On c-Compactness in Topological and Bitopological Spaces

Abstract

The primary goal of this research is to initiate the pairwise c-compact concept in topological and bitopological spaces. This would make us to define the concept of c-compact space with some of its generalization, and present some necessary notions such as the H-closed, the quasi compact and extremely disconnected compact spaces in topological and bitopological spaces. As a consequence, we derive numerous theoretical results that demonstrate the relations between c-separation axioms and the c-compact spaces.

1. Introduction

Compactness owns a significant role in topology and also so for a certain of its more grounded and weaker kinds. Among those kinds is H-closedness, whereby the theory of such kinds was studied by Alexandroff et al. in 1929 [1]. Thirty years after that date, Singal et al. discussed the spaces of nearly-compact type. In 1976, the S-compact space was established as another type of compact spaces [2]. Many other researchers have explored a few other types of compactness from time to time [3]. In this work, we intend to introduce many new theoretical results of the weaker type of compact spaces for the purpose of defining the c-compact space, and then generalizing such space to pairwise c-compact space.

The notion of bitopological spaces is a set endowed with two topologies, and it might be written as $\chi =(\chi ,{\beta }_1,{\beta }_2)$, where ${\beta }_1,{\beta }_2$ are topologies on $\chi$. Typically, if the set is $\chi$ and the topologies are ${\beta }_1$ and ${\beta }_2$, then the bitopological space is referred to as $(\chi ,{\beta }_1,{\beta }_2)$. Corresponding to well-known properties of topological spaces, there are versions for bitopological spaces. We state some of them below for completeness:

• A bitopological space $(\chi ,{\beta }_1,{\beta }_2)$ is pairwise compact if each cover $\{U_i\mid i\in I\}$ of $\chi$ with $U_i\in {\beta }_1\cup {\beta }_2$ contains a finite subcover. In this case, $\{U_i\mid i\in I\}$ must contain at least one member from ${\beta }_1$ and at least one member from ${\beta }_2$.
• A bitopological space $(\chi ,{\beta }_1,{\beta }_2)$ is pairwise Hausdorff if for any two distinct points $x,y\in \chi$ there exist disjoint $U_1\in {\beta }_1$ and $U_2\in {\beta }_2$ with $x\in U_1$ and $y\in U_2$.
• A bitopological space $(\chi ,{\beta }_1,{\beta }_2)$ is pairwise zero-dimensional if opens in $(\chi ,{\beta }_1)$ which are closed in $(\chi ,{\beta }_2)$ form a basis for $(\chi ,{\beta }_1)$, and opens in $(\chi ,{\beta }_2)$ which are closed in $(\chi ,{\beta }_1)$ form a basis for $(\chi ,{\beta }_2)$.

The notion of bitopological space is associated with several previous studies that have been performed on bitopological spaces through which every single one of topologies is just a set of points that satisfies a set of axioms. With some common standard theoretical findings characterized by Tietze extension, the so-called pairwise normal spaces, pairwise regular and pairwise Hausdorff were studied well in 1963 by Kelly [4]. Afterward, Kim (1968) and Patty (1967) carried some further works out in the field of bitopological spaces [5,6]. Expand ability, nearly expand ability and feebly expand ability of bitoplogical spaces were explained by Oudetallah in [7,8]. In addition, the space of pairwise r-compact was defined well in bitopological spaces in [9].

For the reason of that the subject of c-compactness is one of the topological spaces’ subjects, we intend to deeply explore this subject in the bitopological spaces. Accordingly, there will be a lot of theoretical results and findings that can be satisfied in these bitopological spaces. We think that the results derived in this paper can find their applications in some applications in the field of real analysis due to it is known, e.g., that $(\mathbb{R},{\tau }_u)$ doesn’t represent a compact space, but $(\mathbb{R},{\tau }_u,{\tau }_u)$ is a c-bitopological compact space. For instance, this assertion ultimately allows one to apply the Heine-Borel Theorem, which is regarded very important in the field of real analysis. In this article, we intend to propose a new class of compact spaces, named the pairwise c-compact space (or simply p-c-compact) in topological spaces and bitopological spaces. Accordingly, numerous results are generated from this concept related to the H-closed, the quasi compact and extremely disconnected compact spaces in the considered spaces. In addition, numerous other results associated with relations between c-separation axioms and the c-compact spaces are derived as well. However, the rest of this article is organized in the subsequent order: In the next part, we define the p-c-compact space, and then we establish numerous results on the basis of this space. In Section 3, we derive other several theorems associated with the connection of the c-separation axioms with the c-compact spaces. Finally, the last section summarizes the main points of this work.

2. On $\mathit{p}$-$\mathit{c}$-Compact Spaces

In this part, we aim to set a definition for the p-c-compact concept in topological spaces and bitopological spaces. As a consequence, numerous other definitions related to this concept are defined well. Those definitions are then used to derive numerous generalizations and novel results associated with the H-closed, the quasi compact and extremely disconnected compact spaces in topological spaces and bitopological spaces. Herein, it is noteworthy to highlight that all preliminaries stated below, are considered an important part of the contribution of this work. In particular, such preliminaries would help us in establishing Theorems 2 and 3 stated at the end of this section in which the first theorem determines a strong condition that makes the topological space $\chi$ is c-compact, while the second theorem outlines another strong condition that can make the bitopological space $\chi$ is p-compact.

Definition 1

([10]). If $B\subseteq \chi$ and $(\chi ,\beta )$ is a topological space. Then

(i)

If $B={\overline{B}}^{{}^o}$, then B is regular open set of χ.

(ii)

$B=\overline{B^{{}^o}}$ if and only if B is regular closed set of χ.

(iii)

There exists an open set ϝ in which $\mathsf{ϝ}\subseteq B\subseteq \overline{\mathsf{ϝ}}$ if and only if B is a semi-open set in χ.

Definition 2.

Consider $(\chi ,{\beta }_1,{\beta }_2)$ is a bitopological space and $B\subseteq \chi$. We say that

(i)

B is a p-regular open set if $B=In{t}_{{\beta }_1}\left(C{L}_{{\beta }_1}\left(B\right)\right)$ and $B=In{t}_{{\beta }_2}\left(C{L}_{{\beta }_2}\left(B\right)\right)$.

(ii)

B is a p-regular closed set if $B=C{L}_{{\beta }_1}\left(In{t}_{{\beta }_1}\left(B\right)\right)$ and $B=C{L}_{{\beta }_2}\left(In{t}_{{\beta }_2}\left(B\right)\right)$.

(iii)

B is a p-semi-open set if there exists an open set ω in which ${\omega }_{{\beta }_1}\subseteq B\subseteq C{L}_{{\beta }_1}\left(\omega \right)$ and ${\omega }_{{\beta }_2}\subseteq B\subseteq C{L}_{{\beta }_2}\left(\omega \right)$.

Remark 1.

If $(\chi ,{\beta }_1,{\beta }_2)$ is a bitopological space and $B\subseteq \chi$, we have

• If $\chi -B$ is a p-regular open set, then B is called a p-regular closed set.
• If $\chi -B$ is a p-regular closed set, then B is called a p-regular open set.

