Fr Aii

( )

3) Intii

( )

A Frii

( )

A =

φ

4) bii

( )

A ⊂Frii

( )

A

5) Frii

( )

A =bii

( )

A Dii

( )

A

6) Frii

( )

A =D Aii

( )

if and only if A is ii-open set

7) Frii

( )

A =CLii

( )

A CLii

(

X A\

)

8) Frii

( )

A =Frii

(

X A\

)

9) Fr Aii

( )

is ii-closed

10) Fr Fii

(

rii

( )

A

)

⊂Frii

( )

A

11) Frii

(

Intii

( )

A

)

⊂Frii

( )

A

12) Fr Cii

(

Lii

( )

A

)

⊂Frii

( )

A

13) Intii

( )

A =A\Frii

( )

A

Proof.

1) Since CLii

( )

A ⊆CL A

( )

and Int

( )

A ⊆Intii

( )

A , it follows that

( )

\

( )

\

( )

\

( )

ii A ii A ii A CL A ii A

(6)

( )

A Frii

( )

A =Intii

( )

A 

(

CLii

( )

A \Intii

( )

A

)

=φ.

4) Since A⊆CLii

( )

A , we have

( )

\

( )

\

( )

ii A ii ii ii ii

b =A Int A ⊆CL A Int A =Fr A .

5) Since Intii

( )

A Frii

( )

A =Intii

( )

A bii

( )

A D Aii

( )

,

( )

ii A ii A ii

Fr =b D A .

6) Assume that A is ii-open. Then

( )

\

( )

(

( )

\

)

( )

\

(

\

)

ii ii ii ii ii ii ii

Fr A =b A D A Int A =φ D A A =D A A=b X A , by using (5), Theorem 3.6 (2), Theorem 3.8 (4) and Theorem 3.8 (9).

Conversely, suppose that Frii

( )

A =bii

(

X A\

)

. Then

( )

\

(

\

)

(

( )

\

( )

)

\

(

\

)

\

(

\

)

\

( )

ii ii ii ii ii ii

Fr A b X A CL A Int A X A Int X A A Int A

φ= = = _.

by using (4) and (5) of Theorem 3.6, and so A⊆Intii

( )

A . Since Int Aii

( )

⊆A

in general, it follows that Int Aii

( )

= A so from Theorem 3.6 (2) that A is

ii-open set.

7) Frii

( )

A =CLii

( )

A \Intii

( )

A =CLii

( )

A 

(

CLii

(

X A\

)

.

8) It follows from (7).

9)

(

( )

)

(

( )

)

(

)

( )

(

)

(

)

( )

\

\ .

ii ii ii ii ii

CL Fr CL CL CL C

A A X A

A X A

L CL CL CL Fr A

=

⊂ =



 Hence, Fr Aii

( )

is

ii-closed.

10)

(

( )

)

( )

(

)

(

\

( )

)

(

( )

)

( )

ii ii

ii ii ii ii ii ii ii

A

A X A

Fr Fr

CL Fr CL Fr CL Fr A Fr A

= _ ⊂ = .

11) Using Theorem 3.6 (3), we get

( )

(

)

(

( )

)

\

(

( )

)

( )

\

( )

ii ii A ii ii A ii ii A ii ii ii

Fr Int =CL Int Int Int ⊆CL A Int A =Fr A _.

12)

(

_{( )}

( )

)

_{( )}

(

( )

_{( )}

)

\

(

( )

)

( )

\

(

( )

)

\ .

ii ii ii ii ii ii ii ii ii

ii ii ii

Fr CL CL CL Int CL CL Int CL CL In

A A A A A

A t A Fr A

= =

13) A\Frii

( )

A =

(

A\CLii

( )

A

)

\I tnii

( )

A =Intii

( )

A .

The converses of (1) and (4) of Theorem 3.11 are not true in general, as shown by Example

Example 3.12. Consider the topological space

(

X,

τ

)

given in Example 3.3. If A=

{ }

c , then Fr A

( ) { } { }

= a c, ⊄ c =Frii

( )

A , and if B=

{ }

a b, , then

( ) { }

( )

ii B c ii

Fr = ⊄b B .

Definition 3.13. Extii

( )

A =Intii

(

X A\

)

is said to be an ii-exterior of A.

