International Mathematical Forum, Vol. 14, 2019, no. 1, 41 - 48 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2019.913

ii – Open Sets in Topological Spaces

Amir A. Mohammed and Beyda S. Abdullah

Department of Mathematics College of Education University of Mosul, Mosul, Iraq

Copyright © 2019 Amir A. Mohammed and Beyda S. Abdullah. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we introduce a new class of open sets in a topological space called 푖푖 − open sets. We study some properties and several characterizations of this class, also we explain the relation of 푖푖 − open sets with many other classes of open sets. Furthermore, we define 푖푤 − closed sets and 푖푖푤 − closed sets and we give some fundamental properties and relations between these classes and other classes such as 푤 − closed and 훼푤 − closed sets.

Keywords: 훼 − open set, 푤 − closed set, 푖 − open set

1. Introduction

Throughout this paper we introduce and study the concept of 푖푖 − open sets in topological space (푋, 휏). The 푖푖 − open set is defined as follows: A subset 퐴 of a topological space (푋, 휏) is said to be 푖푖 − open if there exist an open set 퐺 in the topology 휏 of X, such that i. 퐺 ≠ ∅,푋 ii. A is contained in the closure of (A∩ 퐺) iii. interior points of A equal G. One of the classes of open sets that produce a topological space is 훼 − open. This class of open sets has been introduced in 1965 [3]. In this paper we prove that the family of 푖푖 − open sets are also produce a topological space (Theorem 2.10). The 푖푖 − open set is a generalization of many classes of open sets we mention some of them namely semi − open sets, 훼 − open sets and 푤 − open sets defined by Levine [2], Njȧstad [3] and Sundaram and John [5], respectively. Further, we study the relation of 푖푖 − open sets with the following sets: 푖 − open set, 푤 − closed set and 훼푤 − closed set introduced by Mohammed and Askander [1], Sundaram and

42 Amir A. Mohammed and Beyda S. Abdullah

John [5] and Parimala et.al [4], respectively. Depending on 푖 − open and 푖푖 − open sets we define 푖푤 − closed and 푖푖푤 − closed sets. We present our work in two sections. In the first one, we define 푖푖 − open sets and we give many related examples, and we investigate the relationship with other classes of open sets. We prove some theorems to discuss their properties. In the second section, we discuss the relationship between iiw-closed sets with closed, 푤 − closed, 푖푤 − closed and 훼푤 − closed sets. Further, we present some fundamental properties of 푖푖푤 − closed sets with classes mentioned above.

2. ii-Open Sets In Topological Space

In this section, we introduce and study the notion of 푖푖 − open sets in a topological space and we obtain some of its basic properties. We recall the following definitions, which are useful in the sequel.

Definition 2.1 A subset 퐴 of a space (푋, 휏) is called 1- Semi − open set [2] if 퐴 ⊆ 푐푙(푖푛푡(퐴)). 2- 훼 − open set [3] if 퐴 ⊆ 푖푛푡(푐푙(푖푛푡(퐴))). 3- 푖 − open set [1] if there exist an open set 퐺 ∈ 휏(푋) such that i. 퐺 ≠ ∅, 푋. ii. 퐴 ⊆ 푐푙(퐴 ∩ 퐺). The complement of an 푖 − open set is an 푖 − closed set. 4- 푤 − closed set [5] if 푐푙(퐴) ⊆ 푈 whenever 퐴 ⊆ 푈 and 푈 is semi − open in (푋, 휏). 5- 훼푤 - closed set [4] if 푤푐푙(퐴) ⊆ 푈 whenever 퐴 ⊆ 푈 and 푈 is 훼 − open in (푋, 휏). The complement of an 훼푤 − closed set is an 훼푤 − open set. 6- 표(푋), 푠표(푋), 훼표(푋), 푖표(푋), 푤푐(푋), 훼푤푐(푋) are family of open, semi − open, 훼 − open, 푖 − open, 푤 − closed, 훼푤 − closed sets respectively.

Definition 2.2 A subset 퐴 of a space (푋, 휏) is called 푖푛푡 − open set if there exist an open set 퐺 ∈ 표(푋) and 퐺 ≠ ∅, 푋, such that 푖푛푡(퐴) = 퐺. The complement of the 푖푛푡 − open set is called 푖푛푡 − closed. We denote the family of all 푖푛푡 − open sets of topological space by 푖푛푡표(푋) and the int-closed sets is denoted by 푖푛푡푐(푋).

