I. INTRODUCTION

Topology is seen in many areas of science [14]. It is applied in biochemistry [5] and information systems [19]. Topology as a mathematical system is fundamentally comprised of open sets together with the operations union and intersection. Over time, open sets were generalized to different varieties. To name some, we have, Stone [20] introduced regular open set. Levine [10] introduced semi-open sets. Njasted [16] introduced $\alpha$ -open sets. Mashhour et al. [13] introduced pre-open sets. Abd El-Monsef et al. [1] introduced $\beta$ -open set.

In the year 1970, Levine [11] introduced generalized closed sets, and achoring on this notion, Andrijevic [3] presented yet another generalization of open sets called $b$ -open sets. This study uses the notion of $b$ -open sets to come up with a new concept called $b^{\ast }$ -open sets.

The concept ideal topological spaces was first seen in [9]. Vaidyanathaswamy [23] investigated this concept in point set topology. Tripathy and Shravan [17, 18], Tripathy and Acharjee [21], Tripathy and Ray [22], Catalan et al. [6] also made investigations on ideal topological spaces.

Several concepts in topology were generalized using this structure. One of which is the concept $b^{\ast }$ -open sets. Using the notion of $b^{\ast }$ -open sets, we introduced the concepts $b^{\ast }$ -compact sets, compatible $b_I^{\ast }$ -compact sets, countably $b_I^{\ast }$ -compact sets, $b_I^{\ast }$ -connected sets, in ideal generalized topological spaces.

Let $W$ be a non-empty set. An ideal $I$ on a set $W$ is a non-empty collection of subsets of $W$ which satisfies:

1. $B\in I$ and $D\subseteq B$ implies $D\in I$.
2. $B\in I$ and $D\in I$ implies $B\cup D\in I$

Let $W$ be a topological space and $B$ be a subset of $W$. We say that $B$ is $b^{\ast }$ -open set if $B=\mathrm{cl}(\mathrm{int}(B))\cup \mathrm{int}(\mathrm{cl}(B))$. For example, consider $W=\{a,b,c\}$ and the topology $\varsigma =\{\varnothing ,\{a\},\{b\},\{a,b\},W\}$ on $W$. Then the $b^{\ast }$ -open subsets are $\varnothing ,\{a,b\},\{c\}$ and $W$.

Let $W$ be a topological space and $B$ be a subset of $W$. The set $B$ is called $b^{\ast }$ -open relative to an ideal $I$ (or $b_I^{\ast }$ -open), if there is an open set $P$ with $P\subseteq \mathrm{Int}(B)$, and a closed set $S$ with $\mathrm{Cl}(B)\subseteq S$ such that

1. $(\mathrm{Int}(S)\cup \mathrm{Cl}(\mathrm{Int}(B)))\mathrm{\backslash }B\in I$, and
2. $B\mathrm{\backslash }(\mathrm{Int}(\mathrm{Cl}(B))\cup \mathrm{Cl}(P))\in I.$

In addition, we say that a set $B$ is a $b_I^{\ast }$ -closed set if $B^C$ is $b_I^{\ast }$ -open.

Consider the ideal space $(\{q,r,s\},\{\mathrm{\varnothing },\{q\},\{r\},\{q,r\},\{q,r,s\}\},\{\mathrm{\varnothing },\{r\}\})$. Then $B=\{r,s\}$ is a $b^{\ast }$ -open with respect to the ideal $I=\{\mathrm{\varnothing },\{r\}\}$. To see this, we let $P$ be the open set $\{r\}$ and $S$ be the closed set $\{r,s\}$. Then $\mathrm{Int}(S)\cup \mathrm{cl}(\mathrm{int}(\{r,s\}))\mathrm{\backslash }\{r,s\}=\mathrm{int}(\{r,s\})\cup \mathrm{cl}(\{r\})\mathrm{\backslash }\{r,s\}=\{r\}\cup \{r,s\}\mathrm{\backslash }\{r,s\}=\{r,s\}\mathrm{\backslash }\{r,s\}=\mathrm{\varnothing }\in I$. Also, $\mathrm{Int}(\mathrm{cl}(\{r,s\})\cup \mathrm{cl}(P)\mathrm{\backslash }\{r,s\}=\mathrm{int}(\{r,s\})\cup \mathrm{cl}(\{r\})\mathrm{\backslash }\{r,s\}=\{r\}\cup \{r,s\}\mathrm{\backslash }\{r,s\}=\{r,s\}\mathrm{\backslash }\{r,s\}=\mathrm{\varnothing }\in I$. This shows that $B=\{r,s\}$ is a $b_I^{\ast }$ -open.

