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

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.

1. Introduction

Let $\left(X,\tau \right)$ be a topological space. A subset $A$ of $X$ is said to be a $\beta$-open set [1] if $A\subseteq \mathrm{cl}\left(int\left(\mathrm{cl}\left(A\right.\right)\right)$. For example, consider the topology $\left(X,\tau \right)=\left(\left\{a,b,c\right\},\left\{\varnothing ,\left\{a\right\}\right.\right.,$ $\left\{a,b\right\}\left\{a,c\right\},X\left\}\right)$. Then $\varnothing$, $X$, $\left\{a,b\right\}$, $\left\{a,c\right\}$ are the $\beta$-open sets of $\left(X,\tau \right)$. A subset $A$ of $X$ is said to be semi-open set [1] if $A\subseteq \mathrm{cl}\left(int\left(A\right)\right)$. A subset $A$ of $X$ is said to be $\alpha$-open set [2] if $A\subseteq int\left(\mathrm{cl}\left(int\left(A\right)\right)\right)$. A subset $A$ of $X$ is said to be pre-open set [3] if $A\subseteq int\left(\mathrm{cl}\left(A\right)\right)$. A subset $A$ of $X$ is said to be regular-open set [4] if $A=int\left(\mathrm{cl}\left(A\right)\right)$. A subset $A$ of $X$ is said to be ${\beta }^{\ast }$-open set [5] if $A\subseteq \mathrm{cl}\left(int\left(\mathrm{cl}\left(A\right)\right)\right)\cup int\left({\mathrm{cl}}_{\delta }\left(A\right)\right)$ (please see [5] for the notation ${\mathrm{cl}}_{\delta }\left(A\right)$). A subset $A$ of $X$ is said to be $\widehat{\beta }$-generalized-closed set [6] if $\mathrm{cl}\left(int\left(\mathrm{cl}\left(A\right)\right)\right)\subseteq O$ whenever $A\subseteq O$ 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) $A\in I$ and $B\subseteq A$ implies $B\in I$, (2) $A\in I$ and $B\in I$ implies $A\cup B\in I$. Let $\left(X,\tau \right)$ be a topological space and $I$ be an ideal in $X$. Then we call $\left(X,\tau ,I\right)$ an ideal topological space. For example, let $X=\left\{a,b,c\right\}$. Then $I=\left\{\varnothing ,\left\{a\right\}\right\}$ is an ideal on $X$. To see this, we note that the subsets of $\varnothing$ is itself, and the subsets of $\left\{a\right\}$ are $\left\{a\right\}$ and $\varnothing$. Note that all of these subsets are in $I$. Next, we observe that $\varnothing \cup \varnothing =\varnothing \in I$, $\varnothing \cup \left\{a\right\}=\left\{a\right\}\in I$ and $\left\{a\right\}\cup \left\{a\right\}=\left\{a\right\}\in I$. Thus, $I=\left\{\varnothing ,\left\{a\right\}\right\}$ 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 $\alpha$-open sets. Mashhour et al. [3] presented pre-open sets. Abd El-Monsef et al. [7] presented $\beta$-open set. Among these generalization, this study focused on one—the $\beta$-open sets.

Abd El-Monsef et al. [7] also presented the concepts $\beta$-continuous and $\beta$-open mappings. They gave some of their properties. Recently, $\beta$-open sets were investigated by many math enthusiast. For example, Abid [16] utilized $\beta$-open sets to gain some properties of non-semi-predense set. Tahiliani [13] presented an operation involving $\beta$-open sets which paved way to the creation of $\beta$-$\gamma$-open sets. Kannan and Nagaveni [6] generalized $\beta$-open set, and named it $\widehat{\beta }$-generalized closed set. Mubarki et al. [5] also generalized $\beta$-open set, and named it ${\beta }^{\ast }$-open set. El-Mabhouh and Mizyed [17] also generalized $\beta$-open set, and named it $\mathit{\beta c}$-open set. Akdag and Ozkan [8] made an investigation of $\beta$-open sets in soft topological spaces. Arockiarani and Arokia Lancy [9] introduced $\mathit{g\beta }$-closed set and $\mathit{gs\beta }$-closed set (these were defined using $\beta$-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, ${\beta }_I$-compactness, ${\beta }_I^{\ast }$-hyperconnectedness and ${\beta }_I$-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 $\beta$-open sets in some ideal topological spaces.

2. ${\beta }_I$-compactness in ideal spaces

In this section, we gave some important properties of ${\beta }_I$-open sets in ${\tau }_{\omega }$-spaces.

Recall, a topological space $\left(X,\tau \right)$ is said to be a ${\tau }_{\omega }$-space if for every subset $A$ of $X$, it is always true that $int\left(\mathrm{cl}\left(A\right)\right)=int\left(\mathrm{cl}\left(intA\right)\right)$. For example, let $X=\left\{w,x,y,z\right\}$. Then ${\tau }_1=\left\{\varnothing ,\left\{w\right\},\left\{w,x\right\},\left\{w,y\right\},\left\{w,x,y\right\},X\right\}$ is a ${\tau }_{\omega }$-space, while ${\tau }_2=\left\{\varnothing ,\left\{w,x,y\right\},X\right\}$ is not. Also, a discrete space is a ${\tau }_{\omega }$-space, while an indiscete space is not.

Lemma 1.1 characterizes $\beta$-open sets in a ${\tau }_{\omega }$-space.

Lemma 1.1. Let $\left(X,\tau \right)$ be a ${\tau }_{\omega }$-space and $I$ be an ideal in $X$. A set $A$ $\left(\subseteq X\right)$ is a $\beta$-open set precisely if there is a set $O\left(\in \tau \right)$ with the property that $O\subseteq A\subseteq \mathrm{cl}\left(int\left(\mathrm{cl}\left(O\right)\right)\right)$.

Proof: Suppose that $A$ is a $\beta$-open set. Then $A\subseteq \mathrm{cl}\left(int\left(\mathrm{cl}\left(A\right)\right)\right)$. Consider $O=int\left(A\right)$ (note that $O$ is open). Since $\left(X,\tau \right)$ is a ${\tau }_{\omega }$-space, $int\left(\mathrm{cl}\left(A\right)\right)=int\left(\mathrm{cl}\left(intA\right)\right)$. Hence, $O\subseteq A\subseteq \mathrm{cl}\left(int\left(\mathrm{cl}\left(A\right)\right)\right)=\mathrm{cl}\left(int\left(\mathrm{cl}\left(int\left(A\right)\right)\right)\right)=\mathrm{cl}\left(int\left(\mathrm{cl}\left(O\right)\right)\right)$.

Conversely, suppose that there is a set $O\left(\in \tau \right)$ with the property that $O\subseteq A\subseteq \mathrm{cl}\left(int\left(\mathrm{cl}\left(O\right)\right)\right)$. Since $O\subseteq A$, we have $\mathrm{cl}\left(O\right)\subseteq \mathrm{cl}\left(A\right)$. And so, $int\left(\mathrm{cl}\left(O\right)\right)\subseteq int\left(\mathrm{cl}\left(A\right)\right)$. Therefore, $\mathrm{cl}\left(int\left(\mathrm{cl}\left(O\right)\right)\right)\subseteq \mathrm{cl}\left(int\left(\mathrm{cl}\left(A\right)\right)\right)$ Thus, $A\subseteq \mathrm{cl}\left(int\left(\mathrm{cl}\left(A\right)\right)\right)$.

□

Next, we define ${\beta }_I$-open set.

Definition 1.1. Let $\left(X,\tau \right)$ be a topological space and $I$ be an ideal in $X$. A subset $A$ of $X$ is called $\beta$-open with respect to the ideal $I$, or a ${\beta }_I$-open set, if there exists an open set $O$ such that $\left(1\right)$ $O-A\in I$, and $\left(2\right)$ $A-\mathit{cl}\left(\mathit{int}\left(\mathit{cl}\left(O\right)\right)\right)\in I$.