Theorem 1.

Consider $(\chi ,{\beta }_1,{\beta }_2)$ is a bitopological space. Each p-open set is a p-semi-open set.

Proof. 

Consider B is a p-regular open set. Then, we have $In{t}_{{\beta }_i}\left(B\right)\subseteq B\subseteq C{L}_{{\beta }_i}\left(B\right)$, for all $i=1,2$. So, we obtain $\omega \subseteq B\subseteq C{L}_{{\beta }_i}\left(\omega \right)$, for all $i=1,2$. Therefore, B is a p-semi-open set. □

Definition 3

([2]). If every open cover of χ has a finite subfamily whose closures cover χ, then the topological space $(\chi ,\beta )$ is called quasi H-closed space.

Definition 4

([2]). If every ${\beta }_i$-open cover of χ has a finite subfamily whose closures cover χ, the bitopological space $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is called p-quasi H-closed space, for all $i=1,2$.

Definition 5

([11]). If every open cover has a finite subfamily such that the interior of the closures of which covers χ, then the space $(\chi ,\beta )$ is called nearly compact space.

Definition 6.

If every ${\beta }_i$-open cover of χ has a finite subfamily so that the interior of closures of which covers χ, then the bitopological space $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is called a p-nearly compact space, for all $i=1,2$.

Definition 7

([2,12]). If every semi-open cover of χ has a finite subfamily whose closure covers χ, then the space $(\chi ,\beta )$ is called S-closed space.

Definition 8.

If every ${\beta }_i$-semi open cover of χ has a finite subfamily whose closure covers χ, then the bitopological space $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is called a p-S-closed space, for all $i=1,2$.

Definition 9.

Consider $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a bitopological space. It is said that χ is a p-c-compact space if for all $i,j=1,2$ and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is a ${\beta }_i$-open cover of A, there exists a finite collection of ${\beta }_i$-open sets ${\omega }_{{\alpha }_1},{\omega }_{{\alpha }_2},\dots ,{\omega }_{{\alpha }_n}$ such that ${\displaystyle A\subset \bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}}$, for all $i=1,2$.

Definition 10

([2,4]). A Housderff space $\chi =(\chi ,\beta )$ is defined as a p-H-closed space if for all open cover $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ of χ, there exists a finite collection ${\left\{{\omega }_{{\alpha }_k}\right\}}_{k=1}^n$ in which ${\displaystyle A\subset \bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}}$.

Definition 11.

A p-Housderff space $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is defined as a p-H-closed space if ${\beta }_i$-open cover $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ has a finite ${\beta }_i$-collection ${\left\{{\omega }_{{\alpha }_k}\right\}}_{k=1}^n$ such that ${\displaystyle A\subset \bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}}$, for all $i,j=1,2$.

Definition 12

([2]). A set A of a bitopological space is defined as regular open set if $Int\left(\overline{A}\right)=A$.

Definition 13.

If $Int\left(\overline{A}\right)=A$ in ${\beta }_1$ and $Int\left(\overline{A}\right)=A$ in ${\beta }_2$, then the subset A of bitopological space $(\chi ,{\beta }_1,{\beta }_2)$ is called a p-regular open set.

Theorem 2.

Consider $\chi =(\chi ,\beta )$ is a topological space. Then, the space χ is c-compact if and only if for all A subset of χ and for every $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ in which ${\omega }_{\alpha }$ is a regular open set and covers A, there exists a finite collection ${\left\{{\omega }_{{\alpha }_k}\right\}}_{k=1}^n$ of $\underset{\sim }{\mathsf{ϝ}}$ in which ${\displaystyle A\subset \bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}}$.

Proof. 

⇒ Consider $\chi$ is a c-compact space. Consider $A\subset \chi$ and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ such that ${\mathsf{ϝ}}_{\alpha }$ is a regular open set and covers A. Then, we have $Int\left(\overline{{\omega }_{\alpha }}\right)={\omega }_{\alpha }$, for all $\alpha \in A$. Since $Int\left(\overline{{\omega }_{\alpha }}\right)$ is open set for all $\alpha \in \wedge$, then by the c-compactness of $\chi$ the result is hold.

⇐ Consider the condition here is to prove that $\chi$ is a c-compact space. For this purpose, we consider that $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is an open cover of A, $\forall A\subset \chi$. So, we have

$${\displaystyle A\subset \bigcup _{\alpha \in \wedge }{\omega }_{\alpha }\subset \bigcup _{\alpha \in \wedge }\overline{{\omega }_{\alpha }}\subset \bigcup _{\alpha \in \wedge }Int\left(\overline{{\omega }_{\alpha }}\right).}$$

Thus, $\{Int\left(\overline{{\omega }_{\alpha }}\right),\alpha \in \wedge \}$ forms an open cover of A called $Int\left(\overline{{\omega }_{\alpha }}\right)=v_{\alpha }$, for all $\alpha \in \wedge$. Therefore, ${\displaystyle A\subset \bigcup _{\alpha \in \wedge }{v}_{\alpha }}$. Consequently, by the conditions of this theorem, we can have ${\displaystyle A\subset \bigcup _{\alpha \in \wedge }{v}_{{\alpha }_k}}$, and hence $\chi$ is a c-compact space. □

Theorem 3.

Consider $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a bitopological space. The space χ is p-space if and only if $\forall A$ subset of χ and for every ${\beta }_i$, $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ such that ${\omega }_{\alpha }$ is regular open set and covers A, there exists a ${\beta }_i$-finite collection ${\left\{{\omega }_{{\alpha }_k}\right\}}_{k=1}^n$ of $\underset{\sim }{\mathsf{ϝ}}$ such that ${\displaystyle A\subset \bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}}$, for all $i=1,2$.

Proof. 

⇒ Consider $\chi$ is a p-c-compact space. Consider $A\subset \chi$ and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ such that ${\omega }_{\alpha }$ is regular open set and ${\beta }_i$ covers A, for all $i=1,2$. So, we have $Int\left(\overline{{\omega }_{\alpha }}\right)={\omega }_{\alpha }$, for all $\alpha \in \wedge$ in ${\beta }_i$, for all $i=1,2$. Now, since $Int\left(\overline{{\omega }_{\alpha }}\right)$ is ${\beta }_i$-open set for all $\alpha \in \wedge$ and for all $i=1,2$, then by the p-c-compactness of $\chi$, the result is hold.

