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

Written By

Glaisa T. Catalan, Michael P. Baldado Jr and Roberto N. Padua

Reviewed: November 5th, 2021Published: January 26th, 2022

DOI: 10.5772/intechopen.101524

IntechOpen
Advanced Topics of TopologyEdited by Francisco Bulnes

From the Edited Volume

Advanced Topics of Topology [Working Title]

Dr. 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 1O−A∈I and 2A−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 Xτbe a topological space. A subset Aof Xis said to be a β-open set [1] if AclintclA. For example, consider the topology Xτ=abca,abac,X}). Then , X, ab, acare the β-open sets of Xτ. A subset Aof Xis said to be semi-openset [1] if AclintA. A subset Aof Xis said to be α-openset [2] if AintclintA. A subset Aof Xis said to be pre-openset [3] if AintclA. A subset Aof Xis said to be regular-openset [4] if A=intclA. A subset Aof Xis said to be β-openset [5] if AclintclAintclδA(please see [5] for the notation clδA). A subset Aof Xis said to be β̂-generalized-closedset [6] if clintclAOwhenever AOand Ois open in X.

An ideal Ion a set Xis a nonempty collection of subsets of Xwhich satisfies the conditions: (1) AIand BAimplies BI, (2) AIand BIimplies ABI. Let Xτbe a topological space and Ibe an ideal in X. Then we call XτIan ideal topological space. For example, let X=abc. Then I=ais an ideal on X. To see this, we note that the subsets of is itself, and the subsets of aare aand . Note that all of these subsets are in I. Next, we observe that =I, a=aIand aa=aI. Thus, I=ais 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.

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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 Xτis said to be a τω-space if for every subset Aof X, it is always true that intclA=intclintA. For example, let X=wxyz. Then τ1=wwxwywxyXis a τω-space, while τ2=wxyXis 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.LetXτbe aτω-space andIbe an ideal inX. A setAXis aβ-open set precisely if there is a setOτwith the property thatOAclintclO.

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

Conversely, suppose that there is a set Oτwith the property that OAclintclO. Since OA, we have clOclA. And so, intclOintclA. Therefore, clintclOclintclAThus, AclintclA.

Next, we define βI-open set.

Definition 1.1.LetXτbe a topological space andIbe an ideal inX. A subsetAofXis calledβ-open with respect to the idealI, or aβI-open set, if there exists an open setOsuch that1OAI, and2AclintclOI.

For example, let X=abc, τ=abcX, and I=b(note that τis a topology on X, and Iis an ideal on X). Then A=bcis a βI-open set. To see this, consider O=bc. Then Ois a open set. Observe that OA=bcbc=I, and AclintclO=bcclintclbc=bcclintbc=bcclbc=bcbc=I. Thus, A=bcis 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.LetXτbe a topological space andIbe an ideal inX. Then the following statements are true.

i. Every open set is aβI-open set.

ii. Every element ofIis aβI-open set.

Proof:iLet Abe an open set. Note that AA=I, and AclintclAAclA=I. Thus, Ais βI-open. iiLet AI. Consider O=. Note that OA=A=I, and AclintclO=A=AI. Thus, Ais βI-open. □

Lemma 1.3.LetXτbe aτω-space andIbe an ideal inX. Then everyβ-open set is aβI-open set.

Proof:Let Abe a β-open set. By Lemma 1.1 there exists an open set Osuch that OAclintclO. Hence OA=I, and AclintclO=I. Thus, Ais βI-open. □

Let Xτbe a topology and Ibe an ideal in X. We say that Iis countably additive if Ai:iNIwhenever Ai:iNis a (countable) family of elements of I.

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

Lemma 1.4.LetXτbe aτω-space andIbe an ideal inX. IfIis not countably additive, then the following statements are equivalent.