For example, let $X=\left\{a,b,c\right\}$, $\tau =\left\{\varnothing ,\left\{a\right\},\left\{b,c\right\},X\right\}$, and $I=\left\{\varnothing ,\left\{b\right\}\right\}$ (note that $\tau$ is a topology on $X$, and $I$ is an ideal on $X$). Then $A=\left\{b,c\right\}$ is a ${\beta }_I$-open set. To see this, consider $O=\left\{b,c\right\}$. Then $O$ is a open set. Observe that $O-A=\left\{b,c\right\}-\left\{b,c\right\}=\varnothing \in I$, and $A-\mathrm{cl}\left(int\left(\mathrm{cl}\left(O\right)\right)\right)=\left\{b,c\right\}-\mathrm{cl}\left(int\left(\mathrm{cl}\left(\left\{b,c\right\}\right)\right)\right)=\left\{b,c\right\}-\mathrm{cl}\left(int\left(\left\{b,c\right\}\right)\right)=\left\{b,c\right\}-\mathrm{cl}\left(\left\{b,c\right\}\right)=\left\{b,c\right\}-\left\{b,c\right\}=\varnothing \in I$. Thus, $A=\left\{b,c\right\}$ is $\beta$-open with respect to the ideal $I$.

Lemma 1.2 says that an open set is a ${\beta }_I$-open set, and an element of the ideal is a ${\beta }_I$-open set. One may see [24] to gain more insights relative to these ideas. While, Lemma 1.3 says that in a ${\tau }_{\omega }$-space a $\beta$-open set is also a ${\beta }_I$-open set.

Lemma 1.2. Let $\left(X,\tau \right)$ be a topological space and $I$ be an ideal in $X$. Then the following statements are true.

i. Every open set is a ${\beta }_I$-open set.

ii. Every element of $I$ is a ${\beta }_I$-open set.

Proof: $\left(i\right)$ Let $A$ be an open set. Note that $A-A=\varnothing \in I$, and $A-\mathrm{cl}\left(int\left(\mathrm{cl}\left(A\right)\right)\right)\subseteq A-\mathrm{cl}\left(A\right)=\varnothing \in I$. Thus, $A$ is ${\beta }_I$-open. $\left(\mathit{ii}\right)$ Let $A\in I$. Consider $O=\varnothing$. Note that $O-A=\varnothing -A=\varnothing \in I$, and $A-\mathrm{cl}\left(int\left(\mathrm{cl}\left(O\right)\right)\right)=A-\varnothing =A\in I$. Thus, $A$ is ${\beta }_I$-open. □

Lemma 1.3. Let $\left(X,\tau \right)$ be a ${\tau }_{\omega }$-space and $I$ be an ideal in $X$. Then every $\beta$-open set is a ${\beta }_I$-open set.

Proof: Let $A$ be a $\beta$-open set. By Lemma 1.1 there exists an open set $O$ such that $O\subseteq A\subseteq \mathrm{cl}\left(int\left(\mathrm{cl}\left(O\right)\right)\right)$. Hence $O-A=\varnothing \in I$, and $A-\mathrm{cl}\left(int\left(\mathrm{cl}\left(O\right)\right)\right)=\varnothing \in I$. Thus, $A$ is ${\beta }_I$-open. □

Let $\left(X,\tau \right)$ be a topology and $I$ be an ideal in $X$. We say that $I$ is countably additive if $\cup \left\{A_i:i\in \mathbb{N}\right\}\in I$ whenever $\left\{A_i:i\in \mathbb{N}\right\}$ is a (countable) family of elements of $I$.

Lemma 1.4 says that in a ${\tau }_{\omega }$-space, if $I$ is the minimal ideal, then the ${\beta }_I$-open sets are precisely the $\beta$-open sets.

Lemma 1.4. Let $\left(X,\tau \right)$ be a ${\tau }_{\omega }$-space and $I$ be an ideal in $X$. If $I$ is not countably additive, then the following statements are equivalent.

1. $I=\left\{\varnothing \right\}$.

2. $A$ is a $\beta$-open set precisely when $A$ is a ${\beta }_I$-open set.

Proof: $\left(i\right)\Rightarrow \left(\mathit{ii}\right)$ Suppose that $I=\left\{\varnothing \right\}$. Let $A$ be a $\beta$-open set. By Lemma 1.3, $A$ is a ${\beta }_I$-open set. For the converse, let $A$ be a ${\beta }_I$-open set and $O$ be an open set with $O-A\in I$ and $A-\mathrm{cl}\left(int\left(\mathrm{cl}\left(O\right)\right)\right)\in I$. Because $I=\left\{\varnothing \right\}$, we have $O-A=\varnothing$ and $A-\mathrm{cl}\left(int\left(\mathrm{cl}\left(O\right)\right)\right)=\varnothing$. Hence, $O\subseteq A$ and $A\subseteq \mathrm{cl}\left(int\left(\mathrm{cl}\left(O\right)\right)\right)$, that is $O\subseteq A\subseteq \mathrm{cl}$ $\left(int\left(\mathrm{cl}\right.\right.\left(O\right.)))$. Therefore, by Lemma 1.1 $A$ is $\beta$-open.

$\left(\mathit{ii}\right)\Rightarrow \left(i\right)$ Suppose that $\left(\mathit{ii}\right)$ holds, and that $I\ne \left\{\varnothing \right\}$. Let $D$ be a non-empty element of $I$. By Lemma 1.2, $D$ is ${\beta }_I$-open. Thus, by assumption $D$ is $\beta$-open. Now, by Lemma 1.1, there exists $O_1\in \tau$ with $O_1\subseteq D\subseteq \mathrm{cl}\left(int\left(\mathrm{cl}\left(O_1\right)\right)\right)$. Since $D$ is an element of $I$ and $O_1\subseteq D$, we have $O_1\in I$. Hence, $O_1\cup D\in I$. By Lemma 1.1, $O_1\cup D$ is a ${\beta }_I$-open set. Hence, by assumption $O_1\cup D$ is a $\beta$-open set. And so, again there exists $O_2\in \tau$ with $O_2\subseteq \left(O_1\cup D\right)\subseteq \mathrm{cl}\left(int\left(\mathrm{cl}\left(O_2\right)\right)\right)$. Since $O_1\cup D\in I$ and $O_2\subseteq O_1\cup D$, we have $O_2\in I$. Hence, $O_1\cup O_2\cup D\in I$. Thus, by Lemma 1.1, $O_1\cup O_2\cup D$ is a ${\beta }_I$-open set. By assumption $O_1\cup O_2\cup D$ is a $\beta$-open set. And so, again there exists $O_3\in \tau$ with $O_3\subseteq \left(O_1\cup O_2\cup D\right)\subseteq \mathrm{cl}\left(int\left(\mathrm{cl}\left(O_3\right)\right)\right)$. Since $O_1\cup O_2\cup D\in I$ and $O_3\subseteq O_1\cup O_2\cup D$, we have $O_3\in I$. Hence, $O_1\cup O_2\cup O_3\cup D\in I$. Continuing in this fashion we obtain a countably infinite subset $\left\{O_1,O_2,O_3,\dots \right\}$ of $I$ with $O_1\cup O_2\cup O_3\cup \cdots \in I$. This is a contradiction since $I$ is not countably additive. Thus, $I=\left\{\varnothing \right\}$. □

Next, we define ${\beta }_I$-compact set, ${\beta }_I$-compact space, compatible ${\beta }_I$-compact set, and compatible ${\beta }_I$-compact space.

