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# βI-Compactness, βI*-Hyperconnectedness and βI-Separatedness in Ideal Topological Spaces

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Reviewed: 05 November 2021 Published: 26 January 2022

DOI: 10.5772/intechopen.101524

From the Edited Volume

Edited by Francisco Bulnes

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## Abstract

Let XτI be an ideal topological space. A subset A of X is said to be β-open if A⊆clintclA, and it is said to be βI-open if there is a set O∈τ with the property 1 O−A∈I and 2 A−clintclO∈I. The set A is called βI-compact if every cover of A by βI-open sets has a finite sub-cover. The set A is said to be cβI-compact, if every cover Oλ:λ∈Λ of A by β-open sets, Λ has a finite subset Λ0 such that A−∪Oλ:λ∈Λ0∈I. The set A is said to be countably βI-compact if every countable cover of A by βI-open sets has a finite sub-cover. An ideal topological space XτI is said to be βI∗-hyperconnected if X−cl∗A∈I for every non-empty βI-open subset A of X. Two subsets A and B of X is said to be βI-separated if clβIA∩B=∅=A∩clβB. Moreover, A is called a βI-connected set if it can’t be written as a union of two βI-separated subsets. An ideal topological space XτI is called βI-connected space if X is βI-connected. In this article, we give some important properties of βI-open sets, βI-compact spaces, cβI-compact spaces, βI∗-hyperconnected spaces, and βI-connected spaces.

### Keywords

• β-open set
• βI-open set
• βI-compact
• cβI-compact
• β*I-hyperconnected

## 1. Introduction

Let Xτ be a topological space. A subset A of X is said to be a β-open set [1] if AclintclA. For example, consider the topology Xτ=abca, abac,X}). Then , X, ab, ac are the β-open sets of Xτ. A subset A of X is said to be semi-open set [1] if AclintA. A subset A of X is said to be α-open set [2] if AintclintA. A subset A of X is said to be pre-open set [3] if AintclA. A subset A of X is said to be regular-open set [4] if A=intclA. A subset A of X is said to be β-open set [5] if AclintclAintclδA (please see [5] for the notation clδA). A subset A of X is said to be β̂-generalized-closed set [6] if clintclAO whenever AO and O is open in X.

An ideal I on a set X is a nonempty collection of subsets of X which satisfies the conditions: (1) AI and BA implies BI, (2) AI and BI implies ABI. Let Xτ be a topological space and I be an ideal in X. Then we call XτI an ideal topological space. For example, let X=abc. Then I=a is an ideal on X. To see this, we note that the subsets of is itself, and the subsets of a are a and . Note that all of these subsets are in I. Next, we observe that =I, a=aI and aa=aI. Thus, I=a is an ideal on X.

For the concepts that were not discussed here please refer to [5, 7, 8, 9, 10, 11, 12, 13].

Topology is a new subject of mathematics, being born in the nineteenth century. However, the involvement of topology is clear in the other branches of math [12].

Topology is also seen in some fields of science. In particular, it is applied in biochemistry [14] and information systems [15].

Topology as a mathematical system is fundamentally comprised of open sets, among others. Open sets were generalized in a couple of different ways over the past. To mention a few, Stone [4] presented regular open set. Levine [1] presented semi-open sets. Najasted [2] presented α-open sets. Mashhour et al. [3] presented pre-open sets. Abd El-Monsef et al. [7] presented β-open set. Among these generalization, this study focused on one—the β-open sets.

Abd El-Monsef et al. [7] also presented the concepts β-continuous and β-open mappings. They gave some of their properties. Recently, β-open sets were investigated by many math enthusiast. For example, Abid [16] utilized β-open sets to gain some properties of non-semi-predense set. Tahiliani [13] presented an operation involving β-open sets which paved way to the creation of β-γ-open sets. Kannan and Nagaveni [6] generalized β-open set, and named it β̂-generalized closed set. Mubarki et al. [5] also generalized β-open set, and named it β-open set. El-Mabhouh and Mizyed [17] also generalized β-open set, and named it βc-open set. Akdag and Ozkan [8] made an investigation of β-open sets in soft topological spaces. Arockiarani and Arokia Lancy [9] introduced -closed set and gsβ-closed set (these were defined using β-open sets).