The succeeding sections present the rudimentary properties of $b_I^{\ast \ast }$ -hyperconnected spaces and $b_I^{\ast }$ -separated spaces.

II. RESULTS

This section presents the results of this study.

a) Preliminary Result: The following Lemmas were established in [4]. They will be used in the proofs of some of the succeeding statements. In particular, Lemma 2.1 is used in Theorem 2.13 and Remark 2.15, while Lemma 2.2 is used in Lemma 2.19.

• Lemma 2.1. [4] Let $(X,\tau ,I)$ be an ideal topological space. Then every $b^{\ast }$ -open set is a $b_I^{\ast }$ -open set.
• Lemma 2.2. [4] Let $(X,\tau ,I)$ be an ideal topological space with $I=\{\varnothing \}$. Then $A$ is a $b^{\ast }$ -open set if and only if $A$ is a $b_I^{\ast }$ -open set. b) $b_I^{\ast \ast }$ -Hyperconnected Ideal Topological Spaces: The concept $\ast$ -hyperconnectedness was introduced by Ekici et al. [8], and the concept $I$ -hyperconnectedness was introduced by Abd El-Monsef et al. [12]. These insights motivated us to create the concept called $b_I^{\ast \ast }$ -hyperconnectedness. One may see [15] to gain more insights on these ideas.

Definition 2.3. Let $(X,\tau )$ be a topological space and $I$ be an ideal on $X$. A function $(\ast {)}^{\ast }(I,\tau ):P(X)\to P(X)$ given by $A^{\ast }(I,\tau )=\{x\in X:A\cup U\notin I\text{ for every }U\in \tau (x)\}$ where $\tau (x)=\{U\in \tau :x\in U\}$ is called a local of $A$ with respect to $\tau$ and $I$.

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

Definition 2.4. Let $(X,\tau )$ be a topological space and $I$ be an ideal on $X$. The Kuratowski closure operator $Cl({)}^{\ast }(I,\tau ):P(X)\to P(X)$ for the topology ${\tau }^{\ast }(I,\tau )$ is given by $Cl(A{)}^{\ast }(I,\tau )=A\cup A^{\ast }$.

Consider the ideal space in the previous example. We have, $Cl(\mathrm{\varnothing }{)}^{\ast }=\mathrm{\varnothing }\cup {\mathrm{\varnothing }}^{\ast }=\mathrm{\varnothing }\cup \mathrm{\varnothing }=\mathrm{\varnothing }$, $Cl(\{a\}{)}^{\ast }=\{a\}\cup \{a{\}}^{\ast }=\{a\}\cup \{c\}=\{a,c\}$, $Cl(\{b\}{)}^{\ast }=\{b\}\cup \{b{\}}^{\ast }=\{b\}\cup \{c\}=\{b,c\}$, $Cl(\{c\}{)}^{\ast }=\{c\}\cup \{c{\}}^{\ast }=\{c\}\cup X=X$, $Cl(\{a,b\}{)}^{\ast }=\{a,b\}\cup \{a,b{\}}^{\ast }=\{a,b\}\cup \{c\}=X$, $Cl(\{a,c\}{)}^{\ast }=\{a,c\}\cup \{a,c{\}}^{\ast }=\{a,c\}\cup X=X$, $Cl(\{b,c\}{)}^{\ast }=\{b,c\}\cup \{b,c{\}}^{\ast }=\{b,c\}\cup X=X$, and $Cl(X{)}^{\ast }=X\cup X^{\ast }=X\cup X=X$.