Definition 1.2. Let $\left(X,\tau ,I\right)$ be an ideal topological space. A subset $A$ of $X$ is said to be ${\beta }_I$-compact if for every cover $\left\{O_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}$ of $A$ by ${\beta }_I$-open sets, $\mathrm{\Lambda }$ has a finite subset ${\mathrm{\Lambda }}_0$, such that $\left\{O_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}$ still covers $A$. A space $X$ is said to be a ${\beta }_I$-compact space if it is ${\beta }_I$-compact as a subset.

Definition 1.3. Let $\left(X,\tau ,I\right)$ be an ideal topological space. A subset $A$ of $X$ is said to be countably ${\beta }_I$-compact if for every countable cover $\left\{O_n:n\in \mathbb{N}\right\}$ of $A$ by ${\beta }_I$-open sets, $\mathbb{N}$ has a finite subset $\left\{i_j:j=1,2,\dots ,k\right\}$ with the property that $\left\{O_{i_j}:j=1,2,\dots ,k\right\}$ still covers $A$. A space $X$ is said to be a countably ${\beta }_I$-compact space if it is countably ${\beta }_I$-compact as a subset.

Definition 1.4. Let $\left(X,\tau ,I\right)$ be an ideal topological space. A subset $A$ of $X$ is said to be compatible ${\beta }_I$-compact, or simply $c{\beta }_I$-compact, if for every cover $\left\{O_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}$ of $A$ by $\beta$-open sets, $\mathrm{\Lambda }$ has a finite subset ${\mathrm{\Lambda }}_0$, such that $A-\cup \left\{O_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}\in I$. An ideal topological space $\left(X,\tau ,I\right)$ is said to be $c{\beta }_I$-compact space if it is $c{\beta }_I$-compact as a subset.

Theorem 1.1 says that in an ideal ${\tau }_{\omega }$-space in which $I$ is the minimal ideal, the notions $\beta$-compact, ${\beta }_I$-compact and $c{\beta }_I$-compact coincides.

Theorem 1.1. Let $\left(X,\tau \right)$ be a ${\tau }_{\omega }$-space and $I=\left\{\varnothing \right\}$. Then the following statements are equivalent.

1. $\left(X,\tau ,I\right)$ is a $\beta$-compact space.

2. $\left(X,\tau ,I\right)$ is a ${\beta }_I$-compact space.

3. $\left(X,\tau ,I\right)$ is a $c{\beta }_I$-compact space.

Proof: $\left(i\right)\Rightarrow \left(\mathit{ii}\right)$ Suppose that $\left(i\right)$ holds. Let $\left\{O_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}$ be a family of $\beta$-open sets that covers $X$. By assumption, $\mathrm{\Lambda }$ has a finite subset, say ${\mathrm{\Lambda }}_0$, with the property that $\left\{O_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}$ still covers $X$. By Lemma 1.3 $\left(\mathit{iii}\right)$, $\left\{O_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}$ is also a family of ${\beta }_I$-open sets. Hence, $\left\{O_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}$ is a finite covering of $X$ by ${\beta }_I$-open sets. Therefore, $X$ is ${\beta }_I$ compact.

$\left(\mathit{ii}\right)\Rightarrow \left(\mathit{iii}\right)$ Suppose that $\left(\mathit{ii}\right)$ holds. Let $\left\{O_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}$ be a family of $\beta$-open sets that covers $X$. Since $I=\left\{\varnothing \right\}$, by Lemma 1.4 $\left\{O_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}$ is also a family of ${\beta }_I$-open sets that covers $X$. By assumption, $\mathrm{\Lambda }$ has a finite subset, say ${\mathrm{\Lambda }}_0$, with the property that $\left\{O_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}$ still covers $X$. Note that $\left\{U_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}$ is also a family of $\beta$-open sets, and $X-\underset{\lambda \in {\mathrm{\Lambda }}_0}{\cup }{O}_{\lambda }=\varnothing \in I$. Therefore, $X$ is $c{\beta }_I$ compact.

$\left(\mathit{iii}\right)\Rightarrow \left(i\right)$ Suppose that $\left(\mathit{iii}\right)$ holds. Let $\left\{O_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}$ be a family of $\beta$-open sets that covers $X$. By assumption, $\mathrm{\Lambda }$ has a finite subset, say ${\mathrm{\Lambda }}_0$, with the property that $X-\underset{\lambda \in {\mathrm{\Lambda }}_0}{\cup }{O}_{\lambda }\in I$. Since $I=\left\{\varnothing \right\}$, $X-\underset{\lambda \in {\mathrm{\Lambda }}_0}{\cup }{O}_{\lambda }=\varnothing$, that is $X\subseteq \underset{\lambda \in {\mathrm{\Lambda }}_0}{\cup }{O}_{\lambda }$. Hence, $\left\{O_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}$ is covering of $X$. Therefore, $X$ is $\beta$ compact. □

Theorem 1.2 presents a characterization of ${\beta }_I$-compact spaces.

Theorem 1.2. Let $\left(X,\tau ,I\right)$ be an ideal topological space. Then the following are equivalent.

1. $\left(X,\tau ,I\right)$ is ${\beta }_I$-compact.

2. If $\left\{F_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}$ is a family of ${\beta }_I$-closed sets with $\cap \left\{F_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}=\varnothing$, then $\mathrm{\Lambda }$ has a finite subset, say ${\mathrm{\Lambda }}_0$, with the property $\cap \left\{F_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}=\varnothing$.

Proof: $\left(i\right)\Rightarrow \left(\mathit{ii}\right)$ Suppose that $\left(i\right)$ holds. Let $\left\{F_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}$ be a family of ${\beta }_I$-closed sets with the property $\cap \left\{F_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}=\varnothing$. Then $\cup \left\{F_{\lambda }^C:\lambda \in \mathrm{\Lambda }\right\}={\left(\cap \left\{F_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}\right)}^C=X$. Hence, $\left\{F_{\lambda }^C:\lambda \in \mathrm{\Lambda }\right\}$ is a family of ${\beta }_I$-open sets which covers of $X$. By assumption, $\mathrm{\Lambda }$ has a finite subset, say ${\mathrm{\Lambda }}_0$, with the property $\cup \left\{F_{\lambda }^C:\lambda \in {\mathrm{\Lambda }}_0\right\}=X$, i.e. $\cap \left\{F_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}=\varnothing$.

$\left(\mathit{ii}\right)\Rightarrow \left(i\right)$ Suppose that $\left(\mathit{ii}\right)$ holds. Let $\left\{O_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}$ be a family of ${\beta }_I$-open sets that covers $X$, i.e. $\cup \left\{U_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}=X$. Then $\cap \left\{O_{\lambda }^C:\lambda \in \mathrm{\Lambda }\right\}={\left(\cup \left\{U_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}\right)}^C=\varnothing$. Note that $O^C$ is ${\beta }_I$-closed since $O$ is ${\beta }_I$-open. By assumption, $\mathrm{\Lambda }$ has a finite subset, say ${\mathrm{\Lambda }}_0$, with the property that $\cap \left\{O_{\lambda }^C:\lambda \in {\mathrm{\Lambda }}_0\right\}=\varnothing$. Thus, $\cup \left\{O_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}={\left(\cap \left\{O_{\lambda }^C:\lambda \in {\mathrm{\Lambda }}_0\right\}\right)}^C=X$, that is $\left\{O_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}$ is a family of ${\beta }_I$-open sets that covers $X$. □

Theorem 1.3 presents a characterization of ${\beta }_I$-compact spaces.

Theorem 1.3. Let $\left(X,\tau \right)$ be a topological space and $I$ be an ideal in $X$. Then the following are equivalent.

1. $\left(X,\tau ,I\right)$ is $c{\beta }_I$-compact.

2. If $\left\{F_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}$ is a family of $\beta$-closed sets with $\cap \left\{F_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}=\varnothing$, then $\mathrm{\Lambda }$ has a finite subset, say ${\mathrm{\Lambda }}_0$, with the property that $\cap \left\{F_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}\in I$.