The notion of ideal topological spaces was introduced by Kuratowski [18]. Later, Vaidyanathaswamy [19] studied the concept in point set topology. Tripathy and Shravan [20, 21], Tripathy and Acharjee [22], Triapthy and Ray [23], among others, were also some of those who studied ideal topological spaces.

This study have important applications in some areas of mathematics. In particular, βI-compactness, βI-hyperconnectedness and βI-separatedness can be investigated in the areas of measure theory, continuum theory and dimension theory just as the parallel notions (compactness, hypercompactness, and separatedness, respectively) were studied in those areas. The purpose of this paper is to introduce and study a notion of connectedness, hypercompactness, and separatedness relative to the family of all β-open sets in some ideal topological spaces.

## 2. βI-compactness in ideal spaces

In this section, we gave some important properties of βI-open sets in τω-spaces.

Recall, a topological space Xτ is said to be a τω-space if for every subset A of X, it is always true that intclA=intclintA. For example, let X=wxyz. Then τ1=wwxwywxyX is a τω-space, while τ2=wxyX is not. Also, a discrete space is a τω-space, while an indiscete space is not.

Lemma 1.1 characterizes β-open sets in a τω-space.

Lemma 1.1. Let Xτ be a τω-space and I be an ideal in X. A set A X is a β-open set precisely if there is a set Oτ with the property that OAclintclO.

Proof: Suppose that A is a β-open set. Then AclintclA. Consider O=intA (note that O is open). Since Xτ is a τω-space, intclA=intclintA. Hence, OAclintclA=clintclintA=clintclO.

Conversely, suppose that there is a set Oτ with the property that OAclintclO. Since OA, we have clOclA. And so, intclOintclA. Therefore, clintclOclintclA Thus, AclintclA.

Next, we define βI-open set.

Definition 1.1. Let Xτ be a topological space and I be an ideal in X. A subset A of X is called β-open with respect to the ideal I, or a βI-open set, if there exists an open set O such that 1 OAI, and 2 AclintclOI.

For example, let X=abc, τ=abcX, and I=b (note that τ is a topology on X, and I is an ideal on X). Then A=bc is a βI-open set. To see this, consider O=bc. Then O is a open set. Observe that OA=bcbc=I, and AclintclO=bcclintclbc=bcclintbc=bcclbc=bcbc=I. Thus, A=bc is β-open with respect to the ideal I.

Lemma 1.2 says that an open set is a βI-open set, and an element of the ideal is a βI-open set. One may see [24] to gain more insights relative to these ideas. While, Lemma 1.3 says that in a τω-space a β-open set is also a βI-open set.

Lemma 1.2. Let Xτ be a topological space and I be an ideal in X. Then the following statements are true.

i. Every open set is a βI-open set.

ii. Every element of I is a βI-open set.

Proof: i Let A be an open set. Note that AA=I, and AclintclAAclA=I. Thus, A is βI-open. ii Let AI. Consider O=. Note that OA=A=I, and AclintclO=A=AI. Thus, A is βI-open. □

Lemma 1.3. Let Xτ be a τω-space and I be an ideal in X. Then every β-open set is a βI-open set.

Proof: Let A be a β-open set. By Lemma 1.1 there exists an open set O such that OAclintclO. Hence OA=I, and AclintclO=I. Thus, A is βI-open. □

Let Xτ be a topology and I be an ideal in X. We say that I is countably additive if Ai:iNI whenever Ai:iN is a (countable) family of elements of I.

Lemma 1.4 says that in a τω-space, if I is the minimal ideal, then the βI-open sets are precisely the β-open sets.

Lemma 1.4. Let Xτ be a τω-space and I be an ideal in X. If I is not countably additive, then the following statements are equivalent.