⇐ Consider the state here is to show that $\chi$ is a p-c-compact space. To this end, we consider $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is a ${\beta }_i$-open cover of A, for all $A\subset \chi$ and for all $i=1,2$. So, we have

$${\displaystyle A\subset \bigcup _{\alpha \in \wedge }{\omega }_{\alpha }\subset \bigcup _{\alpha \in \wedge }\overline{{\omega }_{\alpha }}\subset \bigcup _{\alpha \in \wedge }Int\left(\overline{{\omega }_{\alpha }}\right).}$$

Consequently, $\{Int\left(\overline{{\omega }_{\alpha }}\right),\alpha \in \wedge \}$ forms an open cover of A called $Int\left(\overline{{\omega }_{\alpha }}\right)=v_{\alpha }$, $\forall \alpha \in \wedge$. Therefore, we obtain ${\displaystyle A\subset \bigcup _{\alpha \in \wedge }{v}_{\alpha }}$. Thus, by the conditions of this theorem, we can have ${\displaystyle A\subset \bigcup _{\alpha \in \wedge }{v}_{{\alpha }_k}}$, and therefore $\chi$ is a p-c-compact space. □

3. Relations between $\mathit{c}$-Separation Axioms and $\mathit{c}$-Compact Spaces

In the following content, we continue deriving numerous results theoretically, but this time to demonstrate the relations between c-separation axioms and the c-compact spaces. In what follows, we state two important definitions in relation to the topological and bitopological spaces in which they would be very useful to establish the next theorems.

Definition 14.

Consider $\chi =(\chi ,\beta )$ is a topological space. The space χ is said to be

(i)

$c-T_0$-space if for all $\theta \ne \vartheta \in \chi$, there exists an open set ${\omega }_{\theta }$ in χ in which $\theta \in \overline{{\omega }_{\theta }}$ and $\vartheta \notin \overline{{\omega }_{\theta }}$, and there exists an open set $v_{\vartheta }$ in which $\vartheta \in \overline{v_{\vartheta }}$ and $\theta \notin \overline{v_{\vartheta }}$.

(ii)

c-compact $T_1$-space if for all $\theta \ne \vartheta$ in χ, there exists an open set ${\omega }_{\theta }$ in χ in which $\theta \in \overline{{\omega }_{\theta }}$ and $\vartheta \notin \overline{{\omega }_{\theta }}$, and there exists an open set $v_{\vartheta }$ in χ in which $\vartheta \notin \overline{v_{\vartheta }}$ and $\theta \notin \overline{v_{\vartheta }}$.

(iii)

c-compact $T_2$-space if for all $\theta \ne \vartheta$ in χ, there exist two open sets ${\omega }_{\theta }$ and $v_{\vartheta }$ in χ in which $\theta \in \overline{{\omega }_{\theta }}$, $\vartheta \in \overline{v_{\vartheta }}$ and $\overline{{\omega }_{\theta }}\cap \overline{v_{\vartheta }}=\varphi$.

(iv)

c-regular space if for all $\theta \notin A$ such that A is a closed subset in χ, there exist two open sets ${\omega }_{\theta }$ and $v_{\vartheta }$ in which $\theta \in {\omega }_{\theta }$, $A\subset \overline{v_A}$ and $\overline{{\omega }_{\theta }}\cap \overline{v_{\vartheta }}=\varphi$.

(v)

$c-T_3$-space if χ is a $c-T_1$-space and c-regular space.

(vi)

$c-T_4$-space if χ is c-normal space and c-$T_1$-space.

(vii)

c-normal space if ∀ closed sets A and B in which $A\cap B=\varphi$, there exist two open sets ${\omega }_{\theta }$ and $v_{\vartheta }$ in which $A\subset \overline{{\omega }_{\theta }}$, $B\subset \overline{v_{\vartheta }}$ and $\overline{{\omega }_{\theta }}\cap \overline{v_{\vartheta }}=\varphi$.

Definition 15.

Consider $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a bitopological space. The space χ is said to be

(i)

a p-c-$T_0$-space if for all $\theta \ne \vartheta \in \chi$, $\exists {\omega }_{\theta }$ of a ${\beta }_i$-open set in χ in which $\theta \in \overline{{\omega }_{\theta }}$ and $\vartheta \notin \overline{{\omega }_{\theta }}$, and $\exists v_{\vartheta }$ of a ${\beta }_j$-open set in which $\vartheta \in \overline{v_{\vartheta }}$ and $\theta \notin \overline{v_{\vartheta }}$ for all $i,j=1$.

(ii)

a p-c-compact $T_1$-space if $\forall \theta \ne \vartheta$ in χ, $\exists {\omega }_{\theta }$ of a ${\beta }_i$-open set in χ in which $\theta \in \overline{{\omega }_{\theta }}$ and $\vartheta \notin \overline{{\omega }_{\theta }}$, and $\exists v_{\vartheta }$ of a ${\beta }_j$-open set in χ in which $\vartheta \notin \overline{v_{\vartheta }}$ and $\theta \notin \overline{v_{\vartheta }}$, for all $i,j=1,2$.

(iii)

a p-c-compact $T_2$-space, if for all $\theta \ne \vartheta$ in χ, $\exists {\omega }_{\theta }$ of a ${\beta }_i$-open set and $v_{\vartheta }$ of a ${\beta }_j$-open set in χ in which $\theta \in \overline{{\omega }_{\theta }}$, $\vartheta \in \overline{v_{\vartheta }}$ and $\overline{{\omega }_{\theta }}\cap \overline{v_{\vartheta }}=\varphi$, $\forall i,j=1,2$.

(iv)

a p-c-regular space if for all $\theta \notin A$ and a $\beta i$-closed subset of χ, $\exists {\omega }_{\theta }$ of a ${\beta }_i$-open set and $v_{\vartheta }$ of a ${\beta }_j$-open set in which $\theta \in {\omega }_{\theta }$, $A\subset \overline{v_A}$ and $\overline{{\omega }_{\theta }}\cap \overline{v_{\vartheta }}=\varphi$, for all $i,j=1,2$.

(v)

a p-c-$T_3$-space if χ is a p-c-$T_1$-space and a p-c-regular space.

(vi)

a p-c-normal space if for all A of a ${\beta }_i$-closed set and for all B of a ${\beta }_j$-closed set in which $A\cap B=\varphi$, $\exists {\beta }_i$-open set and ${\beta }_j$-open set in which $A\subset \overline{{\omega }_{\theta }}$, $B\subset \overline{v_{\vartheta }}$ and $\overline{{\omega }_{\theta }}\cap \overline{v_{\vartheta }}=\varphi$, for all $i,j=1,2$.

(vii)

a p-c-$T_4$-space if χ is a p-c-normal space and a p-c-$T_1$-space.

Theorem 4.

Consider B is a c-compact subset in c-$T_2$-space. For all $\theta \ne \vartheta$, there exist two open sets ${\omega }_{\theta }$ and $v_B$ in which $\theta \in \overline{{\omega }_{\theta }}$, $B\subset \overline{v_B}$ and $\overline{{\omega }_{\theta }}\cap \overline{v_A}\doteq \varphi$.

Proof. 