Proof: $\left(i\right)\Rightarrow \left(\mathit{ii}\right)$ Suppose that $\left(i\right)$ holds. Let $\left\{F_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}$ be a family of $\beta$-closed sets such that $\cap \left\{F_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}=\varnothing$. Note that $\cup \left\{F_{\lambda }^C:\lambda \in \mathrm{\Lambda }\right\}={\left(\cap \left\{F_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}\right)}^C=X$. Hence, $\left\{F_{\lambda }^C:\lambda \in \mathrm{\Lambda }\right\}$ is a family of $\beta$-open sets that covers $X$. By assumption, $\mathrm{\Lambda }$ has a finite subset, say ${\mathrm{\Lambda }}_0$, with the property $X-\cup \left\{F_{\lambda }^C:\lambda \in {\mathrm{\Lambda }}_0\right\}\in I$, i.e. $\cap \left\{F_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}\in I$.

$\left(\mathit{ii}\right)\Rightarrow \left(i\right)$ Suppose that $\left(\mathit{ii}\right)$ holds. Let $\left\{O_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}$ be a family of $\beta$-open sets that covers $X$, i.e. $\cup \left\{O_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}=X$. Note that $\cap \left\{O_{\lambda }^C:\lambda \in \mathrm{\Lambda }\right\}={\left(\cup \left\{O_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}\right)}^C=\varnothing$. By assumption, $\mathrm{\Lambda }$ has a finite subset, say ${\mathrm{\Lambda }}_0$, with the property $\cap \left\{O_{\lambda }^C:\lambda \in {\mathrm{\Lambda }}_0\right\}\in I$, i.e. $X-\cup \left\{O_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}\in I$.□

Remark 1.1. [25] Let $\left(X,\tau ,I\right)$ and $\left(Y,\sigma ,J\right)$ be ideal spaces, and $f:X\to Y$ be a fuction. Then:

1. $f\left(I\right)=\left\{f\left(A\right):A\in I\right\}$ is an ideal in $Y$, where $f\left(A\right)=\left\{f\left(a\right):a\in A\right\}$; And,

2. if $f$ is bijective, then $f^{-1}\left(J\right)=\left\{f^{-1}\left(B\right):B\in J\right\}$ is an ideal in $X$, where $f^{-1}\left(B\right)=\left\{f^{-1}\left(b\right):b\in B\right\}$.

Next, we define ${\beta }_I$-open, ${\beta }_I$-irresolute, and ${\beta }_I$-continuous functions.

Definition 1.5. Let $\left(X,\tau ,I\right)$ and $\left(Y,\sigma ,J\right)$ be ideal topological spaces. A function $f:X\to Y$ is said to be

1. $\beta$-open if $f\left(A\right)$ is $\beta$-open for every $\beta$-open set $A$,

2. $\beta$-irresolute if $f^{-1}\left(B\right)$ is $\beta$-open for every $\beta$-open set $B$, and

3. $\beta$-continuous if $f^{-1}\left(B\right)$ is $\beta$-open for every open set $B$.

4. ${\beta }_I$-open if $f\left(A\right)$ is ${\beta }_J$-open for every ${\beta }_I$-open set $A$,

5. ${\beta }_I$-irresolute if $f^{-1}\left(B\right)$ is ${\beta }_I$-open for every ${\beta }_J$-open set $B$, and

6. ${\beta }_I$-continuous if $f^{-1}\left(B\right)$ is ${\beta }_I$-open for every open set $B$.

Theorem 1.4 says that given a $\beta$-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 $\left(X,\tau \right)$ and $\left(Y,\sigma \right)$ be topological spaces, $I$ be an ideal in $X$, and $f:X\to Y$ be a $\beta$-irresolute function. If $X$ is a $c{\beta }_I$-compact space, then $\left(f\left(X\right),\left\{f\left(X\right)\cap B:B\in \sigma \right\},f\left(I\right)\right)$ is a $c{\beta }_{f\left(I\right)}$-compact space.

Proof: Let $\left\{O_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}$ be a family of $\beta$-open sets that covers $f\left(X\right)$. Beacuse $f$ is a $\beta$-irresolute, $\left\{f^{-1}\left(O_{\lambda }\right):\lambda \in \mathrm{\Lambda }\right\}$ is a family of by $\beta$-open sets that covers $X$. Because $X$ is $c{\beta }_I$-compact, $\mathrm{\Lambda }$ has a finite subset, say ${\mathrm{\Lambda }}_0$, with the property $X-\cup \left\{f^{-1}\left(O_{\lambda }\right):\lambda \in {\mathrm{\Lambda }}_0\right\}\in I$. Hence, by Remark 1.1 $f\left(X\right)-\cup \left\{O_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}=f\left(X-\cup \left\{f^{-1}\left(O_{\lambda }\right):\lambda \in {\mathrm{\Lambda }}_0\right\}\right)\in f\left(I\right)$.□

Theorem 1.5. Let $\left(X,\tau \right)$ and $\left(Y,\sigma \right)$ be topological spaces, $J$ be an ideal in $Y$, and $f:X\to Y$ be a $\beta$-open surjection (surjective function). If $Y$ is $c{\beta }_J$-compact, then $X$ is $c{\beta }_{f^{-1}\left(J\right)}$-compact.

Proof: Let $\left\{O_{\lambda }:\lambda \in \mathrm{\Lambda }\right\}$ be a family of $\beta$-open sets that covers $X$. Beacuse $f$ is a $\beta$-open surjection, $\left\{f\left(O_{\lambda }\right):\lambda \in \mathrm{\Lambda }\right\}$ is a family $\beta$-open sets that covers $Y$. Because $Y$ is $c{\beta }_J$-compact, $\mathrm{\Lambda }$ has a finite subset, say ${\mathrm{\Lambda }}_0$, with the property $Y-\cup \left\{f\left(O_{\lambda }\right):\lambda \in {\mathrm{\Lambda }}_0\right\}$$\in J$. Hence, $X-\cup \left\{O_{\lambda }:\lambda \in {\mathrm{\Lambda }}_0\right\}=f^{-1}\left(Y-\cup \left\{f\left(O_{\lambda }\right):\lambda \in {\mathrm{\Lambda }}_0\right\}\right)\in f^{-1}\left(J\right)$.□

The next theorem says that in a ${\tau }_{\omega }$-space and when $I=\{\varnothing$}, the family of all countably ${\beta }_I$-compact space contains all $c{\beta }_I$-compact space.

Theorem 1.6. Let $\left(X,\tau ,I\right)$ be an ideal ${\tau }_{\omega }$-space and $I=\varnothing$. If $X$ is $c{\beta }_I$-compact, then it is also countably ${\beta }_I$-compact.

Proof: Let $\left\{O_n:n\in \mathbb{N}\right\}$ be a countable family ${\beta }_I$-open sets that covers $X$. Because $X$ is $c{\beta }_I$-compact, $\mathbb{N}$ has a finite subset $\left\{i_j:j=1,2,\dots ,k\right\}$ with the property that $X-\cup \left\{O_{i_j}:j=1,2,\dots ,k\right\}\in I$. Because $I=\left\{\varnothing \right\}$, $X=\cup \left\{O_{i_j}:j=1,2,\dots ,k\right\}\in I$. By Lemma 1.4 $\left\{O_{i_j}:j=1,2,\dots ,k\right\}$ is also a family of $\beta$-open sets. Hence, $\left\{O_{i_j}:j=1,2,\dots ,k\right\}$ is a finite subcover of $X$ by $\beta$-open sets. □

3. ${\beta }_I^{\ast }$-hyperconnectedness in ideal spaces

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

Definition 1.6. Let $\left(X,\tau \right)$ be a topological space and $I$ be an ideal on $X$. A function ${\left(\right)}^{\ast }\left(I,\tau \right):P\left(X\right)\to P\left(X\right)$ given by $A^{\ast }\left(I,\tau \right)=\left\{x\in X:A\cup U\notin I\mathit{\text{for\hspace{0.17em}every}}U\in \tau \left(x\right)\right\}$ where $\tau \left(x\right)=\left\{U\in \tau :x\in U\right\}$ is called a local of $A$ with respect to $\tau$ and $I$.