1. I=.

2. A is a β-open set precisely when A is a βI-open set.

Proof: iii Suppose that I=. Let A be a β-open set. By Lemma 1.3, A is a βI-open set. For the converse, let A be a βI-open set and O be an open set with OAI and AclintclOI. Because I=, we have OA= and AclintclO=. Hence, OA and AclintclO, that is OAcl intclO. Therefore, by Lemma 1.1 A is β-open.

iii Suppose that ii holds, and that I. Let D be a non-empty element of I. By Lemma 1.2, D is βI-open. Thus, by assumption D is β-open. Now, by Lemma 1.1, there exists O1τ with O1DclintclO1. Since D is an element of I and O1D, we have O1I. Hence, O1DI. By Lemma 1.1, O1D is a βI-open set. Hence, by assumption O1D is a β-open set. And so, again there exists O2τ with O2O1DclintclO2. Since O1DI and O2O1D, we have O2I. Hence, O1O2DI. Thus, by Lemma 1.1, O1O2D is a βI-open set. By assumption O1O2D is a β-open set. And so, again there exists O3τ with O3O1O2DclintclO3. Since O1O2DI and O3O1O2D, we have O3I. Hence, O1O2O3DI. Continuing in this fashion we obtain a countably infinite subset O1O2O3 of I with O1O2O3I. This is a contradiction since I is not countably additive. Thus, I=. □

Next, we define βI-compact set, βI-compact space, compatible βI-compact set, and compatible βI-compact space.

Definition 1.2. Let XτI be an ideal topological space. A subset A of X is said to be βI-compact if for every cover Oλ:λΛ of A by βI-open sets, Λ has a finite subset Λ0, such that Oλ:λΛ0 still covers A. A space X is said to be a βI-compact space if it is βI-compact as a subset.

Definition 1.3. Let XτI be an ideal topological space. A subset A of X is said to be countably βI-compact if for every countable cover On:nN of A by βI-open sets, N has a finite subset ij:j=12k with the property that Oij:j=12k still covers A. A space X is said to be a countably βI-compact space if it is countably βI-compact as a subset.

Definition 1.4. Let XτI be an ideal topological space. A subset A of X is said to be compatible βI-compact, or simply cβI-compact, if for every cover Oλ:λΛ of A by β-open sets, Λ has a finite subset Λ0, such that AOλ:λΛ0I. An ideal topological space XτI is said to be cβI-compact space if it is cβI-compact as a subset.

Theorem 1.1 says that in an ideal τω-space in which I is the minimal ideal, the notions β-compact, βI-compact and cβI-compact coincides.

Theorem 1.1. Let Xτ be a τω-space and I=. Then the following statements are equivalent.

1. XτI is a β-compact space.

2. XτI is a βI-compact space.

3. XτI is a cβI-compact space.

Proof: iii Suppose that i holds. Let Oλ:λΛ be a family of β-open sets that covers X. By assumption, Λ has a finite subset, say Λ0, with the property that Oλ:λΛ0 still covers X. By Lemma 1.3 iii, Oλ:λΛ0 is also a family of βI-open sets. Hence, Oλ:λΛ0 is a finite covering of X by βI-open sets. Therefore, X is βI compact.

iiiii Suppose that ii holds. Let Oλ:λΛ be a family of β-open sets that covers X. Since I=, by Lemma 1.4 Oλ:λΛ is also a family of βI-open sets that covers X. By assumption, Λ has a finite subset, say Λ0, with the property that Oλ:λΛ0 still covers X. Note that Uλ:λΛ0 is also a family of β-open sets, and XλΛ0Oλ=I. Therefore, X is cβI compact.

iiii Suppose that iii holds. Let Oλ:λΛ be a family of β-open sets that covers X. By assumption, Λ has a finite subset, say Λ0, with the property that XλΛ0OλI. Since I=, XλΛ0Oλ=, that is XλΛ0Oλ. Hence, Oλ:λΛ0 is covering of X. Therefore, X is β compact. □

Theorem 1.2 presents a characterization of βI-compact spaces.