Consider $b\in B$ and $\theta \ne a$. Since $\theta \notin B$ and $\chi$ is c-$T_2$-space. So, there exist two open sets ${\omega }_{\theta }\left(b\right)$ and $v\left(b\right)$ in which $\overline{{\omega }_{\theta }\left(b\right)}$ and $b\in \overline{v\left(b\right)}$ with $\overline{{\omega }_{\theta }\left(b\right)}\cap \overline{v_{{\alpha }_k}\left(b\right)}=\varphi$. So, $\underset{\sim }{v}=\{v\left(b\right):a\in B\}$ is an open cover of B. Due to $\overline{A}\subset \chi$, then there exists a collection ${\left\{v_{{\alpha }_{\theta }}\right\}}_{k=1}^n$ in which ${\displaystyle \overline{B}\subset \bigcup _{k=1}^n\overline{v_{{\alpha }_k}\left(b\right)}}$. Thus, we have $\theta \in \overline{v_{{\alpha }_k}\left(b\right)}$ and $B\subset \overline{v}$, where ${\displaystyle \overline{v}=\bigcup _{k=1}^n\overline{v_{{\alpha }_k}\left(b\right)}}$. As a result, we can obtain

$${\displaystyle \overline{\omega \left(b\right)}\cap \overline{v}=\overline{\omega \left(b\right)}\cap \bigcup _{k=1}^n\overline{v_{{\alpha }_k}\left(b\right)}=\bigcup _{k=1}^n\overline{\omega \left(b\right)}\cap \overline{v_{{\alpha }_k}\left(b\right)}=\bigcup _{k=1}^n\varphi =\varphi ,}$$

and hence the result is hold. □

Theorem 5.

Consider B is a p-c-compact subset of a p-c-$T_2$-space. For all $\theta \ne \vartheta$, $\exists {\beta }_i$-open set ${\omega }_{\theta }$ and ${\beta }_j$-open set $v_B$ in which $\theta \in \overline{{\omega }_{\theta }}$ and $B\subset \overline{v_B}$ with $\overline{{\omega }_{\theta }}\cap \overline{v_B}\doteq \varphi$, for all $i=1,2$.

Proof. 

Consider $b\in B$ and $\theta \ne a$. Since $\theta \notin B$ and $\chi$ is a p-c-$T_2$-space, then $\exists {\beta }_i$-open set ${\omega }_{\theta }\left(b\right)$ and ${\beta }_j$-open set $v\left(b\right)$ in which $\overline{{\omega }_{\theta }\left(b\right)}$ and $b\in \overline{v\left(b\right)}$ with $\overline{{\omega }_{\theta }\left(b\right)}\cap \overline{v\left(b\right)}=\varphi$, for $i\ne j$, $i,j=1,2$. Therefore, $\underset{\sim }{v}=\{v\left(b\right):a\in B\}$ is an open cover of B, and due to $\overline{B}\subset \chi$, so there exists a collection ${\left\{v_{{\alpha }_{\theta }}\right\}}_{k=1}^n$ in which ${\displaystyle \overline{B}\subset \bigcup _{k=1}^n\overline{v_{{\alpha }_k}\left(b\right)}}$. Thus, we obtain $\theta \in \overline{v_{\alpha \left(b\right)}}$ and $B\subset \overline{v}$, where ${\displaystyle \overline{v}=\bigcup _{k=1}^n\overline{v_{\alpha k}\left(b\right)}}$. This implies

$${\displaystyle \overline{\omega \left(b\right)}\cap \overline{v}=\overline{\omega \left(b\right)}\cap \bigcup _{k=1}^n\overline{v_{\alpha \left(b\right)}}=\bigcup _{k=1}^n\overline{\omega \left(b\right)}\cap \overline{v_{\alpha \left(b\right)}}=\bigcup _{k=1}^n\varphi =\varphi .}$$

□

Theorem 6.

Consider A and B are two disjoint c-compact subsets of a p-$T_2$-space $\chi =(\chi ,\beta )$. We can sperate A and B by two disjoint open sets ${\omega }_A$ and $v_B$ in which $A\subset {\omega }_A$ and $B\subset v_B$.

Proof. 

Consider we have two disjoint c-compact subsets A and B. Consider $\chi =(\chi ,\beta )$ is a $T_2$-space. Now, for all $a\in A$, we can obtain $a\notin B$ as $A\cap B=\varphi$. Since B is a c-compact subset of $\chi$, so by Theorem 5, there exist two $B_j$-open sets $\omega \left(a\right)$ and $v\left(B\right)$ in which $a\in \omega \left(a\right)$ and $B\subset v\left(B\right)$ with $\omega \left(a\right)\cap v\left(B\right)=\varphi$. Therefore, $\underset{\sim }{\mathsf{ϝ}}=\{\omega \left(a\right):a\in A\}$ represents an open cover of A, and hence ${\displaystyle A=\bigcup _{a\in A}\left(\omega \left(a\right)\right)}$. Due to A is c-compact subset, there exists a regular open set $\{{\omega }_k\left(a\right):k=1,2,\dots ,n,{\omega }_k\left(a\right)\}$ in which ${\displaystyle A\subset \bigcup _{k=1}^{n}Int\left(\overline{{\omega }_k\left(a\right)}\right)}$, (say ${\displaystyle \bigcup _{k=1}^{n}Int\left(\overline{{\omega }_k\left(a\right)}\right)={\omega }_A}$). So, we have $A\subset {\omega }_A$ and $B\subset v_B$ in which ${\omega }_A,v_B$ are two open sets. Thus, it is enough to show ${\omega }_A\cap v_B=\varphi$. To do so, one might have

$${\displaystyle {\omega }_A\cap v_B=(\bigcup _{k=1}^{n}Int\left(\overline{{\omega }_k\left(a\right)}\right)\cap V_B=\bigcup _{k=1}^{n}Int\left(\overline{{\omega }_k\left(a\right)}\right)\cap V_B.}$$

But we have ${\omega }_k\left(a\right))\cap v_B=\varphi$ and $Int\left(\overline{{\omega }_k\left(a\right)}\right)\subset {\omega }_k\left(a\right)$. Therefore, we get

$$Int\left(\overline{{\omega }_k\left(a\right)}\right)\cap V_B=\varphi ,$$

and hence ${\displaystyle {\omega }_A\cap v_B=\bigcup _{k=1}^n\varphi =\varphi }$. □

Theorem 7.

Consider we have two disjoint $B_i$-c-compact subsets A and B of a p-$T_2$-space $\chi =(\chi ,{\beta }_1,{\beta }_2)$. We can sperate A and B by two disjoint $B_j$-open sets ${\omega }_A$ and $v_B$ in which $A\subset {\omega }_A$ and $B\subset v_B$, for all $i\ne j$, $i,j=1,2$.

Proof. 