Example 1.1. Let $X=\left\{a,b,c\right\}$, $\tau =\left\{\varnothing ,\left\{a\right\},\left\{b\right\},\left\{c\right\},\left\{a,b\right\},\left\{a,c\right\},\left\{b,c\right\},X\right\}$, and $I=\left\{\varnothing ,\left\{a\right\},\left\{b\right\},\left\{a,b\right\}\right\}$ (note that $\tau$ is a topology on $X$ and $I$ is an ideal on $X$). Then, ${\varnothing }^{\ast }=\varnothing$, ${\left\{a\right\}}^{\ast }=\left\{c\right\}$, ${\left\{b\right\}}^{\ast }=\left\{c\right\}$, ${\left\{c\right\}}^{\ast }=X$, ${\left\{a,b\right\}}^{\ast }=\left\{c\right\}$, ${\left\{a,c\right\}}^{\ast }=X$, ${\left\{b,c\right\}}^{\ast }=X$ and $X^{\ast }=X$.

Definition 1.7. Let $\left(X,\tau \right)$ be a topological space and $I$ be an ideal on $X$. The Kuratowski closure operator $\mathit{Cl}{\left(\right)}^{\ast }\left(I,\tau \right):P\left(X\right)\to P\left(X\right)$ for the topology ${\tau }^{\ast }\left(I,\tau \right)$ is given by $\mathit{Cl}{\left(A\right)}^{\ast }\left(I,\tau \right)=A\cup A^{\ast }$.

Example 1.2. Consider the ideal space of Example 3. Then we have, $\mathit{Cl}{\left(\varnothing \right)}^{\ast }=\varnothing \cup {\varnothing }^{\ast }=\varnothing \cup \varnothing =\varnothing$, $\mathit{Cl}{\left(\left\{a\right\}\right)}^{\ast }=\left\{a\right\}\cup {\left\{a\right\}}^{\ast }=\left\{a\right\}\cup \left\{c\right\}=\left\{a,c\right\}$, $\mathit{Cl}{\left(\left\{b\right\}\right)}^{\ast }=\left\{b\right\}\cup {\left\{b\right\}}^{\ast }=\left\{b\right\}\cup \left\{c\right\}=\left\{b,c\right\}$, $\mathit{Cl}{\left(\left\{c\right\}\right)}^{\ast }=\left\{c\right\}\cup {\left\{c\right\}}^{\ast }=\left\{c\right\}\cup X=X$, $\mathit{Cl}{\left(\left\{a,b\right\}\right)}^{\ast }=\left\{a,b\right\}\cup {\left\{a,b\right\}}^{\ast }=\left\{a,b\right\}\cup \left\{c\right\}=X$, $\mathit{Cl}{\left(\left\{a,c\right\}\right)}^{\ast }=\left\{a,c\right\}\cup {\left\{a,c\right\}}^{\ast }=\left\{a,c\right\}\cup X=X$, $\mathit{Cl}{\left(\left\{b,c\right\}\right)}^{\ast }=\left\{b,c\right\}\cup {\left\{b,c\right\}}^{\ast }=\left\{b,c\right\}\cup X=X$, and $\mathit{Cl}{\left(X\right)}^{\ast }=X\cup X^{\ast }=X\cup X=X$.

Definition 1.8. Let $\left(X,\tau \right)$ be a topological space and $I$ be an ideal on $X$. The Kuratowski interior operator $\mathit{Int}{\left(\right)}^{\ast }\left(I,\tau \right):P\left(X\right)\to P\left(X\right)$ for the topology ${\tau }^{\ast }\left(I,\tau \right)$ is given by $\mathit{Int}{\left(A\right)}^{\ast }\left(I,\tau \right)=X-\mathit{Cl}{\left(X-A\right)}^{\ast }$.

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

Definition 1.9. [26] An ideal space $\left(X,\tau ,I\right)$ is called $\ast$-hyperconnected if ${\mathit{cl}}^{\ast }\left(A\right)=X$ for all non-empty open set $A$ $\subseteq X$.

Definition 1.10. [29] An ideal space $\left(X,\tau ,I\right)$ is called $I^{\ast }$-hyperconnected if $X-{\mathit{cl}}^{\ast }\left(A\right)\in I$ for all non-empty open set $A$ $\subseteq X$.

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

Definition 1.11. An ideal topological space $\left(X,\tau ,I\right)$ is said to be ${\beta }_I^{\ast }$-hyperconnected space if $X-{\mathrm{cl}}^{\ast }\left(A\right)\in I$ for every non-empty ${\beta }_I$-open subset $A$ of $X$.

The next theorem says that the family of all ${\beta }_I\ast$-hyperconnected space contains all $I\ast$-hyperconnected space.

Theorem 1.7. Let $\left(X,\tau \right)$ be a topological space, and $I$ be an ideal in $X$. If $X$ is $I\ast$-hyperconnected, then it is ${\beta }_I^{\ast }$-hyperconnected also.

Proof: Let $X$ be $I\ast$-hyperconnected, and $A$ be a non-empty open set. Because $X$ is $I_{\ast }$-hyperconnected, we have $X-\mathit{cl}{\left(A\right)}^{\ast }\in I$ for all non-empty open set $A$ $\subseteq X$. And, because an open set is also a ${\beta }_I$-open set, we have $X-\mathit{cl}{\left(A\right)}^{\ast }\in I$ for all non-empty ${\beta }_I$-open set $A$ $\subseteq X$. Hence, $X$ is ${\beta }_I^{\ast }$-hyperconnected. □

The next lemma is clear.

Lemma 1.5. Let $\left(X,\tau \right)$ 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 $\ast$-hyperconnected and $I^{\ast }$-hyperconnected are equivalent.

Theorem 1.8. [29] Let $\left(X,\tau \right)$ be a topological space, and $I=\left\{\varnothing \right\}$. Then, $X$ is $\ast$-hyperconnected if and only if it is $I^{\ast }$-hyperconnected.

The next remark is clear.

Remark 1.2. If $\left(X,\tau \right)$ 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 $\beta$-open.

To see this, let $A$ be an open set. Since $\tau$ is clopen, $A$ is closed also. Hence, $\mathrm{cl}\left(int\left(\mathrm{cl}\left(A\right)\right)\right)=A$. Thus, $A$ is a $\beta$-open set. Conversely, if $A$ is a $\beta$-open set, then $A\subseteq \mathrm{cl}\left(int\left(\mathrm{cl}\left(A\right)\right)\right)$. This implies that $A$ must be open.

Theorem 1.9 says that in a clopen ${\tau }_{\omega }$-space, with respect to the minimal ideal $I$, the notions ${\beta }_I^{\ast }$-hyperconnected and $I^{\ast }$-hyperconnected are equivalent.

Theorem 1.9. Let $\left(X,\tau \right)$ be a clopen ${\tau }_{\omega }$-space, and $I=\left\{\varnothing \right\}$. Then, $X$ is $I^{\ast }$-hyperconnected if and only if it is ${\beta }_I^{\ast }$-hyperconnected.