Theorem 3.14. For a subset A of a space X, the following statements hold: 1) Ext

( )

A ⊂Extii

( )

A where Ext A

( )

denotes the exterior of A

2) Ext Aii

( )

isii-open

3) Extii

( )

A =Intii

(

X A\

)

=X\CL Aii

( )

4) Ext Extii

(

ii

( )

A

)

=Int CL Aii

(

ii

( )

)

5) If A B⊆ , then Extii

( )

A ⊃Extii

( )

B

6) Extii

(

A B

)

⊂Extii

( )

A Extii

( )

B

7) Extii

(

A B

)

⊃Extii

( )

A Extii

( )

B

8) Ext Xii

( )

=

φ

9) Extii

( )

φ

=X

(7)

( )

A ⊂Ext Extii

(

ii

( )

A

)

12) X =Extii

( )

A Extii

( )

A Frii

( )

A

Proof. 1) It follows from Theorem 3.6 (1). 2) It is straightforward by Theorem 3.6 (6).

3)

(

( )

)

(

( )

)

( )

(

)

(

( )

)

\ \ \

ii ii ii ii

Ext Ext Ext CL

Int CL I

A X A

X X A nt CL A =

= = .

4) Assume that A B⊂ . Then

( )

(

\

)

(

\

)

( )

ii B ii X B ii X ii

Ext =Ext ⊆Ext A =Ext A , by using Theorem 3.6 (7).

5) Applying Theorem 3.6 (8), we get

(

)

(

)

(

) (

)

(

)

(

)

( )

\ \ \

\ \ .

ii ii ii

ii ii ii ii

A B X A B X A X

Ext Int Int

Int Int Ext Ext

B

X A X B A B

= =

⊆ =

  

 

6) Applying Theorem 3.6 (9), we obtain

(

)

(

)

(

) (

)

(

)

(

)

( )

\ \ \

\ \ .

ii ii ii

ii ii ii ii

A B X A B X A X

Ext Int Int

Int Int Ext Ext

B

X A X B A B

= = ⊃ =      . 7) Straightforward. 8) Straightforward.

9)

(

( )

)

(

)

(

)

(

)

(

)

(

)

( )

\ \ \ \ \ \

\ \ .

ii ii ii ii ii ii

ii ii ii ii

Ext X Ext A Ext X Int X A Int X X I X A X A

nt Int Int Int X A Ext A

= =

= = =

10)

( )

(

( )

)

(

)

( )

(

)

(

( )

)

\ \

\ .

ii ii ii ii ii

ii ii ii ii

A A X X

Int Int CL Int Int Int Ext Ext Ext

A

X A A

⊂ =

= =

{

a b, , ,c d

}

and τ=

{

φ,X c d,

{ }

,

}

_{. Thus,}

( )

{

, ,

{ } {

, , , , , , ,

} {

}

iio x = φ X c d b c d a c d _{. If} A=

{ }

a _and B=

{ }

b _{. Then}

( )

ii A

Ext ⊄Ext A . Extii

(

A B

)

≠Extii

( )

A Extii

( )

B and

(

)

( )

ii ii ii

Ext A B ≠Ext A Ext B .

4. A New Decomposition of

ii

-Continuity

We begin by the following definition:

(

,

τ

) (

→ Y,

σ

)

_{is called}_ii-_{continuous if}

( )

1

f− G _is_ii-open_{set in}

(

_X_,

_τ

)

_{for any open set}_G_of

(

_Y_,

_σ

)

_. Theorem 4.2. Let f X:

(

,

τ

) (

→ Y,

σ

)

_{be a function then:}

1) Every continuous function is an ii-continuous, 2) Every ii-continuous function is an i-continuous, 3) Every α-continuous function is an ii-continuous.

Proof. 1) Let G be open set in

(

Y,

σ

)

. Since f is continuous, it follows that

( )

1

f− G _{is open set in}

(

_X_,

_τ

)

_{. But every open set is}_ii-open_set_[4]_{. Hence}

( )

1

(

_X_,

_τ

)

_._T_hus_f_is_ii-_continuous.

2) Let G be open set in

(

Y,

σ

)

. Since f is an ii-continuous, it follows that

( )

1

f− G _{is an}_ii-open_{set in}

(

_X_,

_τ

)

_{. But every}_ii-open_{set is}_i_{-open set}_[4]_. Hence _f−1

( )

_G _is_i_{-open set in}

(

_X_,

_τ

)

_._T_hus_f_is_i-_continuous.