Consider we have two disjoint $B_i-$c$-compact$ subsets A and B, for all $i=1,2$. Consider $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a p-$T_2$-space. For all $a\in A$, we have $a\notin B$as$A\cap B=\varphi$. Since B is $B_i$-c-compact subset of $\chi$, so by Theorem 6, $\exists B_j$-open sets $\omega \left(a\right)$ and $v\left(B\right)$ in which $a\in \omega \left(a\right)$ and $B\subset v\left(B\right)$ with $\omega \left(a\right)\cap v\left(B\right)=\varphi$, for all $i,j=1,2$. Thus, $\underset{\sim }{\mathsf{ϝ}}=\{\omega \left(a\right):a\in A\}$ represents an $B_j$-open cover of A, and so ${\displaystyle A=\bigcup _{a\in A}\left(\omega \left(a\right)\right)}$. Due to A is a $B_i$-c-compact subset, so there exists a regular open set $\{{\omega }_k\left(a\right):k=1,2,\dots ,n,{\omega }_k\left(a\right)\}$ in which ${\displaystyle A\subset \bigcup _{k=1}^{n}Int\left(\overline{{\omega }_k\left(a\right)}\right)}$, (say ${\displaystyle \bigcup _{k=1}^{n}Int\left(\overline{{\omega }_k\left(a\right)}\right)={\omega }_A}$), for all $i=1,2$. Accordingly, $A\subset {\omega }_A$ and $B\subset v_B$ such that ${\omega }_A$ and $v_B$ are two $B_i$-open sets, for all $i=1,2$. From this point, it is enough to show ${\omega }_A\cap v_B=\varphi$. To do so, we have

$${\displaystyle {\omega }_A\cap v_B=(\bigcup _{k=1}^{n}Int\left(\overline{{\omega }_k\left(a\right)}\right)\cap V_B=\bigcup _{k=1}^{n}Int\left(\overline{{\omega }_k\left(a\right)}\right)\cap v_B.}$$

But ${\omega }_k\left(a\right))\cap v_B=\varphi$ and $Int\left(\overline{{\omega }_k\left(a\right)}\right)\subset {\omega }_k\left(a\right)$. Consequently, we have $Int\left(\overline{{\omega }_k\left(a\right)}\right)\cap v_B=\varphi$, and hence

$${\displaystyle {\omega }_A\cap v_B=\bigcup _{k=1}^n\varphi =\varphi .}$$

□

In subsequent paragraphs, we first introduce a specific definition that illustrates the concept of p-extremely disconnected bitopological space, followed by a certain theoretical result associated with such a definition. Afterward, we continue exploring further results in connection with the relationships between the c-separation axioms and the c-compact spaces.

Definition 16.

If every ${\beta }_i$-open set is a ${\beta }_i$-clopen set, then the bitopological space $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is called p-extremely disconnected, for all $i=1,2$.

Theorem 8.

The space $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a p-extremely disconnected compact space if and only if it is a p-c-compact space.

Proof. 

Consider $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is a ${\beta }_i$-open cover of A, for all $i=1,2$. Consider $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a p-extremely disconnected compact space and A be a subset of $\chi$. Then, ${\displaystyle A=\bigcup _{\alpha \in \wedge }{\omega }_{\alpha }}$ and $\chi =(\chi -A)\cup A$ implies

$${\displaystyle \chi =(\chi -A)\cup \left(\bigcup _{\alpha \in \wedge }{\omega }_{\alpha }\right).}$$

As a result, ${\underset{\sim }{U\mathsf{ϝ}}}^{*}=\{\chi -A,{\omega }_{\alpha }:\alpha \in \wedge \}$ represents an open cover of $\chi$. Due to $\chi$ is a p-compact space, then $\chi$ has a ${\beta }_i$-finite subcover, say

$${\displaystyle \chi =(\chi -A)\cup \left(\bigcup _{\alpha \in \wedge }{\omega }_{{\alpha }_k}\right):k=1,2,\dots ,n,}$$

for all $i=1,2$. Consequently, we get ${\displaystyle \chi =(\chi -A)\cup \bigcup _{k=1}^n{\omega }_{{\alpha }_k}}$. Thus, we attain ${\displaystyle A=\bigcup _{k=1}^n{\omega }_{{\alpha }_k}}$. Due to $\chi$ is a p-disconnected space, then $\overline{{\omega }_{{\alpha }_k}}={\omega }_{{\alpha }_k}$, for all $k=1,2,\dots ,n$. Thus, ${\displaystyle A=\bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}}$ for $\chi$ is a p-c-compact space. Now, consider $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a p-extremely disconnected c-compact space. Consider $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is a ${\beta }_i$-open cover of $\chi$, for $i=1,2$ and $A\subset \chi$. So, $\underset{\sim }{\mathsf{ϝ}}$ is a ${\beta }_i$-open cover of A, for all $i=1,2$. It means that ${\displaystyle A=\bigcup _{k=1}^n{\omega }_{{\alpha }_k}}$. But, we have

$${\displaystyle \chi =A\cup (\chi -A)=\left(\bigcup _{k=1}^n{\omega }_{{\alpha }_k}\right)\cup (\chi -A).}$$

So $\{\chi -A,{\omega }_{{\alpha }_k}:k=1,2,\dots ,n\}$ is a finite subcover of $\chi$, and therefore $\chi$ is a p-compact space. □

Theorem 9.

Every compact space $\chi =(\chi ,\beta )$ is a c-compact space.

Proof. 

Consider $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is an open cover of A, where A is a subset of $\chi$. So, $\{{\omega }_{\alpha }:\chi -A:\alpha \in \wedge \}$ forms an open cover of $\chi$. Due to $\chi$ is a compact space, we have ${\displaystyle \chi \subset \left(\bigcup _{k=1}^n{\omega }_{{\alpha }_k}\right)\cup (\chi -A)}$, and so ${\displaystyle A\subset \bigcup _{k=1}^n{\omega }_{{\alpha }_k}\subset \bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}}$. Therefore, $\{{\omega }_{{\alpha }_1},{\omega }_{{\alpha }_2},\dots ,{\omega }_{{\alpha }_n}\}$ is a collection of $\underset{\sim }{\mathsf{ϝ}}$ and ${\displaystyle A\subset \bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}}$. Therefore, $\chi$ is a c-compact space. □

Theorem 10.

Every p-compact space $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a p-c-compact space.

Proof. 

Consider $i=1,2$, and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is a $\beta i$-open cover of A, where A is a subset of $\chi$. So, $\{{\omega }_{\alpha }:\chi -A:\alpha \in \wedge \}$ forms a $\beta i$-open cover of $\chi$, for all $i=1,2$. Due to $\chi$ is a p-compact space, then ${\displaystyle \chi \subset \left(\bigcup _{k=1}^n{\omega }_{{\alpha }_k}\right)\cup (\chi -A)}$, and so ${\displaystyle A\subset \bigcup _{k=1}^n{\omega }_{{\alpha }_k}\subset \bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}}$. Therefore, $\{{\omega }_{{\alpha }_1},{\omega }_{{\alpha }_2},\dots ,{\omega }_{{\alpha }_n}\}$ is a $\beta i$-collection of $\underset{\sim }{\mathsf{ϝ}}$ and ${\displaystyle A\subset \bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}}$, for all $i=1,2$. Hence, $\chi$ is a p-c-compact space. □

Theorem 11.

Every extremely disconnected nearly compact space $\chi =(\chi ,\beta )$ is a c-compact space.

Proof. 