Example 2.3 Let 푋 = {푎, 푏, 푐} and 휏 = {∅, 푋, {푏}, {푏, 푐}}. Here the closed sets in (푋, 휏) are 퐶(푋, 휏) = {∅, 푋, {푎, 푐}, {푎}} 푖푛푡표(푋) = {∅, 푋, {푏}, {푎, 푏}, {푏, 푐}} 푖푛푡푐(푋) = {∅, 푋, {푎, 푐}, {푐}, {푎}} .

ii – open sets in topological spaces 43

Definition 2.4 A subset 퐴 of a space (푋, 휏) is called 푖푖 − open set if there exist an open set 퐺 ∈ 표(푋), such that i. 퐺 ≠ ∅, 푋 ii. 퐴 ⊆ 푐푙(퐴 ∩ 퐺) iii. 푖푛푡(퐴) = 퐺. The complement of the 푖푖 − open set is called 푖푖 − closed set. We denote the family of all 푖푖 − open sets of topological space by 푖푖표(푋).This definition means that 퐴 is 푖푛푡 − open and 푖 − open set.

Example 2.5 Let 푋 = {푎, 푏, 푐} and 휏 = {∅, 푋, {푏}, {푏, 푐}}. Here 퐶(푋, 휏) = {∅, 푋, {푎, 푐}, {푎}} 푖푛푡표(푋) = {∅, 푋, {푏}, {푎, 푏}, {푏, 푐}} 푖표(푋) = {∅, 푋, {푏}, {푐}, {푎, 푐}, {푎, 푏}, {푏, 푐}} 푖푖표(푋) = {∅, 푋, {푏}, {푎, 푏}, {푏, 푐}}

Remark 2.6 Note that for a topological space (푋, 휏) every 푖푖 − open set is 푖 − open and 푖푛푡 − open. Further, every semi − open is 푖푖 − open set. But the converse is not true as shown in the following example.

Example 2.7 Let 푋 = {푎, 푏, 푐} and 휏 = {∅, 푋, {푎}, {푏, 푐}}. Here 퐶(푋, 휏) = {∅, 푋, {푏, 푐}, {푎}} 푖표(푋) = {∅, 푋, {푎}, {푏}, {푐}, {푏, 푐}} 푖푛푡표(푋) = {∅, 푋, {푎}, {푏, 푐}, {푎, 푏}, {푎, 푐}} 푖푖표(푋) = 푆표(푋) = {∅, 푋, {푎}, {푏, 푐}} Let 퐴 = {푏}. Then 퐴 is not 푖푖 − open, but 퐴 is an 푖 − open set. Let 퐵 = {푎, 푏}. Then 퐵 is not 푖푖 − open. However 퐵 is 푖푛푡 − open set.

Theorem 2.8 Every open set is 푖푖 − 표푝푒푛 set. Proof. Let 퐺 be open set in (푋, 휏). Since 퐺 ⊆ 푐푙(퐺 ∩ 퐺) = 푐푙(퐺), it follows that 퐺 is 푖 − open. Also, 퐺 is 푖푛푡 − open because 푖푛푡(퐺) = 퐺. Thus 퐺 is 푖푖 − open set.■

The converse of the above theorem is not true in general as shown in the following example.

Example 2.9 Let 푋 = {푎, 푏, 푐} and 휏 = {∅, 푋, {푏}, {푏, 푐}}. Here 퐶(푋, 휏) = {∅, 푋, {푎, 푐}, {푎}} 푖푛푡표(푋) = {∅, 푋, {푏}, {푎, 푏}, {푏, 푐}} 44 Amir A. Mohammed and Beyda S. Abdullah

푖표(푋) = {∅, 푋, {푏}, {푐}, {푎, 푐}, {푎, 푏}, {푏, 푐}} 푖푖표(푋) = {∅, 푋, {푏}, {푎, 푏}, {푏, 푐}} Now, A= {푎, 푏}, is 푖푖 − open but 퐴 is not open.