(8)

( )

1

f− G _is_α-_{open set in}

(

_X_,

_τ

)

_{. But every}_α-_{open set is}_ii-open_set_[4]_{. Hence}

( )

1

(

_X_,

_τ

)

_{. Thus}_f_{is an}_ii-_continuous. The converse need not be true by the following example. Example 4.3. Let

{

a b, , ,

}

X = c d _,τ=

{

φ,X,

{ } { } { }

a , b , ,ab

}

and

{

a b, , ,

}

Y = c d , σ =

{

φ,Y,

{ } { } { } { } {

a b a b a, , , , , , , ,d a b d

}

and

( )

{

, ,

{ } { } { } { } {

, , , , , , , , , , ,

} {

}

iio x = φ X a b a b a d a b c a db _,

( )

{

{ } { } { } { } { } { } { }

{

} {

}

, , , , , , , , , , , , , , , , , , , , ,

,

, , , ,

io x a b a b a c a d b c b d a b c a b d a c d b

X

c d φ

=

( )

{

, ,

{ } { } { } {

, , , , , ,

} {

, , ,

}

o x φ X a b a b a b c a b d

α = _.

Let f X:

(

,

τ

) (

→ Y,

σ

)

be the identity function then _f−1

( )

{ }

_a ₌

{ }

_a _,

{ }

( )

{ }

1

f− b ₌ b _, _f−1

( )

{ }

_c ₌

{ }

_c _, _f−1

( )

{ }

_d ₌

{ }

_d _{. Then}_f_is_ii-_{continuous, but}_f

is not α-continuous, since for the open set

{ }

a d, in

(

Y,

σ

)

,

{ }

(

)

{ }

1 _{a d}_, _,

f− ₌ a d _{is not}_α_{-open in}

(

_X_,

_τ

)

_{and f}_{is not continuous, since for} the open set

{ }

a d, in

(

Y,

σ

)

, _f−1

(

{ }

_{a d}_,

)

₌

{ }

_{a d}_, _{is not open in}

(

_X_,

_τ

)

_.

Now when f X:

(

,

τ

) (

→ Y,

σ

)

be defined by _f−1

( )

{ }

_a ₌

{ }

_b _, _f−1

( )

{ }

_b ₌

{ }

_a _,

{ }

( )

{ }

1

f− c ₌ c _, _f−1

( )

{ }

_d ₌

{ }

_d _{we get}_f_is _i_{-continuous, but}_f_{is not}

ii-continuous, since for the open set

{ }

a d, in

(

Y,

σ

)

, _f−1

(

{ }

_{a d}_,

)

₌

{ }

_{b d}_, _is

not ii-open in

(

X,

τ

)

.

Theorem 4.4. Let f X:

(

,

τ

) (

→ Y,

σ

)

_{be a function then every}

semi-continuous function is an ii-continuous.

Proof. Let G be open set in

(

Y,

σ

)

_{. Since}_f_{is semi-continuous, it follows that}

( )

1

f− G _{is semi-open set in}

(

_X_,

_τ

)

_{. But every semi-open set is}_ii-open_set_[4]_. Hence _f−1

( )

_G _is_ii-open_{set in}

(

_X_,

_τ

)

_._T_hus_f_{is an}_ii-_continuous.

(

,

τ

) (

→ Y,

σ

)

_{is called}_int-_{continuous if}

( )

1

f− G _is_int-_{open set in}

(

_X_,

_τ

)

_{for any open set}_G_in

(

_Y_,

_σ

)

_. Theorem 4.6. Let f X:

(

,

τ

) (

→ Y,

σ

)

_{be a function then:}

1) Every continuous function is int-continuous, 2) Every ii-continuous function is int-continuous, 3) Every α-continuous function is int-continuous.

Proof. 1) Let G be open set in

(

Y,

σ

)

. Since f is continuous, it follows that

( )

1

f− G _{is open set in}

(

_X_,

_τ

)

_{. But every open set is}_int-_{open set}_[4]_{. Hence}

( )

1

f− G _is_int-_{open set in}

(

_X_,

_τ

)

_{. Thus}_f_is_int-_continuous.