  1. I=.

  2. Ais aβ-open set precisely whenAis aβI-open set.

Proof:iiiSuppose that I=. Let Abe a β-open set. By Lemma 1.3, Ais a βI-open set. For the converse, let Abe a βI-open set and Obe an open set with OAIand AclintclOI. Because I=, we have OA=and AclintclO=. Hence, OAand AclintclO, that is OAclintclO. Therefore, by Lemma 1.1 Ais β-open.

iiiSuppose that iiholds, and that I. Let Dbe a non-empty element of I. By Lemma 1.2, Dis βI-open. Thus, by assumption Dis β-open. Now, by Lemma 1.1, there exists O1τwith O1DclintclO1. Since Dis an element of Iand O1D, we have O1I. Hence, O1DI. By Lemma 1.1, O1Dis a βI-open set. Hence, by assumption O1Dis a β-open set. And so, again there exists O2τwith O2O1DclintclO2. Since O1DIand O2O1D, we have O2I. Hence, O1O2DI. Thus, by Lemma 1.1, O1O2Dis a βI-open set. By assumption O1O2Dis a β-open set. And so, again there exists O3τwith O3O1O2DclintclO3. Since O1O2DIand O3O1O2D, we have O3I. Hence, O1O2O3DI. Continuing in this fashion we obtain a countably infinite subset O1O2O3of Iwith O1O2O3I. This is a contradiction since Iis 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.LetXτIbe an ideal topological space. A subsetAofXis said to beβI-compact if for every coverOλ:λΛofAbyβI-open sets,Λhas a finite subsetΛ0, such thatOλ:λΛ0still coversA. A spaceXis said to be aβI-compact space if it isβI-compact as a subset.

Definition 1.3.LetXτIbe an ideal topological space. A subsetAofXis said to be countablyβI-compact if for every countable coverOn:nNofAbyβI-open sets,Nhas a finite subsetij:j=12kwith the property thatOij:j=12kstill coversA. A spaceXis said to be a countablyβI-compact space if it is countablyβI-compact as a subset.

Definition 1.4.LetXτIbe an ideal topological space. A subsetAofXis said to be compatibleβI-compact, or simplycβI-compact, if for every coverOλ:λΛofAbyβ-open sets,Λhas a finite subsetΛ0, such thatAOλ:λΛ0I. An ideal topological spaceXτIis said to becβI-compact space if it iscβI-compact as a subset.

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

Theorem 1.1.LetXτbe aτω-space andI=. Then the following statements are equivalent.

  1. XτIis aβ-compact space.

  2. XτIis aβI-compact space.

  3. XτIis acβI-compact space.

Proof:iiiSuppose that iholds. Let Oλ:λΛbe a family of β-open sets that covers X. By assumption, Λhas a finite subset, say Λ0, with the property that Oλ:λΛ0still covers X. By Lemma 1.3 iii, Oλ:λΛ0is also a family of βI-open sets. Hence, Oλ:λΛ0is a finite covering of Xby βI-open sets. Therefore, Xis βIcompact.

iiiiiSuppose that iiholds. Let Oλ:λΛbe a family of β-open sets that covers X. Since I=, by Lemma 1.4 Oλ:λΛis also a family of βI-open sets that covers X. By assumption, Λhas a finite subset, say Λ0, with the property that Oλ:λΛ0still covers X. Note that Uλ:λΛ0is also a family of β-open sets, and XλΛ0Oλ=I. Therefore, Xis cβIcompact.

iiiiSuppose that iiiholds. Let Oλ:λΛbe a family of β-open sets that covers X. By assumption, Λhas a finite subset, say Λ0, with the property that XλΛ0OλI. Since I=, XλΛ0Oλ=, that is XλΛ0Oλ. Hence, Oλ:λΛ0is covering of X. Therefore, Xis βcompact. □

Theorem 1.2 presents a characterization of βI-compact spaces.

Theorem 1.2.LetXτIbe an ideal topological space. Then the following are equivalent.