Consider $(\chi ,\beta )$ is an extremely disconnected nearly compact space. Consider $A\subset \chi$ and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is an open cover of A, so ${\displaystyle A\subset \bigcup _{\alpha \in \wedge }{\omega }_{\alpha }}$ and $\chi -A$ covers it set. Thus, $\{{\omega }_{\alpha },\chi -A:\alpha \in \wedge \}$ forms an open cover of $\chi$. But $\chi$ is extremely disconnected, then ${\omega }_{\alpha }$ is a clopen set $\forall \alpha \in \wedge$. Thus, we have ${\omega }_{\alpha }=\overline{{\omega }_{\alpha }}$, and so ${\omega }_{\alpha }^{\circ }=\overline{{\omega }_{\alpha }}$. Hence, we get ${\omega }_{\alpha }=\overline{{\omega }_{\alpha }^{\circ }}$, which gives ${\displaystyle \chi \subset \left(\bigcup _{\alpha \in \wedge }\overline{{\omega }_{\alpha }^{\circ }}\right)\cup (\chi -A)}$. Now, since $\chi$ is a nearly compact space, we have ${\displaystyle \chi \subset \left(\bigcup _{\alpha \in \wedge }\overline{{\omega }_{\alpha }^{\circ }}\right)\cup (\chi -A)}$, and so we have

$${\displaystyle A\subset \left(\bigcup _{k=1}^n\overline{{\omega }_{\alpha }^{\circ }}\right)=\subset \left(\bigcup _{k=1}^n\overline{{\omega }_{\alpha }^{\circ }}\right).}$$

Hence, $\chi$ is a c-compact space. □

Theorem 12.

Every p-extremely disconnected nearly compact space $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a p-c-compact space.

Proof. 

Consider $(\chi ,{\beta }_1,{\beta }_2)$ is a p-extremely disconnected nearly compact space. Consider that $A\subset \chi$ and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is a $\beta i$-open cover of A, for all $i=1,2$. So, ${\displaystyle A\subset \bigcup _{\alpha \in \wedge }{\omega }_{\alpha }}$ and $\chi -A$ covers it set. So, we have $\{{\omega }_{\alpha },\chi -A:\alpha \in \wedge \}$ forms a $\beta i$-open cover of $\chi$, for all $i=1,2$. But, $\chi$ is p-extremely disconnected, which implies that ${\omega }_{\alpha }$ is a $\beta i$-clopen set for all $\alpha \in \wedge$ and for all $i=1,2$. Therefore, ${\omega }_{\alpha }=\overline{{\omega }_{\alpha }}$, and so ${\omega }_{\alpha }^{\circ }=\overline{{\omega }_{\alpha }}$. Thus, we get ${\omega }_{\alpha }=\overline{{\omega }_{\alpha }^{\circ }}$, and consequently we obtain ${\displaystyle \chi \subset \left(\bigcup _{\alpha \in \wedge }\overline{{\omega }_{\alpha }^{\circ }}\right)\cup (\chi -A)}$. Now, since $\chi$ is a p-nearly compact space, then ${\displaystyle \chi \subset \left(\bigcup _{\alpha \in \wedge }\overline{{\omega }_{\alpha }^{\circ }}\right)\cup (\chi -A)}$. This immediately gives

$${\displaystyle A\subset \bigcup _{k=1}^n\overline{{\omega }_{\alpha }^{\circ }}.}$$

Hence, $\chi$ is a p-c-compact space. □

Theorem 13.

Every quasi H-closed space $\chi =(\chi ,\beta )$ is a c-compact space.

Proof. 

Consider $(\chi ,\beta )$ is a quasi H-closed space. Consider $A\subset \chi$ and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is an open cover of A. As a consequence, $\{{\omega }_{\alpha },\chi -A\}:\alpha \in \wedge \}$ forms an open cover of $\chi$, which is a quasi H-closed. Consequently, we have ${\displaystyle \chi \subset \left(\bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}\right)\cup \left(\overline{\chi -A}\right)}$. Now, due to $\left(\overline{\chi -A}\right)$ covers $\chi -A$, then ${\displaystyle A\subset \left(\bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}\right)}$, and hence $\chi$ is a c-compact space. □

Theorem 14.

Every quasi p-H-closed space $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a p-c-compact space.

Proof. 

Consider $(\chi ,{\beta }_1,{\beta }_2)$ is a p-quasi H-closed space. Consider $A\subset \chi$ and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is a $\beta i$-open cover of A, for all $i=1,2$. Then, $\{{\omega }_{\alpha },\chi -A:\alpha \in \wedge \}$ forms a p-open cover of $\chi$. Due to $\chi$ is quasi H-closed, we have ${\displaystyle \chi \subset \left(\bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}\right)\cup \left(\overline{\chi -A}\right)}$. Also, due to $\left(\overline{\chi -A}\right)$ covers $\chi -A$, then ${\displaystyle A\subset \left(\bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}\right)}$, and hence $\chi$ is a p-c-compact space. □

Theorem 15.

If $\chi =(\chi ,\beta )$ is an H-closed and S-closed space, then it is a c-compact space.

Proof. 

Consider $\chi =(\chi ,\beta )$ is an H-closed and S-closed space. Consider A is a subset of $\chi$ and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is an open cover of A. Then, there exists $\alpha \in \wedge$ in which ${\omega }_{\alpha }\subset A$$\subset \overline{{\omega }_{\alpha }}$ as $\chi$ is an S-closed space. Thus, we have

$${\displaystyle \bigcup _{\alpha \in \wedge }{\omega }_{\alpha }\subset A\subset \bigcup _{\alpha \in \wedge }\overline{{\omega }_{\alpha }}.}$$

This implies that $\{\overline{{\omega }_{\alpha }}:\alpha \in \wedge \}$ forms a cover of $\chi$, which is also a closure of $\chi$. In the same regard, since $\chi$ is an H-closed space, we get ${\displaystyle A\subset \bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}}$, and hence $\chi$ is a c-compact space. □

Theorem 16.

Consider $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a p-H-closed and a p-S-closed space, then it is a c-compact space.

Proof. 

Consider $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a p-H-closed and a p-S-closed space. Consider A is a subset of $\chi$ and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is a $\beta i$-open cover of A, for all $i=1,2$. So, $\exists \alpha \in \wedge$ in which ${\omega }_{\alpha }\subset A$ as $\chi$ is a p-S-closed space. As a consequence ${\displaystyle \bigcup _{\alpha \in \wedge }{\omega }_{\alpha }\subset A\subset \bigcup _{\alpha \in \wedge }\overline{{\omega }_{\alpha }}}$. Therefore, $\{\overline{{\omega }_{\alpha }}:\alpha \in \wedge \}$ forms a $\beta i$-H-closed cover of $\chi$, for all $i=1,2$. Thus, because of $\chi$ is p-H-closed space, then ${\displaystyle A\subset \bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}}$. Hence, $\chi$ is a p-c-compact. □

Theorem 17.

If $\chi =(\chi ,\beta )$ is an extremely disconnected space, then the statements below are equivalent:

(i)

χ is c-compact with respect to the closed subspace.

(ii)

χ is nearly compact.

(iii)

χ is a quasi H-closed space.

Proof. 