Definition 2.5. Let $(X,\tau )$ be a topological space and $I$ be an ideal on $X$. The Kuratowski interior operator $\mathrm{Int}()\ast (I,\tau ):P(X)\to P(X)$ for the topology ${\tau }^{\ast }(I,\tau )$ is given by $\mathrm{Int}(A{)}^{\ast }(I,\tau )=X-Cl(X-A{)}^{\ast }$.

Definition 2.6. An ideal space $(X,\tau ,I)$ is called $\ast$ -hyperconnected [8] if $cl(A{)}^{\ast }=X$ for all non-empty open set $A\subseteq X$.

Definition 2.8. An ideal topological space $(X,\tau ,I)$ is said to be $b_I^{\ast \ast }$ -hyperconnected space if $X-cl(A{)}^{\ast }\in I$ for every non-empty $b_I^{\ast }$ -open subset $A$ of $X$.

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

Theorem 2.9. Let $(X,\tau ,I)$ be an ideal topological space. If $X$ is $I$ -hyperconnected, then it is $b_I^{\ast \ast }$ -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 $X-cl(A{)}^{\ast }\in I$ for all non-empty open set $A\subseteq X$. And, because an open set is also a $b_I^{\ast }$ -open set, we have $X-cl(A{)}^{\ast }\in I$ for all non-empty $b_I^{\ast }$ -open set $A\subseteq X$. Hence, $X$ is $b_I^{\ast \ast }$ -hyperconnected.

The next lemma is clear.

Lemma 2.10. Let $(X,\tau )$ be a topological space. Then the intersection of any family of ideals on $X$ is an ideal on $X$.

Theorem 2.11 is taken from [2]. It says that when $I$ is the minimal ideal, then the notions --hyperconnected and $I^{\ast }$ -hyperconnected are equivalent.

Theorem 2.11 [2] Let $(X,\tau )$ be a clopen ideal topological space with $I=\{\varnothing \}$. Then, $X$ is $\ast$ -hyperconnected if and only if it is $I^{\ast }$ -hyperconnected.

The next remark is clear.

Remark 2.12. If $(X,\tau )$ 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 $b^{\ast }$ -open.

Theorem 2.13 says that in a clopen space, with respect to the minimal ideal $I$, the notions $b_I^{\ast \ast }$ -hyperconnected and $I^{\ast }$ -hyperconnected are equivalent.

Theorem 2.13 Let $(X,\tau ,I)$ be a clopen ideal topological space with $I=\{\varnothing \}$. Then, $X$ is $I^{\ast }$ -hyperconnected if and only if it is $b_I^{\ast \ast }$ -hyperconnected.

Proof. Suppose that $X$ is $I^{\ast }$ -hyperconnected. Let $A$ be a non-empty element of $\tau$. Then $X-c{l}^{\ast }(A)\in I$. By Remark 2.12 and Lemma 2.2, every open set is precisely $b_I^{\ast }$ -open. Thus, $X-c{l}^{\ast }(A)\in I$ for all $b_I^{\ast }$ -open set $A(\ne \mathrm{\varnothing })$. Therefore, $X$ is $b_I^{\ast \ast }$ -hyperconnected also. Conversely, suppose that $X$ is $b_I^{\ast \ast }$ -hyperconnected. Let $A$ be a non-empty $b_I^{\ast }$ -open set. Then $X-c{l}^{\ast }(A)\in I$. By Remark 2.12 and Lemma 2.2, $b_I^{\ast }$ -open set is precisely open. Thus, $X-cl(A{)}^{\ast }\in I$ for all open set $A(\ne \mathrm{\varnothing })$. Therefore, $X$ is $I^{\ast }$ -hyperconnected also.

Corollary 2.14 says that in a clopen ideal topological space, relative to the minimal ideal $I$, the notions $b_I^{\ast \ast }$ -hyperconnected, $I^{\ast }$ -hyperconnected, and $\ast$ -hyperconnected are equivalent.

Corollary 2.14. Let $(X,\tau ,I)$ be a clopen ideal topological space with $I=\{\varnothing \}$. Then the following statements are equivalent.

i. $X$ is $\ast$ -hyperconnected. ii. $X$ is $I^{\ast }$ -hyperconnected. iii. $X$ is $b_I^{\ast \ast }$ -hyperconnected.