  1. XτIisβI-compact.

  2. IfFλ:λΛis a family ofβI-closed sets withFλ:λΛ=, thenΛhas a finite subset, sayΛ0, with the propertyFλ:λΛ0=.

Proof:iiiSuppose that iholds. Let Fλ:λΛbe a family of βI-closed sets with the property Fλ:λΛ=. Then FλC:λΛ=Fλ:λΛC=X. Hence, FλC:λΛis a family of βI-open sets which covers of X. By assumption, Λhas a finite subset, say Λ0, with the property FλC:λΛ0=X, i.e. Fλ:λΛ0=.

iiiSuppose that iiholds. Let Oλ:λΛbe a family of βI-open sets that covers X, i.e. Uλ:λΛ=X. Then OλC:λΛ=Uλ:λΛC=. Note that OCis βI-closed since Ois βI-open. By assumption, Λhas a finite subset, say Λ0, with the property that OλC:λΛ0=. Thus, Oλ:λΛ0=OλC:λΛ0C=X, that is Oλ:λΛ0is a family of βI-open sets that covers X. □

Theorem 1.3 presents a characterization of βI-compact spaces.

Theorem 1.3.LetXτbe a topological space andIbe an ideal inX. Then the following are equivalent.

  1. XτIiscβI-compact.

  2. IfFλ:λΛis a family ofβ-closed sets withFλ:λΛ=, thenΛhas a finite subset, sayΛ0, with the property thatFλ:λΛ0I.

Proof:iiiSuppose that iholds. Let Fλ:λΛbe a family of β-closed sets such that Fλ:λΛ=. Note that FλC:λΛ=Fλ:λΛC=X. Hence, FλC:λΛis a family of β-open sets that covers X. By assumption, Λhas a finite subset, say Λ0, with the property XFλC:λΛ0I, i.e. Fλ:λΛ0I.

iiiSuppose that iiholds. Let Oλ:λΛbe a family of β-open sets that covers X, i.e. Oλ:λΛ=X. Note that OλC:λΛ=Oλ:λΛC=. By assumption, Λhas a finite subset, say Λ0, with the property OλC:λΛ0I, i.e. XOλ:λΛ0I.□

Remark 1.1.[25] LetXτIandYσJbe ideal spaces, andf:XYbe a fuction. Then:

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

  2. if fis bijective, then f1J=f1B:BJis an ideal in X, where f1B=f1b:bB.

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

Definition 1.5.LetXτIandYσJbe ideal topological spaces. A functionf:XYis said to be

  1. β-open iffAisβ-open for everyβ-open setA,

  2. β-irresolute iff1Bisβ-open for everyβ-open setB, and

  3. β-continuous iff1Bisβ-open for every open setB.

  4. βI-open iffAisβJ-open for everyβI-open setA,

  5. βI-irresolute iff1BisβI-open for everyβJ-open setB, and

  6. βI-continuous iff1BisβI-open for every open setB.

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.LetXτandYσbe topological spaces,Ibe an ideal inX, andf:XYbe aβ-irresolute function. IfXis acβI-compact space, thenfXfXB:BσfIis acβfI-compact space.

Proof:Let Oλ:λΛbe a family of β-open sets that covers fX. Beacuse fis a β-irresolute, f1Oλ:λΛis a family of by β-open sets that covers X. Because Xis cβI-compact, Λhas a finite subset, say Λ0, with the property Xf1Oλ:λΛ0I. Hence, by Remark 1.1 fXOλ:λΛ0=fXf1Oλ:λΛ0fI.□

Theorem 1.5.LetXτandYσbe topological spaces,Jbe an ideal inY, andf:XYbe aβ-open surjection (surjective function). IfYiscβJ-compact, thenXiscβf1J-compact.

Proof:Let Oλ:λΛbe a family of β-open sets that covers X. Beacuse fis a β-open surjection, fOλ:λΛis a family β-open sets that covers Y. Because Yis cβJ-compact, Λhas 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.LetXτIbe an idealτω-space andI=. IfXiscβI-compact, then it is also countablyβI-compact.