(i→ii)

Consider $\chi$ is c-compact and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is an open cover of $\chi$. Now, $\forall B\subset \chi$, we have $\underset{\sim }{\mathsf{ϝ}}$ is a cover of B. This means ${\displaystyle B\subset \bigcup _{\alpha \in \wedge }{\omega }_{\alpha }}$. But $\chi$ is a c-compact space, so ${\displaystyle B\subset \bigcup _{k=1}^n{\omega }_{\alpha }}$. Due to $\chi$ is an extremely disconnected space, so $\chi -B$ and ${\omega }_{{\alpha }_k}$ are clopen sets, for all $k=1,2,\dots ,n$. Thus, $\overline{{\omega }_{{\alpha }_k}^{\circ }}={\omega }_{{\alpha }_k}$, for all $k=1,2,\dots ,n$. This implies $B\subset \overline{{\omega }_{{\alpha }_k}^{\circ }}$, which means ${\displaystyle \chi \subset \left(\bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}^{\circ }}\right)\cup \left(\overline{\chi -B}\right)}$. Consequently, $\{\overline{{\omega }_{{\alpha }_k}^{\circ }}\overline{\chi -B},k=1,2,\dots ,n\}$ is a subcover of $\underset{\sim }{\mathsf{ϝ}}$ that covers $\chi$. Thus, $\chi$ is nearly compact space.

(ii→iii)

Consider $\chi$ is a nearly compact space and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is an open cover of $\chi$, so it has a finite subcover of interior of closure set, say $\{\overline{{\omega }_{{\alpha }_k}^{\circ }}\overline{\chi -B},k=1,2,\dots ,n\}$. With the use of nearly compactness of $\chi$, we attain ${\displaystyle \chi \subset \left(\bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}^{\circ }}\right)}$. Since $\chi$ is an extremely disconnected space, then ${\omega }_{{\alpha }_k}$ is a clopen set, for all $k=1,2,\dots ,n$. Hence, $\overline{{\omega }_{{\alpha }_k}^{\circ }}=\overline{{\omega }_{{\alpha }_k}}$, and so ${\displaystyle \chi \subset \bigcup _{k=1}^n\overline{{\omega }_{\alpha }}}$. This means that $\chi$ is an H-closed space.

(iii→i)

Consider $\chi$ is an H-closed space and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is an open cover of B, where B is a closed subspace of $\chi$. Thus, $\underset{\sim }{\mathsf{ϝ}}$ is an open cover of B. Since $\chi$ is an H-closed space, so ${\displaystyle B\subset \chi \subset \bigcup _{k=1}^n\overline{{\omega }_{\alpha }}}$. Also, due to $\chi$ is an extremely disconnected space, we obtain ${\displaystyle B\subset \bigcup _{k=1}^n\overline{{\omega }_{\alpha }}}$. Hence, $\chi$ is a c-compact space.

□

Theorem 18.

Consider $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a p-extremely disconnected space, then the following are equivalent:

(i)

χ is p-c-compact with respect to $\beta i$-closed subspace, for all $i=1,2$.

(ii)

χ is p-nearly compact.

(iii)

χ is a p-quasi H-closed space.

Proof. 

(i→ii)

Consider $\chi$ is p-c-compact and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is a $\beta i$-open cover of $\chi$, for all $i=1,2$. Now, for all $\beta i$-closed set $B\subset \chi$, we have $\underset{\sim }{\mathsf{ϝ}}$ is a $\beta i$-cover of B, for all $i=1,2$. This means ${\displaystyle B\subset \bigcup _{\alpha \in \wedge }{\omega }_{\alpha }}$. But $\chi$ is a p-c-compact, which implies ${\displaystyle B\subset \bigcup _{k=1}^n{\omega }_{\alpha }}$. In this regard, since $\chi$ is a p-extremely disconnected space, so $\chi -B$ and ${\omega }_{{\alpha }_k}$ are $\beta i$-clopen sets, for all $k=1,2,\dots ,n$ and for all $i=1,2$. Thus, one might get $\overline{{\omega }_{{\alpha }_k}^{\circ }}={\omega }_{{\alpha }_k}$, for all $k=1,2,\dots ,n$. As a result, $B\subset \overline{{\omega }_{{\alpha }_k}^{\circ }}$, which immediately yields ${\displaystyle \chi \subset \left(\bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}^{\circ }}\right)\cup \left(\overline{\chi -B}\right)}$. As a result, we have $\{\overline{{\omega }_{{\alpha }_k}^{\circ }}\overline{\chi -B},k=1,2,\dots ,n\}$ is a $\beta i$-subcover of $\underset{\sim }{\mathsf{ϝ}}$ of interior of closure of $\beta i$-open set that covers $\chi$, for all $i=1,2$. Hence, $\chi$ is a p-nearly compact space.

(ii→iii)

Consider $\chi$ is a p-nearly compact space and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is a $\beta i$-open cover of $\chi$, for all $i=1,2$. Then, it possesses a finite $\beta i$-subcover of interior of closure set, say $\{\overline{{\omega }_{{\alpha }_k}^{\circ }}\overline{\chi -B},k=1,2,\dots ,n\}$, for all $i=1,2$. By p-nearly compactness of $\chi$, we have ${\displaystyle \chi \subset \left(\bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}^{\circ }}\right)}$. Since $\chi$ is a p-extremely disconnected space, then ${\omega }_{{\alpha }_k}$ is a $\beta i$-clopen set, for all $k=1,2,\dots ,n$ and for all $i=1,2$. Hence, we have $\overline{{\omega }_{{\alpha }_k}^{\circ }}=\overline{{\omega }_{{\alpha }_k}}$, which consequently leads to ${\displaystyle \chi \subset \bigcup _{k=1}^n\overline{{\omega }_{\alpha }}}$. Therefore, $\chi$ is a p-H-closed space.

(iii→i)

Consider $\chi$ is a p-H-closed space and $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is an $\beta i$-open cover of B in which B is a $\beta i$-closed subspace of $\chi$, for all $i=1,2$. Therefore, $\underset{\sim }{\mathsf{ϝ}}$ is a $\beta i$-open cover of B, for all $i=1,2$. Due to $\chi$ is a p-H-closed space, then ${\displaystyle B\subset \chi \subset \bigcup _{k=1}^n\overline{{\omega }_{\alpha }}}$. Also, due to $\chi$ is a p-extremely disconnected space, then ${\displaystyle B\subset \bigcup _{k=1}^n\overline{{\omega }_{\alpha }}}$, and hence $\chi$ is a p-c-compact space.

□

Theorem 19.

The c-compactness possesses a hereditary property with respect to the closed subspace.

Proof. 

Consider $\chi$ is a c-compact space. Consider B is a closed subspace of $\chi$ and C is a subset of B. Consider $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is an open cover of $\beta$. Thus, we have $\chi =C\cup (\chi -C)$. Because of $C\subset B$, then $\chi =C\cup (\chi -B)$. Therefore, we obtain ${\displaystyle \chi =\left(\bigcup _{\alpha \in \wedge }{\omega }_{\alpha }\right)\cup (\chi -B)}$. This means that $\{{\omega }_{\alpha },\chi -B:\alpha \in \wedge \}$ forms an open cover. Since $\chi$ is c-compact, then every subset of $\chi$ might be covered by a finite subcover of closure of subset of $\underset{\sim }{\mathsf{ϝ}}$. So, we have ${\displaystyle \beta \subset \bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}}$, and hence B is a c-compact space. □

Theorem 20.