Theorem 2.10 If (푋, 휏) is a topological space then (푋, 푖푖표(푥)) is also topological space. Proof. Let 퐴 be 푖푖 − open set and let 푥 ∈ 퐴, we will prove the existence of open set, say 퐺 ∈ 표(푋) such that 푥 ∈ 퐺 ⊆ 퐴. Since 퐴 is 푖푖 − open, it follows that there exist an open set 퐺 ∈ 표(푋) such that i. 퐺 ≠ ∅, 푋 ii. 퐴 ⊆ 푐푙(퐴 ∩ 퐺) iii. 푖푛푡(퐴) = 퐺. Therefore, 푥 ∈ 퐴 implies that 푥 ∈ 푐푙(퐴) and 푥 ∈ 푐푙(퐺). If 퐺 is closed, then 푥 ∈ 퐺 ⊆ 퐴 and 퐴 is open. If 퐺 is not closed and 푥 ∉ 퐺 then 푥 ∉ 푐푙(퐺) because 퐺 is not closed. Since 푥 ∈ 퐴 implies that 푥 ∈ 푐푙(퐴) ∩ 푐푙(퐺) implies that 푥 ∈ 푐푙(퐺). This implies a contradiction. Therefore 퐴 is open.■

Example 2.11 Let 푋 = {푎, 푏, 푐} and 휏 = {∅, 푋, {푏}, {푏, 푐}}. Here 퐶(푋, 휏) = {∅, 푋, {푎, 푐}, {푎}} 푖푖표(푋) = {∅, 푋, {푏}, {푎, 푏}, {푏, 푐}} Note that (푋, 휏) is a topological space and (푋, 푖푖표(푥)) is a topological space.

Theorem 2.12 Every 훼 − 표푝푒푛 set is 푖푖 − 표푝푒푛. Proof. Let (푋, 휏) be a topological space and 퐴 ⊆ 푋 be 훼 − open set. Since 퐴 ⊆ 푖푛푡(푐푙(푖푛푡(퐴))) ⊆ 푐푙(푖푛푡(퐴)). Therefore 퐴 is semi − open. Since, there exist an open set, say, 퐺 ≠ ∅, 푋 satisfying int(A)  G, it follows that 푖푛푡(퐴) ⊆ 퐺 ∩ 퐴. Therefore 퐴 ⊆ 푐푙(퐴 ∩ 퐺). Thus, 퐴 is 푖 − open. We shall prove that 푖푛푡(퐴) = 퐺. Note that if 푖푛푡(퐴) ≠ 퐺, for all 퐺 ∈ 표(푋), then 푐푙(푖푛푡(퐴)) ≠ 푐푙(G). From above inclusions we conclude that 퐴 ⊆ 푐푙(푖푛푡(퐴) ∩ 퐴 ∩ 퐺). This implies that 퐴 ⊄ 푐푙(퐺). That is a contradiction. Therefore, 퐴 is 푖푖 − open set.■ The converse of the above theorem is not true in general as shown in the following example.

Example 2.13 Let 푋 = {푎, 푏, 푐, 푑} and 휏 = {∅, 푋, {푎}, {푏, 푐, 푑}}. 퐶(푋, 휏) = {∅, 푋, {푎}, {푏, 푐, 푑}} 훼표(푋) = {∅, 푋, {푎}, {푏, 푐, 푑}} 푖푖표(푋) = {∅, 푋, {푎}, {푏}, {푐}, {푑}, {푏, 푐}, {푏, 푑}, {푐, 푑}, {푏, 푐, 푑}} Now, 퐴 = {푐} is 푖푖 − open but not 훼 − open.

ii – open sets in topological spaces 45

Corollary 2.14 Every 훼 − 표푝푒푛 set is 푖푛푡 − 표푝푒푛. Proof. Clear.■ The converse of the above corollary is not true in general as shown in the following example.

Example 2.15 Let 푋 = {푎, 푏, 푐} and 휏 = {∅, 푋, {푎}, {푏, 푐}}. Here 퐶(푋, 휏) = {∅, 푋, {푏, 푐}, {푎}} 푖푛푡표(푋) = {∅, 푋, {푎}, {푏, 푐}, {푎, 푏}, {푎, 푐}} 훼표(푋) = {∅, 푋, {푎}, {푏, 푐}} Let 퐴 = {푎, 푏}. Then 퐴 is not 훼 − open. However 퐴 is 푖푛푡 − open set.