Theorem 1.2. Let XτI be an ideal topological space. Then the following are equivalent.

1. XτI is βI-compact.

2. If Fλ:λΛ is a family of βI-closed sets with Fλ:λΛ=, then Λ has a finite subset, say Λ0, with the property Fλ:λΛ0=.

Proof: iii Suppose that i holds. Let Fλ:λΛ be a family of βI-closed sets with the property Fλ:λΛ=. Then FλC:λΛ=Fλ:λΛC=X. Hence, FλC:λΛ is a family of βI-open sets which covers of X. By assumption, Λ has a finite subset, say Λ0, with the property FλC:λΛ0=X, i.e. Fλ:λΛ0=.

iii Suppose that ii holds. Let Oλ:λΛ be a family of βI-open sets that covers X, i.e. Uλ:λΛ=X. Then OλC:λΛ=Uλ:λΛC=. Note that OC is βI-closed since O is βI-open. By assumption, Λ has a finite subset, say Λ0, with the property that OλC:λΛ0=. Thus, Oλ:λΛ0=OλC:λΛ0C=X, that is Oλ:λΛ0 is a family of βI-open sets that covers X. □

Theorem 1.3 presents a characterization of βI-compact spaces.

Theorem 1.3. Let Xτ be a topological space and I be an ideal in X. Then the following are equivalent.

1. XτI is cβI-compact.

2. If Fλ:λΛ is a family of β-closed sets with Fλ:λΛ=, then Λ has a finite subset, say Λ0, with the property that Fλ:λΛ0I.

Proof: iii Suppose that i holds. Let Fλ:λΛ be a family of β-closed sets such that Fλ:λΛ=. Note that FλC:λΛ=Fλ:λΛC=X. Hence, FλC:λΛ is a family of β-open sets that covers X. By assumption, Λ has a finite subset, say Λ0, with the property XFλC:λΛ0I, i.e. Fλ:λΛ0I.

iii Suppose that ii holds. Let Oλ:λΛ be a family of β-open sets that covers X, i.e. Oλ:λΛ=X. Note that OλC:λΛ=Oλ:λΛC=. By assumption, Λ has a finite subset, say Λ0, with the property OλC:λΛ0I, i.e. XOλ:λΛ0I.□

Remark 1.1. [25] Let XτI and YσJ be ideal spaces, and f:XY be a fuction. Then:

1. fI=fA:AI is an ideal in Y, where fA=fa:aA; And,

2. if f is bijective, then f1J=f1B:BJ is an ideal in X, where f1B=f1b:bB.

Next, we define βI-open, βI-irresolute, and βI-continuous functions.

Definition 1.5. Let XτI and YσJ be ideal topological spaces. A function f:XY is said to be

1. β-open if fA is β-open for every β-open set A,

2. β-irresolute if f1B is β-open for every β-open set B, and

3. β-continuous if f1B is β-open for every open set B.

4. βI-open if fA is βJ-open for every βI-open set A,

5. βI-irresolute if f1B is βI-open for every βJ-open set B, and

6. βI-continuous if f1B is βI-open for every open set B.

Theorem 1.4 says that given a β-irresolute function, if the domain is compatibly compact, then so is the image of f. On the other hand, Theorem 1.5 say that given an open surjection, if the co-domain is compatibly compact, then so is the domain.

Theorem 1.4. Let Xτ and Yσ be topological spaces, I be an ideal in X, and f:XY be a β-irresolute function. If X is a cβI-compact space, then fXfXB:BσfI is a cβfI-compact space.

Proof: Let Oλ:λΛ be a family of β-open sets that covers fX. Beacuse f is a β-irresolute, f1Oλ:λΛ is a family of by β-open sets that covers X. Because X is cβI-compact, Λ has a finite subset, say Λ0, with the property Xf1Oλ:λΛ0I. Hence, by Remark 1.1 fXOλ:λΛ0=fXf1Oλ:λΛ0fI.□

Theorem 1.5. Let Xτ and Yσ be topological spaces, J be an ideal in Y, and f:XY be a β-open surjection (surjective function). If Y is cβJ-compact, then X is cβf1J-compact.