Proof:Let On:nNbe a countable family βI-open sets that covers X. Because Xis cβI-compact, Nhas a finite subset ij:j=12kwith the property that XOij:j=12kI. Because I=, X=Oij:j=12kI. By Lemma 1.4 Oij:j=12kis also a family of β-open sets. Hence, Oij:j=12kis a finite subcover of Xby β-open sets. □

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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.LetXτbe a topological space andIbe an ideal onX. A functionIτ:PXPXgiven byAIτ=xX:AUIfor everyUτxwhereτx=Uτ:xUis called a local ofAwith respect toτandI.

Example 1.1.LetX=abc,τ=abcabacbcX, andI=abab(note thatτis a topology onXandIis an ideal onX). Then,=,a=c,b=c,c=X,ab=c,ac=X,bc=XandX=X.

Definition 1.7.LetXτbe a topological space andIbe an ideal onX. The Kuratowski closure operatorClIτ:PXPXfor the topologyτIτis given byClAIτ=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, andClX=XX=XX=X.

Definition 1.8.LetXτbe a topological space andIbe an ideal onX. The Kuratowski interior operatorIntIτ:PXPXfor the topologyτIτis given byIntAIτ=XClXA.

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

Definition 1.9.[26] An ideal spaceXτIis called-hyperconnected ifclA=Xfor all non-empty open setAX.

Definition 1.10.[29] An ideal spaceXτIis calledI-hyperconnected ifXclAIfor all non-empty open setAX.

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

Definition 1.11.An ideal topological spaceXτIis said to beβI-hyperconnected space ifXclAIfor every non-emptyβI-open subsetAofX.

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

Theorem 1.7.LetXτbe a topological space, andIbe an ideal inX. IfXisI-hyperconnected, then it isβI-hyperconnected also.

Proof:Let Xbe I-hyperconnected, and Abe a non-empty open set. Because Xis I-hyperconnected, we have XclAIfor all non-empty open set AX. And, because an open set is also a βI-open set, we have XclAIfor all non-empty βI-open set AX. Hence, Xis βI-hyperconnected. □

The next lemma is clear.

Lemma 1.5.LetXτbe a topological space. Then the intersection of any family of ideals onXis an ideal onX.

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

Theorem 1.8.[29] LetXτbe a topological space, andI=. Then,Xis-hyperconnected if and only if it isI-hyperconnected.

The next remark is clear.

Remark 1.2.IfXτis a clopen topological space (a space in which every open set is also closed), thenAis open if and only ifAisβ-open.

To see this, let Abe an open set. Since τis clopen, Ais closed also. Hence, clintclA=A. Thus, Ais a β-open set. Conversely, if Ais a β-open set, then AclintclA. This implies that Amust 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.LetXτbe a clopenτω-space, andI=. Then,XisI-hyperconnected if and only if it isβI-hyperconnected.

Proof:Suppose that Xis I-hyperconnected. Let Ais a non-empty element of τ. Then XclAI. By Remark 1.2 and Lemma 1.5, every open set is absolutely βI-open. Thus, XclAIfor all βI-open set A. Therefore, Xis βI-hyperconnected also. Conversely, suppose that Xis βI-hyperconnected. Let Abe a non-empty βI-open set. Then XclAI. By Remark 1.2 and Lemma 1.5, every βI-open set is absolutely open. Thus, XclAIfor all open set A. Therefore, Xis 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.LetXτbe a clopenτω-space andI=. Then the following statements are equivalent.

  1. XisI-hyperconnected.

  2. XisβI-hyperconnected.

  3. XisβI-hyperconnected.

Theorem 1.3 may be an important property.

Remark 1.3.If an idealτωspaceXτis aβI-hyperconnected space, thenXclAIfor every non-emptyβ-open subsetAofX.

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

Theorem 1.10 is a characterization of βI-hyperconnected space.

Theorem 1.10.LetXτbe an topological space andIbe an ideal inX. Then the following statements are equivalent.