Proof: Suppose that $X$ is $I^{\ast }$-hyperconnected. Let $A$ is a non-empty element of $\tau$. Then $X-{\mathit{cl}}^{\ast }\left(A\right)\in I$. By Remark 1.2 and Lemma 1.5, every open set is absolutely ${\beta }_I$-open. Thus, $X-{\mathit{cl}}^{\ast }\left(A\right)\in I$ for all ${\beta }_I$-open set $A\left(\ne \varnothing \right)$. Therefore, $X$ is ${\beta }_I^{\ast }$-hyperconnected also. Conversely, suppose that $X$ is ${\beta }_I^{\ast }$-hyperconnected. Let $A$ be a non-empty ${\beta }_I$-open set. Then $X-{\mathit{cl}}^{\ast }\left(A\right)\in I$. By Remark 1.2 and Lemma 1.5, every ${\beta }_I$-open set is absolutely open. Thus, $X-\mathit{cl}{\left(A\right)}^{\ast }\in I$ for all open set $A\left(\ne \varnothing \right)$. Therefore, $X$ is $I^{\ast }$-hyperconnected also. □

Corollary 1.1 says that in a clopen ${\tau }_{\omega }$-space, relative to the minimal ideal $I$, the notions ${\beta }_I^{\ast }$-hyperconnected, $I^{\ast }$-hyperconnected, and $\ast$-hyperconnected are equivalent.

Corollary 1.1. Let $\left(X,\tau \right)$ be a clopen ${\tau }_{\omega }$-space and $I=\left\{\varnothing \right\}$. Then the following statements are equivalent.

1. $X$ is $I^{\ast }$-hyperconnected.

2. $X$ is ${\beta }_I^{\ast }$-hyperconnected.

3. $X$ is ${\beta }_I^{\ast }$-hyperconnected.

Theorem 1.3 may be an important property.

Remark 1.3. If an ideal ${\tau }_{\omega }$ space $\left(X,\tau ,\left\{\varnothing \right\}\right)$ is a ${\beta }_I\ast$-hyperconnected space, then $X-{\mathrm{cl}}^{\ast }\left(A\right)\in I$ for every non-empty $\beta$-open subset $A$ of $X$.

To see this, let $A$ is a non-empty $\beta$-open set. By Lemma 1.4 $A$ is ${\beta }_I$-open. Since $X$ is ${\beta }_I^{\ast }$-hyperconnected, $X-{\mathit{cl}}^{\ast }\left(A\right)\in I$.

Theorem 1.10 is a characterization of ${\beta }_I^{\ast }$-hyperconnected space.

Theorem 1.10. Let $\left(X,\tau \right)$ be an topological space and $I$ be an ideal in $X$. Then the following statements are equivalent.

1. $X$ is a ${\beta }_I^{\ast }$-hyperconnected space.

2. $\mathrm{Int}{\left(A\right)}^{\ast }\in I$ for all proper ${\beta }_I$-closed subset $A$ of $X$.

Proof: $\left(i\right)\Rightarrow \left(\mathit{ii}\right)$ Suppose that $\left(i\right)$ holds. Let $B$ be ${\beta }_I$-closed. Then $X-B$ is ${\beta }_I$-open. Since $B\ne X$, $X-B\ne \varnothing$. Hence, by assumption we have $\mathrm{Int}{\left(B\right)}^{\ast }=X-\mathrm{Cl}{\left(X-B\right)}^{\ast }\in I$.

$\left(\mathit{ii}\right)\Rightarrow \left(i\right)$ Suppose that $\left(\mathit{ii}\right)$ holds. Let $A$ $\left(\ne X\right)$ be a non-empty ${\beta }_I$-open set. Then $X-A$ is a non-empty ${\beta }_I$-open set. Hence, by assumption we have $X-\mathrm{cl}{\left(A\right)}^{\ast }=X-\mathrm{cl}{\left(X-\left(X-A\right)\right)}^{\ast }=int{\left(X-A\right)}^{\ast }\in I$. Thus, $X$ is ${\beta }_I^{\ast }$-hyperconnected. □

4. ${\beta }_I$-separatedness in ideal spaces

In this section, we present the concepts ${\beta }_I$-separated sets and ${\beta }_I$-connected sets. We also present some of their important properties.

Let $\left(X,\tau ,I\right)$ be an ideal topological space and $A$ be a subset of $X$. The $\beta$-closure of $A$, denoted by ${\mathrm{cl}}_{\beta }\left(A\right)$, is the smallest $\beta$-closed set that contains $A$. The ${\beta }_I$-closure of $A$, denoted by ${\mathrm{cl}}_{{\beta }_I}\left(A\right)$, is the smallest ${\beta }_I$-closed set that contains $A$.

Next, we define ${\beta }_I$-separated set, ${\beta }_I$-connected set, and ${\beta }_I$-connected space.

Definition 1.12. Let $\left(X,\tau ,I\right)$ be an ideal topological space. A pair of subsets, say $A$ and $B$, of $X$ is said to be ${\beta }_I$-separated if ${\mathit{cl}}_{{\beta }_I}\left(A\right)\cap B=\varnothing =A\cap {\mathit{cl}}_{\beta }\left(B\right)$.

Definition 1.13. Let $\left(X,\tau ,I\right)$ be an ideal topological space and $A$ be a subset of $X$. Then $A$ is said to be ${\beta }_I$-connected if it cannot be expressed as a union of two ${\beta }_I$-separated sets. A topological space $X$ is said to be ${\beta }_I$-connected if it is ${\beta }_I$-connected as a subset.

Recall, a topological space $\left(X,\tau \right)$ is said to be a ${\tau }_{\zeta }$-space if for every pair of subsets $A$ and $B$ of $X$, it is always true that $\mathrm{cl}\left(A\cap B\right)=\mathrm{cl}\left(A\right)\cap \mathrm{cl}\left(B\right)$ and $int\left(A\cap B\right)=int\left(A\right)\cap int\left(B\right)$. For example, a discrete space is a ${\tau }_{\zeta }$-space, while an indiscete space is not. Also, if $X=\left\{a,b,c\right\}$, then $\tau =\left\{\varnothing ,\left\{a\right\},X\right\}$ is not a ${\tau }_{\zeta }$-space. Let $\left(X,\tau \right)$ be a ${\tau }_{\zeta }$-space and $I$ be an ideal in $X$. Then we call $\left(X,\tau ,I\right)$ an ideal ${\tau }_{\zeta }$-space.

Lemma 1.6 present sufficient conditions for two sets to be ${\beta }_I$-separated.

Lemma 1.6. Let $\left(X,\tau \right)$ be a topological space and $I$ be an ideal in $X$. If $A$ $\left(\ne \varnothing \right)$ is $\beta$-open and $B$ $\left(\ne \varnothing \right)$ is ${\beta }_I$-open with $A\cap B=\varnothing$, then they are ${\beta }_I$-separated.

Proof: Suppose that $A$ and $B$ is not ${\beta }_I$-separated, that is ${\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap B\ne \varnothing$ or $A\cap {\mathrm{cl}}_{\beta }\left(B\right)\ne \varnothing$. Because $A\cap B=\varnothing$, we have $A\subseteq B^C$ and $B\subseteq A^C$. If $A$ is $\beta$-open, then $A^C$ is $\beta$-closed. Similarly, if $B$ is ${\beta }_I$-open, then $B^C$ is ${\beta }_I$-closed. Thus, $B^C\cap B\supseteq {\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap B\ne \varnothing$, or $A\cap A^C\supseteq A\cap {\mathrm{cl}}_{\beta }\left(B\right)\ne \varnothing$. A contradiction. □

Lemma 1.7 says that in a ${\tau }_{\omega }$-space every ${\beta }_I$-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 $\left(X,\tau \right)$ be a topology and $I$ be an ideal in $X$. If $X$ is ${\beta }_I$-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 $\tau$ with $X=A\cup B$. Note that $A$ and $B$ are $\beta$-open and ${\beta }_I$-open also. Because $A=B^C$ and $B=A^C$, $A$ and $B$ are also $\beta$-closed and ${\beta }_I$-closed. And so, $A={\mathrm{cl}}_{{\beta }_I}\left(A\right)$ and $B={\mathrm{cl}}_{\beta }\left(B\right)$. Thus, ${\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap B=A\cap B=\varnothing$ and $A\cap {\mathrm{cl}}_{\beta }\left(B\right)=A\cap B=\varnothing$. This implies that $X$ is ${\beta }_I$-separated, that is $X$ is not ${\beta }_I$-connected. □

Remark 1.4. Let $\left(X,\tau \right)$ be a topology and $I$ be an ideal in $X$. If $Y\subseteq X$, then $I_Y=\left\{Y\cap A:A\in I\right\}$ is an ideal in the relative topology $\left(Y,{\tau }_Y\right)$.