Proof: Let Oλ:λΛ be a family of β-open sets that covers X. Beacuse f is a β-open surjection, fOλ:λΛ is a family β-open sets that covers Y. Because Y is cβJ-compact, Λ has a finite subset, say Λ0, with the property YfOλ:λΛ0J. Hence, XOλ:λΛ0=f1YfOλ:λΛ0f1J.□

The next theorem says that in a τω-space and when I={}, the family of all countably βI-compact space contains all cβI-compact space.

Theorem 1.6. Let XτI be an ideal τω-space and I=. If X is cβI-compact, then it is also countably βI-compact.

Proof: Let On:nN be a countable family βI-open sets that covers X. Because X is cβI-compact, N has a finite subset ij:j=12k with the property that XOij:j=12kI. Because I=, X=Oij:j=12kI. By Lemma 1.4 Oij:j=12k is also a family of β-open sets. Hence, Oij:j=12k is a finite subcover of X by β-open sets. □

## 3. βI∗-hyperconnectedness in ideal spaces

The concept -hyperconnectedness was introduced by Ekici et al. [26], and the concept I-hyperconnectedness was introduced by Abd El-Monsef et al. [27]. These insights motivated us to create the concept called βI-hyperconnectedness. One may see [28] to gain more insights on these ideas.

Definition 1.6. Let Xτ be a topological space and I be an ideal on X. A function Iτ:PXPX given by AIτ=xX:AUIfor everyUτx where τx=Uτ:xU is called a local of A with respect to τ and I.

Example 1.1. Let X=abc, τ=abcabacbcX, and I=abab (note that τ is a topology on X and I is an ideal on X). Then, =, a=c, b=c, c=X, ab=c, ac=X, bc=X and X=X.

Definition 1.7. Let Xτ be a topological space and I be an ideal on X. The Kuratowski closure operator ClIτ:PXPX for the topology τIτ is given by ClAIτ=AA.

Example 1.2. Consider the ideal space of Example 3. Then we have, Cl===, Cla=aa=ac=ac, Clb=bb=bc=bc, Clc=cc=cX=X, Clab=abab=abc=X, Clac=acac=acX=X, Clbc=bcbc=bcX=X, and ClX=XX=XX=X.

Definition 1.8. Let Xτ be a topological space and I be an ideal on X. The Kuratowski interior operator IntIτ:PXPX for the topology τIτ is given by IntAIτ=XClXA.

Definition 1.9 is taken from [26], while Definition 1.10 is taken from [29].

Definition 1.9. [26] An ideal space XτI is called -hyperconnected if clA=X for all non-empty open set A X.

Definition 1.10. [29] An ideal space XτI is called I-hyperconnected if XclAI for all non-empty open set A X.

A notion similar to Definition 1.9 and Definition 1.10 is presented next.

Definition 1.11. An ideal topological space XτI is said to be βI-hyperconnected space if XclAI for every non-empty βI-open subset A of X.

The next theorem says that the family of all βI-hyperconnected space contains all I-hyperconnected space.

Theorem 1.7. Let Xτ be a topological space, and I be an ideal in X. If X is I-hyperconnected, then it is βI-hyperconnected also.

Proof: Let X be I-hyperconnected, and A be a non-empty open set. Because X is I-hyperconnected, we have XclAI for all non-empty open set A X. And, because an open set is also a βI-open set, we have XclAI for all non-empty βI-open set A X. Hence, X is βI-hyperconnected. □

The next lemma is clear.

Lemma 1.5. Let Xτ be a topological space. Then the intersection of any family of ideals on X is an ideal on X.

Theorem 1.8 is taken from [29]. It says that when I is the minimal ideal, then the notions -hyperconnected and I-hyperconnected are equivalent.