(

Y,

σ

)

. Since f is ii-continuous, it follows that

( )

1

f− G _{is an}_ii-open_{set in}

(

_X_,

_τ

)

_{. But every}_ii-open_{set is}_int-_{open set}_[4]_. Hence _f−1

( )

_G _is_int-_{open set in}

(

_X_,

_τ

)

_{. Thus}_f_is_int-_continuous.

(

Y,

σ

)

. Since f is α-continuous, it follows that

( )

1

f− G _is_α-_{open set in}

(

_X_,

_τ

)

_{. But every}_α-_{open set is}_int-_{open set}_[4]_{. Hence}

( )

1

(9)

DOI: 10.4236/oalib.1105604 9 Open Access Library Journal Example 4.7. Let X =

{

a b c, ,

}

_, τ=

{

φ, ,X

{ } { }

a b c, ,

}

_and Y =

{

a b c, ,

}

_,

{ } { }

{

, ,Y a a c, ,

}

σ = φ _and into x

( )

=

{

φ,X,

{ } { } { } { }

a a b b, , , ,c a c, ,

}

_,

( )

{

, ,

{ } { }

, ,

}

iio x =αo x = φ X a bc _. _Let f X:

(

,

τ

) (

→ Y,

σ

)

_{be the identity}

function then _f−1

( )

{ }

_a ₌

{ }

_a _, _f−1

( )

{ }

_b ₌

{ }

_b _, _f−1

( )

{ }

_c ₌

{ }

_c _. _Then _f_is

int-continuous, but f is not ii-continuous, since for the open set

{ }

a c, in

(

Y,

σ

)

, _f−1

(

{ }

_{a c}_,

)

₌

{ }

_{a c}_, _{is not}_ii-open_in

(

_X_,

_τ

)

_and_f_{is not continuous,}

since for the open set

{ }

a c, in

(

Y,

σ

)

, _f−1

(

{ }

_{a c}_,

)

₌

{ }

_{a c}_, _{is not open in}

(

X,

τ

)

and f is not α-continuous, since for the open set

{ }

a c, in

(

Y,

σ

)

,

{ }

(

)

{ }

1 _{a c}_, _,

f− ₌ a c _{is not}_α-_open.

Definition 4.8. A subset A of X is called weakly ii-open set if A is ii-open set and A CL Int A⊆

(

( )

_A

)

_.

Theorem 4.9. A subset A of a space X is α-open set if and only if A is weakly

ii-open.

Proof. Let A be α-open set. Since A Int CL Int A⊆

(

( )

)

_and A CL A⊆

( )

. Therefore A CL Int A⊆

(

( )

)

_CL A

( )

_{, this implies that}A CL Int A⊆

(

( )

_A

)

_. Now, put G Int A=

( )

where

G

≠

φ

,

X

, then A is ii-open set. Therefore, A is weakly ii-open set.

Conversely, Let A be weakly ii-open set, then there exist an open set

,

G

≠

φ

X

, such that G Int A=

( )

satisfying A CL Int A⊆

(

( )

_A

)

_and _A_is

ii-open set. Since A CL Int A⊆

(

( )

_A

)

_{, this implies that} A CL Int A⊆

(

( )

)

_and

( )

(

( )

)

Int A ⊆Int CL Int A . Since A is ii-open set, using (2) from Theorem (3.6), we get A Int A=

( )

. Therefore A Int CL Int A⊆

(

( )

)

. Thus A is α -open set.

As a summary the following Figure 1 shows the relations among

[image:9.595.210.538.469.691.2]

semi-continuous, ii-continuous, i-continuous, int-continuous, α-continuous and continuous.

Figure 1. Relations among semi-continuous, ii-continuous, i-continuous, int-continuous,

(10)

DOI: 10.4236/oalib.1105604 10 Open Access Library Journal Corollary 4.10. A function f X:

(

,

τ

) (

→ Y,

σ

)

_is_{α-continuous}_{if and only if}

it is weakly ii-continuous.

Proof. Clear from Theorem 4.9.

5. ii

-Separating Axioms

In this section we define T0ii and T1ii spaces for ii-open sets and we determine

them by giving many examples. Specially, we define T1, T1α and T1i spaces to

compare them with T1ii space.