Theorem 2.15 may be an important property.

Remark 2.15. If an ideal topological space $(X,\tau ,\{\mathrm{\varnothing }\})$ is a $b_I^{\ast }$ -hyperconnected space, then $X-c{I}^{\ast }(A)\in I$ for every non-empty $b^{\ast }$ -open subset $A$ of $X$.

To see this, let $A$ is a non-empty $b$ -open set. Then by Lemma 2.2 $A$ is $b_I^{\ast }$ -open. Since $X$ is $b_I^{\ast \ast }$ -hyperconnected, $X-c{l}^{\ast }(A)\in I$.

Theorem 2.16 is a characterization of $b_I^{\ast \ast }$ -hyperconnected space.

Theorem 2.16. Let $(X,\tau ,I)$ be an ideal topological space. Then the following statements are equivalent.

i. $X$ is a $b_I^{\ast \ast }$ -hyperconnected space. ii. $int(A{)}^{\ast }\in I$ for all $b_I^{\ast }$ -closed proper subset $A$ of $X$.

• Proof. $(i)\Rightarrow (ii)$ Let $B$ be $b_I^{\ast }$ -closed. Then $X-B$ is $b_I^{\ast }$ -open. Since $B\ne X$, $X-B\ne \mathrm{\varnothing }$. Hence, by assumption we have $\mathrm{int}(B{)}^{\ast }=X-\mathrm{cl}(X-B{)}^{\ast }\in I$.
• $(ii)\Rightarrow (i)$ Let $A$ $(\ne X)$ be a non-empty $b_I^{\ast }$ -open set. Then $X-A$ is a non-empty $b_I^{\ast }$ -open set. Hence, by assumption, we have $X-\mathrm{cl}(A{)}^{\ast }=X-\mathrm{cl}(X-(X-A){)}^{\ast }=\mathrm{int}(X-A{)}^{\ast }\in I$. Thus, $X$ is $b_I^{\ast \ast }$ -hyperconnected.

c) $b_I^{\ast }$ -Separated Ideal Topological Spaces: In this section, we present the concepts $b_I^{\ast }$ -separated sets and $b_I^{\ast }$ -connected sets. We also present some of their important properties.

Definition 2.17. Let $(X,\tau ,I)$ be an ideal topological space and $A$ be a subset of $X$. The $b^{\ast }$ -closure of $A$, denoted by $c{l}_{b^{\ast }}(A)$, is the smallest $b^{\ast }$ -closed set that contains $A$. The $b_I^{\ast }$ -closure of $A$, denoted by $c{l}_{b_I^{\ast }}(A)$, is the smallest $b_I^{\ast }$ -closed set that contains $A$.

Next, we define $b_I^{\ast }$ -separated sets, $b_I^{\ast }$ -connected sets, and $b_I^{\ast }$ -connected spaces.

Definition 2.18. Let $(X,\tau ,I)$ be an ideal topological space. A pair of subsets, say $A$ and $B$, of $X$ is said to be $b_I^{\ast }$ -separated if $c{l}_{b_I^{\ast }}(A)\cap B=\mathrm{\varnothing }=A\cap c{l}_{b_I^{\ast }}(B)$. A subset $A$ of $X$ is said to be $b_I^{\ast }$ -connected if it cannot be expressed as a union of two $b_I^{\ast }$ -separated sets. The topological space $X$ is said to be $b_I^{\ast }$ -connected if it is $b_I^{\ast }$ -connected as a subset.

Lemma 2.19 says that every $b_I^{\ast }$ -connected space is connected. Recall, a space is connected if it cannot be written as a union of two non-empty open sets.