Theorem 1.8. [29] Let Xτ be a topological space, and I=. Then, X is -hyperconnected if and only if it is I-hyperconnected.

The next remark is clear.

Remark 1.2. If Xτ is a clopen topological space (a space in which every open set is also closed), then A is open if and only if A is β-open.

To see this, let A be an open set. Since τ is clopen, A is closed also. Hence, clintclA=A. Thus, A is a β-open set. Conversely, if A is a β-open set, then AclintclA. This implies that A must be open.

Theorem 1.9 says that in a clopen τω-space, with respect to the minimal ideal I, the notions βI-hyperconnected and I-hyperconnected are equivalent.

Theorem 1.9. Let Xτ be a clopen τω-space, and I=. Then, X is I-hyperconnected if and only if it is βI-hyperconnected.

Proof: Suppose that X is I-hyperconnected. Let A is a non-empty element of τ. Then XclAI. By Remark 1.2 and Lemma 1.5, every open set is absolutely βI-open. Thus, XclAI for all βI-open set A. Therefore, X is βI-hyperconnected also. Conversely, suppose that X is βI-hyperconnected. Let A be a non-empty βI-open set. Then XclAI. By Remark 1.2 and Lemma 1.5, every βI-open set is absolutely open. Thus, XclAI for all open set A. Therefore, X is I-hyperconnected also. □

Corollary 1.1 says that in a clopen τω-space, relative to the minimal ideal I, the notions βI-hyperconnected, I-hyperconnected, and -hyperconnected are equivalent.

Corollary 1.1. Let Xτ be a clopen τω-space and I=. Then the following statements are equivalent.

1. X is I-hyperconnected.

2. X is βI-hyperconnected.

3. X is βI-hyperconnected.

Theorem 1.3 may be an important property.

Remark 1.3. If an ideal τω space Xτ is a βI-hyperconnected space, then XclAI for every non-empty β-open subset A of X.

To see this, let A is a non-empty β-open set. By Lemma 1.4 A is βI-open. Since X is βI-hyperconnected, XclAI.

Theorem 1.10 is a characterization of βI-hyperconnected space.

Theorem 1.10. Let Xτ be an topological space and I be an ideal in X. Then the following statements are equivalent.

1. X is a βI-hyperconnected space.

2. IntAI for all proper βI-closed subset A of X.

Proof: iii Suppose that i holds. Let B be βI-closed. Then XB is βI-open. Since BX, XB. Hence, by assumption we have IntB=XClXBI.

iii Suppose that ii holds. Let A X be a non-empty βI-open set. Then XA is a non-empty βI-open set. Hence, by assumption we have XclA=XclXXA=intXAI. Thus, X is βI-hyperconnected. □

## 4. βI-separatedness in ideal spaces

In this section, we present the concepts βI-separated sets and βI-connected sets. We also present some of their important properties.

Let XτI be an ideal topological space and A be a subset of X. The β-closure of A, denoted by clβA, is the smallest β-closed set that contains A. The βI-closure of A, denoted by clβIA, is the smallest βI-closed set that contains A.

Next, we define βI-separated set, βI-connected set, and βI-connected space.

Definition 1.12. Let XτI be an ideal topological space. A pair of subsets, say A and B, of X is said to be βI-separated if clβIAB==AclβB.

Definition 1.13. Let XτI be an ideal topological space and A be a subset of X. Then A is said to be βI-connected if it cannot be expressed as a union of two βI-separated sets. A topological space X is said to be βI-connected if it is βI-connected as a subset.

Recall, a topological space Xτ is said to be a τζ-space if for every pair of subsets A and B of X, it is always true that clAB=clAclB and intAB=intAintB. For example, a discrete space is a τζ-space, while an indiscete space is not. Also, if X=abc, then τ=aX is not a τζ-space. Let Xτ be a τζ-space and I be an ideal in X. Then we call XτI an ideal τζ-space.

Lemma 1.6 present sufficient conditions for two sets to be βI-separated.

Lemma 1.6. Let Xτ be a topological space and I be an ideal in X. If A is β-open and B is βI-open with AB=, then they are βI-separated.