  1. Xis aβI-hyperconnected space.

  2. IntAIfor all properβI-closed subsetAofX.

Proof:iiiSuppose that iholds. Let Bbe βI-closed. Then XBis βI-open. Since BX, XB. Hence, by assumption we have IntB=XClXBI.

iiiSuppose that iiholds. Let AXbe a non-empty βI-open set. Then XAis a non-empty βI-open set. Hence, by assumption we have XclA=XclXXA=intXAI. Thus, Xis βI-hyperconnected. □

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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 XτIbe an ideal topological space and Abe 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.LetXτIbe an ideal topological space. A pair of subsets, sayAandB, ofXis said to beβI-separated ifclβIAB==AclβB.

Definition 1.13.LetXτIbe an ideal topological space andAbe a subset ofX. ThenAis said to beβI-connected if it cannot be expressed as a union of twoβI-separated sets. A topological spaceXis said to beβI-connected if it isβI-connected as a subset.

Recall, a topological space Xτis said to be a τζ-space if for every pair of subsets Aand Bof X, it is always true that clAB=clAclBand intAB=intAintB. For example, a discrete space is a τζ-space, while an indiscete space is not. Also, if X=abc, then τ=aXis not a τζ-space. Let Xτbe a τζ-space and Ibe an ideal in X. Then we call XτIan ideal τζ-space.

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

Lemma 1.6.LetXτbe a topological space andIbe an ideal inX. IfAisβ-open andBisβI-open withAB=, then they areβI-separated.

Proof:Suppose that Aand Bis not βI-separated, that is clβIABor AclβB. Because AB=, we have ABCand BAC. If Ais β-open, then ACis β-closed. Similarly, if Bis βI-open, then BCis 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.LetXτbe a topology andIbe an ideal inX. IfXisβI-connected, then it is connected.

Proof:Suppose that to the contrary Xis not connected. Let Aand Bbe non-empty disjoint elements of τwith X=AB. Note that Aand Bare β-open and βI-open also. Because A=BCand B=AC, Aand Bare also β-closed and βI-closed. And so, A=clβIAand B=clβB. Thus, clβIAB=AB=and AclβB=AB=. This implies that Xis βI-separated, that is Xis not βI-connected. □

Remark 1.4.LetXτbe a topology andIbe an ideal inX. IfYX, thenIY=YA:AIis an ideal in the relative topologyYτY.

To see this, for the first property, let BIYand AB. Then ABY. Now, if AIY, then there exist CIsuch that YC=A. Note that ABC. Hence, A,BI. Thus, A=YAIY. Next, for the second, let D,EIY. Then DYand EY. if DIY, then there exist FIsuch that YF=D. Similarly, if EIY, then there exist GIsuch that YG=E. Since Iis 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.LetXτIbe an idealτζ-space andYbe a clopen (a set that is open and closed at the same time) set. IfAis aβI-open subset ofXτI, thenAYis aβIY-open set inYτYIY.

Proof:Let Abe a βI-open set in XτI. Then there exists an open set Usuch that UAIand AclintclUI. Let U=UY. Then

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

Moreover, since Xis a τζ-space and Yis clopen

AYclintclU=AYclintclUY=AYclintclUY=AclintclUYIY.E2

This shows that AYis βIY-open in YτYIY. □

Corollary 1.2.LetXτIbe an idealτζ-space andYbe a clopen set. IfAis aβI-closed subset ofXτI, thenAYis aβIY-closed set inYτYIY.

Proof:If Ais βI-closed, then ACis βI-open. By Theorem 1.11, ACYis βIY-open. Hence, AY=ACYCis βIY-closed in Y. □

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

Remark 1.5.LetXτIbe an ideal topological space andYX. ThenIY=AY:AIis a subset ofI.