Lemma 2.19. Let $(X,\tau )$ be a ${\tau }_{\zeta }$ -space (a topological space in which every element is $b^{\ast }$ -open also) and $I$ be an ideal in $X$. If $X$ is $b_I^{\ast }$ -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 $X=A\cup B$. By Lemma 2.1, $A$ and $B$ are $b_I^{\ast }$ -open sets also. Because $A=B^C$ and $B=A^C$, $A$ and $B$ are also $b_I^{\ast }$ -closed. And so, $A={\mathrm{cl}}_{b_I^{\ast }}(A)$ and $B={\mathrm{cl}}_{b_I^{\ast }}(B)$. Thus, ${\mathrm{cl}}_{b_I^{\ast }}(A)\cap B=A\cap B=\mathrm{\varnothing }$ and $A\cap {\mathrm{cl}}_{b_I^{\ast }}(B)=A\cap B=\mathrm{\varnothing }$. This implies that $X$ is $b_I^{\ast }$ -separated, that is $X$ is not $b_I^{\ast }$ -connected, a contradiction.

Remark 2.20. Let $(X,\tau )$ be a topology and $I$ be an ideal in $X$. If $Y\subseteq X$, then $I_Y=\{Y\cap A:A\in I\}$ is an ideal in the relative topology $(Y,{\tau }_Y)$.

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, $D\cup E=(D\cup E)\cap Y\in I_Y$.

The next statement, Theorem 2.24, presents a way to construct $b_I^{\ast }$ -open sets in a subspace.

Theorem 2.21. Let $(X,\tau ,I)$ be an ideal topological space, and $Y\subseteq X$. If $A$ is a $b_I^{\ast }$ -open subset of $X$, then $A\cap Y$ is a $b_{I_Y}^{\ast }$ -open set in $Y$

Proof. Let $A$ be a $b_I^{\ast }$ -open set in $X(\tau ,I)$. Then there exists an open set $O$ with $O\subseteq \mathrm{int}(A)$, and a closed set $F$ with $\mathrm{cl}(A)\subseteq F$ such that $\mathrm{int}(F)\cup \mathrm{cl}(\mathrm{int}(A))\setminus A\in I$, and $A\setminus \mathrm{int}(\mathrm{cl}(A))\cup \mathrm{cl}(O)\in I$. Let $O^{\prime }=O\cap Y$, and $F^{\prime }=F\cap Y$. Then $O^{\prime }$ is open in $Y$, and $F^{\prime }$ is closed in $Y$. Also, $\mathrm{int}(F)\cup \mathrm{cl}(\mathrm{int}(A))\setminus A\supseteq (\mathrm{int}(F)\cup \mathrm{cl}(\mathrm{int}(A)))\cap Y\setminus (A\cap Y)\supseteq (\mathrm{int}(F))\cap Y\cup \mathrm{cl}(\mathrm{int}(A)))\cap Y\setminus (A\cap Y)\supseteq \mathrm{int}(F\cap Y)\cup \mathrm{cl}(\mathrm{int}(A\cap Y))\setminus (A\cap Y)\supseteq \mathrm{int}(F^{\prime })\cup \mathrm{cl}(\mathrm{int}(A\cap Y))\setminus (A\cap Y)$, and $A\setminus \mathrm{int}(\mathrm{cl}(A))\cup \mathrm{cl}(O)\supseteq (A\cap Y)\setminus (\mathrm{int}(\mathrm{cl}(A))\cup \mathrm{cl}(O))\cap Y\supseteq (A\cap Y)\setminus \mathrm{int}(\mathrm{cl}(A))\cap Y\cup \mathrm{cl}(O)\cap Y\supseteq (A\cap Y)\setminus \mathrm{int}(\mathrm{cl}(A\cap Y))\cup \mathrm{cl}(O)\supseteq (A\cap Y)\setminus (\mathrm{int}(\mathrm{cl}(A)\cap Y))\cup (\mathrm{cl}(O^{\prime }))$. Hence, by heredity $\mathrm{int}(F^{\prime })\cup \mathrm{cl}(\mathrm{int}(A\cap Y))\setminus (A\cap Y)\in I$, and $(A\cap Y)\setminus \mathrm{int}(\mathrm{cl}(A)\cap Y)\cup (\mathrm{cl}(O^{\prime }))\in I$. This shows that $A\cap Y$ is a $b_{I_Y}^{\ast }$ -open set in $Y$.