Remark 2.16 The following diagram shows that the relationships of ii-open sets with other sets.

Semi-open

ii-open set 휶-open set i-open set

int-open

open set

3. 풊풘 − 퐂퐥퐨퐬퐞퐝 And 풊풊풘 − 퐂퐥퐨퐬퐞퐝 Sets In Topological Space

We introduce and study the notion of 푖푤 − closed and 푖푖푤 − closed sets in topological space and obtain some of its basic properties. The proof of main results in this section is similar to that in [4].

Definition 3.1 A subset 퐴 of (푋, 휏) is called a 푖푤 − closed set if 푤푐푙(퐴) ⊆ 푈 whenever 퐴 ⊆ 푈 and 푈 is 푖 − open in (푋, 휏). The complement of an 푖푤 − closed set is 푖푤 − open set.

46 Amir A. Mohammed and Beyda S. Abdullah

Theorem 3.2 Every 푖푤 − 푐푙표푠푒푑 set is 푤 − 푐푙표푠푒푑 set. Proof. Let 퐴 be an 푖푤 − closed set in (푋, 휏) and 푈 is be any semi − open set in 푋 such that 퐴 ⊆ 푈. Since every semi − open set is 푖 − open [1]. Since 퐴 is closed, 푤푐푙(퐴) ⊆ 푐푙(퐴) ⊆ 푈. This shows that 퐴 is 푤 − closed set in (푋, 휏).■ The converse of the above theorem is not true in general as shown in the following example.

Example 3.3 Let 푋 = {푎, 푏, 푐, } and 휏 = {∅, 푋, {푎}, {푏, 푐}}. 퐶(푋, 휏) = {∅, 푋, {푎}, {푏, 푐, }} 푤푐(푋, 휏) = {∅, 푋, {푎}, {푏}, {푐}, {푎, 푏}, {푎, 푐}, {푏, 푐}} let 퐴 = {푎, 푏}.Then , 퐴 is a 푤 − closed set but it is not 푖푤 − closed set.

Theorem 3.4 Every 푖푤 − 푐푙표푠푒푑 set is 훼푤 − 푐푙표푠푒푑 set. Proof. Let 퐴 be an 푖푤 − closed set in (푋, 휏) and 푈 is be any 훼 − open set in 푋 such that 퐴 ⊆ 푈. Since every 훼 − open set is 푖 − open [1]. Since 퐴 is closed, 푤푐푙(퐴) ⊆ 푐푙(퐴) ⊆ 푈. This shows that 퐴 is 훼푤 − closed set in (푋, 휏).■ The converse of the above theorem is not true in general as shown in the following example.

Example 3.5 Let 푋 = {푎, 푏, 푐, } and 휏 = {∅, 푋, {푏}, {푏, 푐}}. 퐶(푋, 휏) = {∅, 푋, {푎}, {푎, 푐, }} 훼푤푐(푋, 휏) = {∅, 푋, {푎}, {푎, 푏}, {푎, 푐}} let 퐴 = {푎, 푏}. Then 퐴 is 훼푤 − closed but it is not 푖푤 − closed set.

Theorem 3.6 If 퐴 is 푖 − 표푝푒푛 and 푖푤 − 푐푙표푠푒푑 set then 퐴 is w-closed. Proof. Let 푈 be semi-open such that 퐴 ⊆ 푈. We shall prove that 푐푙(퐴) ⊆ 푈. Since 푈 is semi − open, this implies that 푈 is 푖 − open. By assumption we have 푐푙(퐴) ⊆ 푈. That is 퐴 is 푤 − closed.■

Definition 3.7 A subset 퐴 of (푋, 휏) is called 푖푖푤 − closed set if 푤푐푙(퐴) ⊆ 푈 whenever 퐴 ⊆ 푈 and 푈 is 푖푖 − open in (푋, 휏). The complement of an 푖푖푤 − closed set is an 푖푖푤 − open set. We denote the family of all 푖푖푤 − closed sets of topological space by 푖푖푤푐(푋).