Proof:If Ais a βIY-open set in Y, then there exists an open set OτYwith OAIYand AclintclOIY. Because τYτand by Remark 1.5, there exists an open set Oτwith OAIand AclintclOI. Thus, Ais 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.LetXτIbe an idealτζ-space,YXbe clopen, andτYτ. IfAY, thenAisβIY-open inYτYIYif and only if it isβI-open inXτI.

Proof:If Ais a βIY-open set in Y, then there exists an open set OτYwith OAIYand AclintclOIY. Because τYτand by Remark 1.5, there exists an open set Oτwith OAIand AclintclOI. Thus, Ais 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.LetXτIbe an idealτζ-space,Ybe clopen, andτYτ. IfAX, thenclβIYAY=clβIAY.

Proof:Since clβIAis a βI-closed set in X, by Lemma 1.8 clβIAYis a βIY-closed set in Y. Hence, clβIYAYclβIYclβIAY=clβIAY. But, clβIYAY=clβIAYclβIAclβIY=clβIAY. Therefore, clβIYAY=clβIAY. □

Definition 1.14.LetXτIbe an ideal topological space, andYτYIYbe a subspace. A pair of subsets, sayAandB, ofXis said to beβIY-separated inYifclβIYAB==AclβYB, whereclβYB=clβBY.

Definition 1.15.LetXτIbe an ideal topological space, andYτYIYbe a subspace. A subsetAofYis said to beβIY-connected if it cannot be expressed as a union of twoβIY-separated sets. The subspaceYis said to beβIY-connected if it isβIY-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.LetXτbe aτζ-space,Ibe an ideal,YXbe clopen, andτYτ. IfAandBareβI-separated inX, then they areβIY-separated inY.

Proof:If Aand Bare βI-separated in X, then by Theorem 1.12 =clβIAB=clβIABY=clβIAYB=clβIYABand =AclβB=AclβBY=AclβBY=AclβYB. Thus, Aand Bare βIY-separated. □

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

Remark 1.6.LetXτbe a topological space andIbe an ideal. IfXisβI-separated (say,X=ABwithA,B, andclβIAB==AclβB), thenAisβ-open whileBisβI-open.

To see this, if Aand Bis βI-separated, then clβIAB=and AclβB=. Hence, AC=clβBand BC=clβIA. Thus, ACis β-closed and BCis βI-closed. Accordingly, Ais β-open and Bis βI-open.

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

Theorem 1.14.LetXτbe a topological space andIbe an idealX. Then,XisβI-connected if and only if it cannot be expressed as a union of two a non-empty disjoint sets in which one is aβ-open set and the other is aβI-open set.

Proof:Suppose that Xis βI-connected, and we can express Xas 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=Xand AB=, then AC=Band BC=A. Since Ais β-open, Bis β-closed. Also, since Bis βI-open, Ais βI-closed. Hence, clβIAB=AB=and AclβB=AB=. Thus, the pair Aand Bis 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.LetXτbe a topological space,Ibe an idealX, andAbe aβI-connected set. IfAHGwhereHandGis a pair ofβI-separated sets, then eitherAHorAG.

Proof:Suppose that to the contrary, A=AHAGwith AHand AG. Since Hand Gis a pair of βI-separated sets, clβIAHAGclβIHG=and AHclβAGHclβG=. Thus, clβIAHAG=and AHclβAG=. Therefore, Acan be expressed as a union of two βI-separated sets AHand AG. A contradiction. □

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

Theorem 1.16.LetXτbe a topological space,Ibe an ideal inX, and,AandBbeβI-separated sets. IfCA(C) andDB(D), thenCandDare alsoβI-separated.

Proof:Suppose that Aand Bare βI-separated. Then clβIAB=and AclβB=. Thus, clβICDclβIAB=and CclβD=AclβB=. Hence, clβICD==CclβD. Therefore, Cand Dis βI-separated. □

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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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Notes

  • ✠ Deceased.

Written By

Glaisa T. Catalan, Michael P. Baldado Jr and Roberto N. Padua

Reviewed: November 5th, 2021Published: January 26th, 2022