To see this, for the first property, let $B\in I_Y$ and $A\subseteq B$. Then $A\subseteq B\subseteq Y$. Now, if $A\in I_Y$, then there exist $C\in I$ such that $Y\cap C=A$. Note that $A\subseteq B\subseteq C$. Hence, $A,B\in I$. Thus, $A=Y\cap A\in I_Y$. Next, for the second, let $D,E\in I_Y$. Then $D\subseteq Y$ and $E\subseteq Y$. if $D\in I_Y$, then there exist $F\in I$ such that $Y\cap F=D$. Similarly, if $E\in I_Y$, then there exist $G\in I$ such that $Y\cap G=E$. Since $I$ is an ideal, $F\cup G\in I$. Now, because $D\cup E\subseteq F\cup G$, $D\cup E\in I$. Thus, $D\cup E=\left(D\cup E\right)\cap Y\in I_Y$.

The next statement, Theorem 1.11, presents a way to construct ${\beta }_I$-open sets in a subspace.

Theorem 1.11. Let $\left(X,\tau ,I\right)$ be an ideal ${\tau }_{\zeta }$-space and $Y$ be a clopen (a set that is open and closed at the same time) set. If $A$ is a ${\beta }_I$-open subset of $X$ $\left(\tau ,I\right)$, then $A\cap Y$ is a ${\beta }_{I_Y}$-open set in $Y$ $\left({\tau }_Y,I_Y\right)$.

Proof: Let $A$ be a ${\beta }_I$-open set in $X$ $\left(\tau ,I\right)$. Then there exists an open set $U^{\prime }$ such that $U^{\prime }-A\in I$ and $A-\mathrm{cl}\left(int\left(\mathrm{cl}\left(U^{\prime }\right)\right)\right)\in I$. Let $U=U^{\prime }\cap Y$. Then

$$\begin{array}{c}U-\left(A\cap Y\right)=U\cap {\left(A\cap Y\right)}^C\\ =\left(U^{\prime }\cap Y\right)\cap \left(A^C\cup Y^C\right)\\ =\left(U^{\prime }\cap Y\cap A^C\right)\cup \left(U^{\prime }\cap Y\cap Y^C\right)\\ =U^{\prime }\cap Y\cap A^C\\ =\left(U^{\prime }-A\right)\cap Y\in I_Y.\end{array}$$E1

Moreover, since $X$ is a ${\tau }_{\zeta }$-space and $Y$ is clopen

$$\begin{array}{c}\left(A\cap Y\right)-\mathrm{cl}\left(int\left(\mathrm{cl}\left(U\right)\right)\right)=\left(A\cap Y\right)-\mathrm{cl}\left(int\left(\mathrm{cl}\left(U^{\prime }\cap Y\right)\right)\right)\\ =\left(A\cap Y\right)-\mathrm{cl}\left(int\left(\mathrm{cl}\left(U^{\prime }\right)\right)\right)\cap Y\\ =\left[A-\mathrm{cl}\left(int\left(\mathrm{cl}\left(U^{\prime }\right)\right)\right)\right]\cap Y\in I_Y.\end{array}$$E2

This shows that $A\cap Y$ is ${\beta }_{I_Y}$-open in $Y\left({\tau }_Y,I_Y\right)$. □

Corollary 1.2. Let $\left(X,\tau ,I\right)$ be an ideal ${\tau }_{\zeta }$-space and $Y$ be a clopen set. If $A$ is a ${\beta }_I$-closed subset of $X$ $\left(\tau ,I\right)$, then $A\cap Y$ is a ${\beta }_{I_Y}$-closed set in $Y$ $\left({\tau }_Y,I_Y\right)$.

Proof: If $A$ is ${\beta }_I$-closed, then $A^C$ is ${\beta }_I$-open. By Theorem 1.11, $A^C\cap Y$ is ${\beta }_{I_Y}$-open. Hence, $A\cap Y={\left(A^C\cap Y\right)}^C$ is ${\beta }_{I_Y}$-closed in $Y$. □

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

Remark 1.5. Let $\left(X,\tau ,I\right)$ be an ideal topological space and $Y\subseteq X$. Then $I_Y=\left\{A\cap Y:A\in I\right\}$ is a subset of $I$.

Proof: If $A$ is a ${\beta }_{I_Y}$-open set in $Y$, then there exists an open set $O\in {\tau }_Y$ with $O-A\in I_Y$ and $A-\mathrm{cl}\left(int\left(\mathrm{cl}\left(O\right)\right)\right)\in I_Y$. Because ${\tau }_Y\subseteq \tau$ and by Remark 1.5, there exists an open set $O\in \tau$ with $O-A\in I$ and $A-\mathrm{cl}\left(int\left(\mathrm{cl}\left(O\right)\right)\right)\in I$. Thus, $A$ is ${\beta }_I$-open in $X\left(\tau ,I\right)$.

The converse follows from Theorem 1.11. □

The next statement, Lemma 1.8, characterizes ${\beta }_I$-open sets in subspaces.

Lemma 1.8. Let $\left(X,\tau ,I\right)$ be an ideal ${\tau }_{\zeta }$-space, $Y$ $\left(\subseteq X\right)$ be clopen, and ${\tau }_Y\subseteq \tau$. If $A\subseteq Y$, then $A$ is ${\beta }_{I_Y}$-open in $Y\left({\tau }_Y,I_Y\right)$ if and only if it is ${\beta }_I$-open in $X\left(\tau ,I\right)$.

Proof: If $A$ is a ${\beta }_{I_Y}$-open set in $Y$, then there exists an open set $O\in {\tau }_Y$ with $O-A\in I_Y$ and $A-\mathrm{cl}\left(int\left(\mathrm{cl}\left(O\right)\right)\right)\in I_Y$. Because ${\tau }_Y\subseteq \tau$ and by Remark 1.5, there exists an open set $O\in \tau$ with $O-A\in I$ and $A-\mathrm{cl}\left(int\left(\mathrm{cl}\left(O\right)\right)\right)\in I$. Thus, $A$ is ${\beta }_I$-open in $X\left(\tau ,I\right)$.

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 $\left(X,\tau ,I\right)$ be an ideal ${\tau }_{\zeta }$-space, $Y$ be clopen, and ${\tau }_Y\subseteq \tau$. If $A\subseteq X$, then ${\mathrm{cl}}_{{\beta }_{I_Y}}\left(A\cap Y\right)={\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap Y$.