Definition 5.1. A topological space X is called

1) T0ii if a, b are to distinct points in X, there exists ii-open set U such that

either a U∈ _and b U∉ , and b U∈ _and a U∉ .

2) T1ii if

a b X

,

∈

and a b≠ , there exist ii-open sets U, V containing a, b

respectively, such that b U∉ and a V∉ .

{ }

a b, _,_τii _{= =}_τ

{

_φ, ,_X

{ } { }

_a , _b

}

(

_X_,

_τ

)

_and

(

_X_,_τii

)

are topological spaces.

1)

a b X

,

∈

(a b≠ ) there exists

{ }

_a _∈_τii _{such that} _a_∈

{ }

_a _, _b_∉

{ }

_a _.

Therefore

(

X,

τ

)

is T0ii.

2)

a b X

,

∈

₍a b≠ ) there exists

{ } { }

_{a b}, _∈_τii_{such that}_a_∈

{ }

_a _, _{b b}_∈

{ }

_.

Therefore

(

X,

τ

)

is T1ii.

Theorem 5.3.

1) Every T0-space is T0ii-space,

4) Every T1ii-space is T0ii-space.

Proof. (1), (2), (3) and (4) follow using the fact that every open set is ii-open [4]. The converse needs not to be true by the following example.

Example 5.4. Let

{

, ,

}

X = a b c , τ=

{

φ, ,X a

{ }

}

_and_τii ₌

{

_φ,_X,

{ } { } { }

_{a a}, ,_b , ,_a_c

}

_.

(

X,

τ

)

and

(

_X,_τii

)

_{are topological spaces.}

(

X,

τ

)

_{is not}T₀-space because,

b c X

,

∈

₍b c≠ _{) there is no open set}_G such that b G∈ , c G∉ .

(

X,

τ

)

is T0ii-space because,

a b X

,

∈

{ }

a ∈τii such

that a∈

{ }

a , b∉

{ }

a .

,

a c X∈ (a c≠ _{) there exists}

{ }

_a _∈_τii_{such that} _a_∈

{ }

_a _, _c_∉

{ }

_a _.

,

b c X

∈

₍b c≠ _{) there exists}

{ }

_{a b}, _∈_τii_{such that} _b_∈

{ }

_{a b}_, _, _c_∉

{ }

_{a b}_, _.

(

X,

τ

)

is not T1-space because,

a b X

,

∈

(a b≠ ) there exists X∈τ such that a X∈ , b X∈ .

(

X,

τ

)

is not T1ii-space because,

b a X

,

∈

{ }

a b, ∈τii

such that a∈

{ }

a b, _, b∈

{ }

a b, _.

Theorem 5.5. Every T1α-space is T1ii-space.

(11)

DOI: 10.4236/oalib.1105604 11 Open Access Library Journal b V∈ _{. Since every}_α_{-open set is}_ii_{-open set}_[4]_,_U_,_V_{is an}_ii_{-open set in}_X_. Hence X is T1ii-space.

Theorem 5.6. Every T1ii-space is T1i-space.

Proof. Let X be a T1ii-space. Let a, b be two distinct points in X. Since X is

1ii

T -space there exist two ii-open sets U, V in X such that a U∈ , b U∉ , a V∉ , b V∈ . Since every ii-open set is i-open set [4], U, V is an i-open set in X. Hence X is T1i-space.

The converse needed not to be true by the following example. Example 5.7. Let X =

{

a b c, ,

}

, τ=

{

φ, ,X

{ } { }

a b c, ,

}

_and.

{ } { } { } { }

{

, , , , , ,

}

i _X _{a b} _c _b_c

τ = φ _. _τii ₌

{

_φ,_{X a b c},

{ } { }

, ,

}

_.

(

X,

τ

)

and

(

_X,_τii

)

_{are topological spaces.}

(

X,

τ

)

_is T₁_i-space because,

a b X

,

∈

₍a b≠ _{) there exists}

{ } { }

_{a b}, _∈_τi

such that a∈

{ }

a , b∉

{ }

a and b b∈

{ }

, a∉

{ }

b .

,

a c X∈ (a c≠ _{) there exists}

_{{ } { }}

_a , _c _∈_τi_{such that} _a_∈

{ }

_a _, _c_∉

{ }

_a _and

{ }

c∈ c , a∉

{ }

c .