Proof: Suppose that A and B is not βI-separated, that is clβIAB or AclβB. Because AB=, we have ABC and BAC. If A is β-open, then AC is β-closed. Similarly, if B is βI-open, then BC is βI-closed. Thus, BCBclβIAB, or AACAclβB. A contradiction. □

Lemma 1.7 says that in a τω-space every βI-connected space is connected. Recall, a space is connected if it cannot be written as a union of two non-empty open sets.

Lemma 1.7. Let Xτ be a topology and I be an ideal in X. If X is βI-connected, then it is connected.

Proof: Suppose that to the contrary X is not connected. Let A and B be non-empty disjoint elements of τ with X=AB. Note that A and B are β-open and βI-open also. Because A=BC and B=AC, A and B are also β-closed and βI-closed. And so, A=clβIA and B=clβB. Thus, clβIAB=AB= and AclβB=AB=. This implies that X is βI-separated, that is X is not βI-connected. □

Remark 1.4. Let Xτ be a topology and I be an ideal in X. If YX, then IY=YA:AI is an ideal in the relative topology YτY.

To see this, for the first property, let BIY and AB. Then ABY. Now, if AIY, then there exist CI such that YC=A. Note that ABC. Hence, A,BI. Thus, A=YAIY. Next, for the second, let D,EIY. Then DY and EY. if DIY, then there exist FI such that YF=D. Similarly, if EIY, then there exist GI such that YG=E. Since I is an ideal, FGI. Now, because DEFG, DEI. Thus, DE=DEYIY.

The next statement, Theorem 1.11, presents a way to construct βI-open sets in a subspace.

Theorem 1.11. Let XτI be an ideal τζ-space and Y be a clopen (a set that is open and closed at the same time) set. If A is a βI-open subset of X τI, then AY is a βIY-open set in Y τYIY.

Proof: Let A be a βI-open set in X τI. Then there exists an open set U such that UAI and AclintclUI. Let U=UY. Then

UAY=UAYC=UYACYC=UYACUYYC=UYAC=UAYIY.E1

Moreover, since X is a τζ-space and Y is clopen

AYclintclU=AYclintclUY=AYclintclUY=AclintclUYIY.E2

This shows that AY is βIY-open in YτYIY. □

Corollary 1.2. Let XτI be an ideal τζ-space and Y be a clopen set. If A is a βI-closed subset of X τI, then AY is a βIY-closed set in Y τYIY.

Proof: If A is βI-closed, then AC is βI-open. By Theorem 1.11, ACY is βIY-open. Hence, AY=ACYC is βIY-closed in Y. □

The next remark is clear. We shall be using it in showing some of the succeeding theorems.

Remark 1.5. Let XτI be an ideal topological space and YX. Then IY=AY:AI is a subset of I.

Proof: If A is a βIY-open set in Y, then there exists an open set OτY with OAIY and AclintclOIY. Because τYτ and by Remark 1.5, there exists an open set Oτ with OAI and AclintclOI. Thus, A is βI-open in XτI.

The converse follows from Theorem 1.11. □

The next statement, Lemma 1.8, characterizes βI-open sets in subspaces.

Lemma 1.8. Let XτI be an ideal τζ-space, Y X be clopen, and τYτ. If AY, then A is βIY-open in YτYIY if and only if it is βI-open in XτI.

Proof: If A is a βIY-open set in Y, then there exists an open set OτY with OAIY and AclintclOIY. Because τYτ and by Remark 1.5, there exists an open set Oτ with OAI and AclintclOI. Thus, A is βI-open in XτI.

The converse follows from Theorem 1.11. □

The next statement, Theorem 1.12, provides a way of determining the closure of a set in the subspace.

Theorem 1.12. Let XτI be an ideal τζ-space, Y be clopen, and τYτ. If AX, then clβIYAY=clβIAY.