Example 3.8 Let 푋 = {푎, 푏, 푐, } and 휏 = {∅, 푋, {푏}, {푏, 푐}}. 퐶(푋, 휏) = {∅, 푋, {푎}, {푎, 푐, }} , 푖푖푤푐(푋, 휏) = {∅, 푋, {푎}, {푎, 푐}} ii – open sets in topological spaces 47

Theorem 3.9 Every 푖푖푤 − 푐푙표푠푒푑 set is 훼푤 − 푐푙표푠푒푑 set. Proof. Let 퐴 be an 푖푖푤 − closed set in (푋, 휏) and 푈 is be any 훼 − open set in 푋 such that 퐴 ⊆ 푈. Since every 훼 − open set is 푖푖 − open. Since 퐴 is closed, 푤푐푙(퐴) ⊆ 푐푙(퐴) ⊆ 푈. This shows that 퐴 is 훼푤 − closed set in (푋, 휏).■ The converse of the above theorem is not true in general as shown in the following example.

Example 3.10 Let 푋 = {푎, 푏, 푐, } and 휏 = {∅, 푋, {푏}, {푏, 푐}}. 퐶(푋, 휏) = {∅, 푋, {푎}, {푎, 푐, }} 푖푖푤푐(푋, 휏) = {∅, 푋, {푎}, {푎, 푐}} 훼푤푐(푋, 휏) = {∅, 푋, {푎}, {푎, 푏}, {푎, 푐}} Let 퐴 = {푎, 푏}. Then 퐴 is 훼푤 − closed set but it is not 푖푖푤 − closed set.

Theorem 3.11 Every 푖푤 − 푐푙표푠푒푑 set is a 푖푖푤 − 푐푙표푠푒푑 set. Proof. Let 퐴 be an 푖푤 − closed set in (푋, 휏) and U is be any 푖푖 − open set in 푋 such that 퐴 ⊆ 푈. Since every 푖푖 − open set is 푖 − open. Since 퐴 is closed, 푤푐푙(퐴) ⊆ 푐푙(퐴) ⊆ 푈. This show that 퐴 is 푖푖푤 − closed set in (푋, 휏).■

Theorem 3.12 If A is 푖푖 − 표푝푒푛 and 푖푖푤 − 푐푙표푠푒푑 set then 퐴 is 푤 − 푐푙표푠푒푑. Proof. Since 퐴 ⊆ 퐴 and 퐴 is 푖푖 − open and 푖푖푤 − closed, we have 푤푐푙(퐴) ⊆ 퐴. Thus 푤푐푙(퐴) = 퐴. Hence 퐴 is 푤 − closed set in X.■

Theorem 3.13 Union of two 푖푖푤 − 푐푙표푠푒푑 sets are 푖푖푤 − 푐푙표푠푒푑 set. Proof. Let 퐴 and 퐵 be two 푖푖푤 − closed sets. Let 퐺 be any 푖푖 − open set in (푋, 휏), such that 퐴 ∪ 퐵 ⊆ 퐺. Then 퐴 ⊆ 퐺 and 퐵 ⊆ 퐺. Since 퐴 and 퐵 are 푖푖푤 − closed set, 푤푐푙(퐴) ⊆ 퐺 and 푤푐푙(퐵) ⊆ 퐺. Therefore 푤푐푙(퐴) ∪ 푤푐푙(퐵) = 푤푐푙(퐴 ∪ 퐵) ⊆ 퐺. Hence 퐴 ∪ 퐵 is 푖푖푤 − closed set.■

Remark 3.14 The following diagram shows that the relationships of 푖푖푤 − closed sets with other known existing sets.

풊흎-closed

흎-closed 휶흎-closed 풊풊흎-closed 48 Amir A. Mohammed and Beyda S. Abdullah

References

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[3] O. Njȧstad, On some classes of nearly open sets, Pacific Journal of Mathematics, 15 (1965), no. 3, 961 - 970. https://doi.org/10.2140/pjm.1965.15.961

[4] M. Parimala, R. Udhayakumar, R. Jeevitha and V. Biju, On 훼휔-closed set in topological spaces, International Journal of Pure and Applied Mathematics, 115 (2017), no. 5, 1049 - 1056.

[5] P. Sundaram and M.S. John, On ω-closed sets in topology, Acts Ciencica Indian Mathematics, 26 (2000), no. 4, 389 - 392.

Received: January 23, 2019; Published: February 4, 2019