Proof: Since ${\mathrm{cl}}_{{\beta }_I}\left(A\right)$ is a ${\beta }_I$-closed set in $X$, by Lemma 1.8 ${\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap Y$ is a ${\beta }_{I_Y}$-closed set in $Y$. Hence, ${\mathrm{cl}}_{{\beta }_{I_Y}}\left(A\cap Y\right)$ $\supseteq {\mathrm{cl}}_{{\beta }_{I_Y}}\left({\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap Y\right)$ $={\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap Y$. But, ${\mathrm{cl}}_{{\beta }_{I_Y}}\left(A\cap Y\right)=$ ${\mathrm{cl}}_{{\beta }_I}\left(A\cap Y\right)\subseteq$ ${\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap {\mathrm{cl}}_{{\beta }_I}\left(Y\right)=$ ${\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap Y$. Therefore, ${\mathrm{cl}}_{{\beta }_{I_Y}}\left(A\cap Y\right)={\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap Y$. □

Definition 1.14. Let $\left(X,\tau ,I\right)$ be an ideal topological space, and $\left(Y,{\tau }_Y,I_Y\right)$ be a subspace. A pair of subsets, say $A$ and $B$, of $X$ is said to be ${\beta }_{I_Y}$-separated in $Y$ if ${\mathit{cl}}_{{\beta }_{I_Y}}\left(A\right)\cap B=\varnothing =A\cap {\mathit{cl}}_{{\beta }_Y}\left(B\right)$, where ${\mathit{cl}}_{{\beta }_Y}\left(B\right)={\mathit{cl}}_{\beta }\left(B\right)\cap Y$.

Definition 1.15. Let $\left(X,\tau ,I\right)$ be an ideal topological space, and $\left(Y,{\tau }_Y,I_Y\right)$ be a subspace. A subset $A$ of $Y$ is said to be ${\beta }_{I_Y}$-connected if it cannot be expressed as a union of two ${\beta }_{I_Y}$-separated sets. The subspace $Y$ is said to be ${\beta }_{I_Y}$-connected if it is ${\beta }_{I_Y}$-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 $\left(X,\tau \right)$ be a ${\tau }_{\zeta }$-space, $I$ be an ideal, $Y$ $\left(\subseteq X\right)$ be clopen, and ${\tau }_Y\subseteq \tau$. If $A$ and $B$ are ${\beta }_I$-separated in $X$, then they are ${\beta }_{I_Y}$-separated in $Y$.

Proof: If $A$ and $B$ are ${\beta }_I$-separated in $X$, then by Theorem 1.12 $\varnothing ={\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap B=\left({\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap B\right)\cap Y=\left({\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap Y\right)\cap B={\mathrm{cl}}_{{\beta }_{I_Y}}\left(A\right)\cap B$ and $\varnothing =A\cap {\mathrm{cl}}_{\beta }\left(B\right)=\left(A\cap {\mathrm{cl}}_{\beta }\left(B\right)\right)\cap Y=A\cap \left({\mathrm{cl}}_{\beta }\left(B\right)\cap Y\right)=A\cap {\mathrm{cl}}_{{\beta }_Y}\left(B\right)$. Thus, $A$ and $B$ are ${\beta }_{I_Y}$-separated. □

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

Remark 1.6. Let $\left(X,\tau \right)$ be a topological space and $I$ be an ideal. If $X$ is ${\beta }_I$-separated (say, $X=A\cup B$ with $A\ne \varnothing$, $B\ne \varnothing$, and ${\mathit{cl}}_{{\beta }_I}\left(A\right)\cap B=\varnothing$ $=A\cap {\mathit{cl}}_{\beta }\left(B\right)$), then $A$ is $\beta$-open while $B$ is ${\beta }_I$-open.

To see this, if $A$ and $B$ is ${\beta }_I$-separated, then ${\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap B=\varnothing$ and $A\cap {\mathrm{cl}}_{\beta }\left(B\right)=\varnothing$. Hence, $A^C={\mathrm{cl}}_{\beta }\left(B\right)$ and $B^C={\mathrm{cl}}_{{\beta }_I}\left(A\right)$. Thus, $A^C$ is $\beta$-closed and $B^C$ is ${\beta }_I$-closed. Accordingly, $A$ is $\beta$-open and $B$ is ${\beta }_I$-open.

The next statement, Theorem 1.14, characterizes ${\beta }_I$-connected spaces.

Theorem 1.14. Let $\left(X,\tau \right)$ be a topological space and $I$ be an ideal $X$. Then, $X$ is ${\beta }_I$-connected if and only if it cannot be expressed as a union of two a non-empty disjoint sets in which one is a $\beta$-open set and the other is a ${\beta }_I$-open set.

Proof: Suppose that $X$ is ${\beta }_I$-connected, and we can express $X$ as a union of two non-empty disjoint $\beta$-open set and ${\beta }_I$-open set, say $A\cup B=X$ (with $A$, a $\beta$-open set, and $B$, a ${\beta }_I$-open set) and $A\cap B=\varnothing$. If $A\cup B=X$ and $A\cap B=\varnothing$, then $A^C=B$ and $B^C=A$. Since $A$ is $\beta$-open, $B$ is $\beta$-closed. Also, since $B$ is ${\beta }_I$-open, $A$ is ${\beta }_I$-closed. Hence, ${\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap B=A\cap B=\varnothing$ and $A\cap {\mathrm{cl}}_{\beta }\left(B\right)=A\cap B=\varnothing$. Thus, the pair $A$ and $B$ is ${\beta }_I$-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 $\left(X,\tau \right)$ be a topological space, $I$ be an ideal $X$, and $A$ be a ${\beta }_I$-connected set. If $A\subseteq H\cup G$ where $H$ and $G$ is a pair of ${\beta }_I$-separated sets, then either $A\subseteq H$ or $A\subseteq G$.

Proof: Suppose that to the contrary, $A=\left(A\cap H\right)\cup \left(A\cap G\right)$ with $A\cap H\ne \varnothing$ and $A\cap G\ne \varnothing$. Since $H$ and $G$ is a pair of ${\beta }_I$-separated sets, ${\mathrm{cl}}_{{\beta }_I}\left(A\cap H\right)\cap \left(A\cap G\right)\subseteq {\mathrm{cl}}_{{\beta }_I}\left(H\right)\cap \left(G\right)=\varnothing$ and $\left(A\cap H\right)\cap {\mathrm{cl}}_{\beta }\left(A\cap G\right)\subseteq H\cap {\mathrm{cl}}_{\beta }\left(G\right)=\varnothing$. Thus, ${\mathrm{cl}}_{{\beta }_I}\left(A\cap H\right)\cap \left(A\cap G\right)=\varnothing$ and $\left(A\cap H\right)\cap {\mathrm{cl}}_{\beta }\left(A\cap G\right)=\varnothing$. Therefore, $A$ can be expressed as a union of two ${\beta }_I$-separated sets $A\cap H$ and $A\cap G$. A contradiction. □

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

Theorem 1.16. Let $\left(X,\tau \right)$ be a topological space, $I$ be an ideal in $X$, and, $A$ and $B$ be ${\beta }_I$-separated sets. If $C\subseteq A$ ($C\ne \varnothing$) and $D\subseteq B$ ($D\ne \varnothing$), then $C$ and $D$ are also ${\beta }_I$-separated.

Proof: Suppose that $A$ and $B$ are ${\beta }_I$-separated. Then ${\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap B=\varnothing$ and $A\cap {\mathrm{cl}}_{\beta }\left(B\right)=\varnothing$. Thus, ${\mathrm{cl}}_{{\beta }_I}\left(C\right)\cap D\subseteq {\mathrm{cl}}_{{\beta }_I}\left(A\right)\cap B=\varnothing$ and $C\cap {\mathrm{cl}}_{\beta }\left(D\right)=A\cap {\mathrm{cl}}_{\beta }\left(B\right)=\varnothing$. Hence, ${\mathrm{cl}}_{{\beta }_I}\left(C\right)\cap D=\varnothing =C\cap {\mathrm{cl}}_{\beta }\left(D\right)$. Therefore, $C$ and $D$ is ${\beta }_I$-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 $\beta$-open sets, ${\beta }_I$-compact spaces, ${\beta }_I^{\ast }$-hyperconnected spaces, ${\beta }_I$-separated spaces. Also, some properties focusing on generalizing ideals in ideal topological space theory were realized.