Proof: Since clβIA is a βI-closed set in X, by Lemma 1.8 clβIAY is a βIY-closed set in Y. Hence, clβIYAY clβIYclβIAY =clβIAY. But, clβIYAY= clβIAY clβIAclβIY= clβIAY. Therefore, clβIYAY=clβIAY. □

Definition 1.14. Let XτI be an ideal topological space, and YτYIY be a subspace. A pair of subsets, say A and B, of X is said to be βIY-separated in Y if clβIYAB==AclβYB, where clβYB=clβBY.

Definition 1.15. Let XτI be an ideal topological space, and YτYIY be a subspace. A subset A of Y is said to be βIY-connected if it cannot be expressed as a union of two βIY-separated sets. The subspace Y is said to be βIY-connected if it is βIY-connected as a subset.

The next statement, Theorem 1.13, says that if two sets are separated in the mother space, then they are also separated in the subspace.

Theorem 1.13. Let Xτ be a τζ-space, I be an ideal, Y X be clopen, and τYτ. If A and B are βI-separated in X, then they are βIY-separated in Y.

Proof: If A and B are βI-separated in X, then by Theorem 1.12 =clβIAB=clβIABY=clβIAYB=clβIYAB and =AclβB=AclβBY=AclβBY=AclβYB. Thus, A and B are βIY-separated. □

The next statement, Remark 1.6, says that if two non-empty sets, which expresses X as a disjoint union, is βI-separated, then one must be β-open and the other must be βI-open.

Remark 1.6. Let Xτ be a topological space and I be an ideal. If X is βI-separated (say, X=AB with A, B, and clβIAB= =AclβB), then A is β-open while B is βI-open.

To see this, if A and B is βI-separated, then clβIAB= and AclβB=. Hence, AC=clβB and BC=clβIA. Thus, AC is β-closed and BC is βI-closed. Accordingly, A is β-open and B is βI-open.

The next statement, Theorem 1.14, characterizes βI-connected spaces.

Theorem 1.14. Let Xτ be a topological space and I be an ideal X. Then, X is βI-connected if and only if it cannot be expressed as a union of two a non-empty disjoint sets in which one is a β-open set and the other is a βI-open set.

Proof: Suppose that X is βI-connected, and we can express X as a union of two non-empty disjoint β-open set and βI-open set, say AB=X (with A, a β-open set, and B, a βI-open set) and AB=. If AB=X and AB=, then AC=B and BC=A. Since A is β-open, B is β-closed. Also, since B is βI-open, A is βI-closed. Hence, clβIAB=AB= and AclβB=AB=. Thus, the pair A and B is βI-separated. This is a contradiction.

The converse follows from Remark 1.6. □

The next statement, Theorem 1.15, says that two separated set cannot contain portions of a connected set.

Theorem 1.15. Let Xτ be a topological space, I be an ideal X, and A be a βI-connected set. If AHG where H and G is a pair of βI-separated sets, then either AH or AG.

Proof: Suppose that to the contrary, A=AHAG with AH and AG. Since H and G is a pair of βI-separated sets, clβIAHAGclβIHG= and AHclβAGHclβG=. Thus, clβIAHAG= and AHclβAG=. Therefore, A can be expressed as a union of two βI-separated sets AH and AG. A contradiction. □

The next statement, Theorem 1.16, says that subsets of each of two separated sets are also separated.

Theorem 1.16. Let Xτ be a topological space, I be an ideal in X, and, A and B be βI-separated sets. If CA (C) and DB (D), then C and D are also βI-separated.

Proof: Suppose that A and B are βI-separated. Then clβIAB= and AclβB=. Thus, clβICDclβIAB= and CclβD=AclβB=. Hence, clβICD==CclβD. Therefore, C and D is βI-separated. □

## 5. Conclusion

With the important concepts and results which intertwined with those introduced by other authors, this chapter is very interesting. The construction of the different theorems were realized using the definitions or properties of β-open sets, βI-compact spaces, βI-hyperconnected spaces, βI-separated spaces. Also, some properties focusing on generalizing ideals in ideal topological space theory were realized.

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