,

b c X

∈

₍b c≠ ) there exists

{ } { }

_{c b}, _∈_τi_{such that}_c_∈

{ }

_c _, _{b c}_∉

{ }

_and

{ }

b b∈ , c b∉

{ }

.

(

X,

τ

)

is not T1ii -space because,

b c X

,

∈

( c b≠ ) there exists

{ } { }

_{b c}, _∈_τii_{such that} _c_∈

{ }

_{b c}_, _, _b_∈

{ }

_{b c}_, _.

Theorem 5.8. A space X is T0ii if and only if CLii

( )

{ }

x ≠CLii

( )

{ }

y for

every pair of distinct points x, y of X.

Proof. Let X be a T0ii-space. Let

x y X

,

∈

such that x y≠ , then there exists

an ii-open set U containing one of the points but not the other, then x U∈ and

y U

∉

. Then X U\ is ii-closed set containing y but not x. But CLii

( )

{ }

y is

the smallest ii-closed set containing y. Therefore CL_ii

( )

{ }

y ⊂X U\ _{and hence}

{ }

( )

ii

x∉CL y _{. Thus} CL_ii

( )

{ }

x ≠CL_ii

( )

{ }

y _.

Conversely, Suppose for any

x y X

,

∈

_with x y≠ _, CL_ii

( )

{ }

x ≠CL_ii

( )

{ }

y _. Let z X∈ such that z∈CLii

( )

{ }

x but z∉CLii

( )

{ }

y . If x∈CLii

( )

{ }

y then

{ }

( )

{ }

ii ii

CL x ⊂CL y _{and hence} z∈CL_ii

( )

{ }

y _{. This is contradiction.} There-fore x∉CLii

( )

{ }

y . That is x∈X\CLii

( )

{ }

y . Therefore X CL\ ii

( )

{ }

y is ii-open set containing x but not y. Hence X is an T0ii-space.

Theorem 5.9. A space

(

X,

τ

)

is T1ii-space if and only if the singletons are

ii-closed sets.

Proof. Let X be T1ii-space and let x X∈ , to prove that

{ }

x is ii-closed set.

We will prove X x\

{ }

is ii-open set in X. Let y∈X\

{ }

x , implies x y≠ _and since X is T1ii-space then their exist two ii-open sets U, V such that x U∉ ,

{ }

\

y V∈ ⊂ X x _{. Since} y V∈ ⊂X x\

{ }

_{, then} X x\

{ }

is ii-open set. Hence

{ }

x is ii-closed set.

Conversely, Let

x y X

≠ ∈

then

{ } { }

x , y are ii-closed sets. That is X x\

{ }

is ii-open set clearly, x X x∉ \

{ }

and y X x∈ \

{ }

. Similarly X y\

{ }

is ii-open set, y X y∉ \

{ }

and x X y∈ \

{ }

. Hence X is an T1ii-space.

As a consequence the following Figure 2 shows the relations among T0, T0ii,

1

(12)

[image:12.595.283.475.70.215.2]

DOI: 10.4236/oalib.1105604 12 Open Access Library Journal Figure 2. Relations among T0, T0ii, T1, T1ii and T1α.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this pa-per.

References

[1] Caldas, M. (2003) A Note on Some Applications of α-Open Sets. International Journal of Mathematics and Mathematical Sciences, 2, 125-130.

https://doi.org/10.1155/S0161171203203161

[2] Njastad, O. (1965) On Some Classes of Nearly Open Sets. Pacific Journal of Mathe-matics, 15, 961-970.https://doi.org/10.2140/pjm.1965.15.961

[3] Askander, S.W. (2016) Oni-Separation Axioms. International Journal of Scientific and Engineering Research, 7, 367-373. https://www.ijser.org

[4] Mohammed, A.A. and Abdullah, B.S. (2019) II-Open Sets in Topological Spaces.

International Mathematical Forum, 14, 41-48.

https://doi.org/10.12988/imf.2019.913

[5] Levine, N. (1963) Semi-Open Sets and Semi-Continuity in Topological Spaces. The American Mathematical Monthly, 70, 36-41.https://doi.org/10.2307/2312781 [6] Mashhour, A.S., Hasanein, I.A. and Ei-Deeb, S.N. (1983) α-Continuous and α-Open