The p-c-compactness possesses a hereditary property with respect to $\beta i$-closed subspace, for all $i=1,2$.

Proof. 

Consider $\chi$ is a p-c-compact space. Consider B is a $\beta i$-closed subspace of $\chi$ and C is a $\beta i$-subset of B, for all $i=1,2$. Consider $\underset{\sim }{\mathsf{ϝ}}=\{{\omega }_{\alpha }:\alpha \in \wedge \}$ is an $\beta i$-open cover of $\beta$, for all $i=1,2$. Then, $\chi =C\cup (\chi -C)$. Since $C\subset B$, then $\chi =C\cup (\chi -B)$. This consequently yields that ${\displaystyle \chi =\left(\bigcup _{\alpha \in \wedge }{\omega }_{\alpha }\right)\cup (\chi -B)}$, which gives $\{{\omega }_{\alpha },\chi -B:\alpha \in \wedge \}$ forms a $\beta i$-open cover, for all $i=1,2$. Now, since $\chi$ is a p-c-compact space, then every subset of $\chi$ might be covered by a finite $\beta i$-subcover of closure of subset of $\underset{\sim }{\mathsf{ϝ}}$, for all $i=1,2$. So, we have ${\displaystyle \beta \subset \bigcup _{k=1}^n\overline{{\omega }_{{\alpha }_k}}}$, which means that B is a p-c-compact space. □

Theorem 21.

If $\chi =(\chi ,\beta )$ is a c-compact, $c-T_2$- and c-extremely disconnected space, then χ is $c-T_4$-space.

Proof. 

Consider $\chi =(\chi ,\beta )$ is a c-compact and c-$T_2$-space. It is clearly that $\chi =(\chi ,\beta )$ is a c-$T_1$-space. Now, consider A and B be c-closed subsets of $\chi$ in which $A\cap B=\varphi$. Due to $\chi$ is a c-compact space, so by Theorem 20, A and B are two c-compact subsets of c-$T_2$-space $\chi$. Also, by Theorem 20, there exist two c-open sets ${\omega }_A$ and $v_B$ in which $A\subseteq \overline{{\omega }_A}$ and $B\subseteq \overline{v_B}$ with $\overline{{\omega }_A}\cap \overline{v_B}=\varphi$. Thus, $\chi$ is a c-$T_4$-space. □

Theorem 22.

If $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a p-c-compact, p-c-$T_2$- and p-c-extremely disconnected space, then χ is a p-c-$T_4$-space.

Proof. 

Consider $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a p-c-compact and p-c-$T_2$-space. It is quite clear that $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a p-c-$T_1$-space. Now, consider A and B are two ${\beta }_i$-c-closed subsets of $\chi$ in which $A\cap B=\varphi$, for all $i=1,2$. As a consequence, due to $\chi$ is a p-c-compact space, then with the use of Theorem 21, A and B are ${\beta }_i$-c-compact subsets of p-c-$T_2$-space, for all $i=1,2$. So, with the use of Theorem 21, there exist two ${\beta }_i$-c-open sets ${\omega }_A$ and $v_B$ in which $A\subseteq \overline{{\omega }_A}$ and $B\subseteq \overline{v_B}$ with $\overline{{\omega }_A}\cap \overline{v_B}=\varphi$, for all $i=1,2$. Thus, $\chi$ is a p-c-$T_4$-space. □

Theorem 23.

Consider $\chi =(\chi ,\beta )$ is a c-$T_2$- and c-extremely disconnected space. Every subset of χ is closed set.

Proof. 

Consider A is a c-compact subset of $\chi$ and $\vartheta \notin A$. So, with the use of Theorem 22, there exist two c-open sets ${\omega }_{\theta }$ and $v_A$ in which $\theta \in {\omega }_{\theta }$ and $A\subseteq v_A$ with ${\omega }_{\theta }\cap v_A=\varphi$. Consequently, we have $\omega \subseteq {\left(v_A\right)}^c$, and because of $A\subseteq v$ yields $v^c\subseteq A^c$, then we have $\theta \in {\omega }_{\theta }\subseteq v^c\subset A^c$. As a result, due to $\omega$ is a c-open set, then $A^c$ is r-open set, which means that A is c-closed set. □

Theorem 24.

Consider $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a p-c-$T_2$- and p-c-extremely disconnected space. Every subset of χ is a ${\beta }_i$-closed set, for all $i=1,2$.

Proof. 

Consider A is a ${\beta }_i$-c-compact subset of $\chi$ and $\theta \notin A$, for all $i=1,2$. So, with the help of Theorem 23, there exist two ${\beta }_i$-c-open sets ${\omega }_{\theta }$ and $v_A$ in which $\theta \in {\omega }_{\theta }$ and $A\subseteq v_A$ with ${\omega }_{\theta }\cap v_A=\varphi$, for all $i=1,2$. Therefore, we have $\omega \subseteq {\left(v_A\right)}^c$. Because of $A\subseteq v$ leads to $v^c\subseteq A^c$, then we have $\theta \in {\omega }_{\theta }\subseteq v^c\subset A^c$. Also, due to $\omega$ is a ${\beta }_i$-c-open set, then $A^c$ is a ${\beta }_i$-c-open set, for all $i=1,2$. Thus, A is ${\beta }_i$-r-closed set, for all $i=1,2$. □

Theorem 25.

If $\chi =(\chi ,\beta )$ is a c-compact, c-$T_2$- and c-extremely disconnected space, then every subset of χ is c-compact if and only if it is c-closed set.

Proof. 

⇒ Consider A is a c-compact subset of $\chi$, so with the help of Theorem 24, A is c-closed set.

⇐ Consider A is a c-closed of a c-$T_2$-extremely disconnected space, then by Theorem 24, A is c-compact. □

Theorem 26.

If $\chi =(\chi ,{\beta }_1,{\beta }_2)$ is a p-c-compact, p-c-$T_2$- and p-c-extremely disconnected space, then every ${\beta }_i$-subset of χ is a p-c-compact if and only if it is a ${\beta }_i$-c-closed set, for all $i=1,2$.

Proof. 

⇒ Consider A is a p-c-compact subset of $\chi$, so with the help of Theorem 25, A is a ${\beta }_i$-c-closed set, for all $i=1,2$.

⇐ Consider A is a ${\beta }_i$-c-closed of a p-c-compact, p-c-$T_2$-extremely disconnected space, so with the help of Theorem 25, A is a ${\beta }_i$-c-compact, for all $i=1,2.$ □

4. Conclusions

In this work, we have initiated a novel concept, named the p-c-compact in topological and bitopological spaces. Accordingly, we have defined the concept of c-compact space and inferred some novel generalizations and results related to the H-closed, the quasi compact and extremely disconnected compact spaces in topological and bitopological spaces. In addition, we have derived several theoretical results that demonstrate the relations between c-separation axioms and the c-compact spaces. However, this study can be extended to the c-compactness in tritopological space $(\chi ,{\beta }_1,{\beta }_2,{\beta }_3)$, where ${\beta }_1$, ${\beta }_2$ and ${\beta }_3$ are topologies on $\chi$. Based on this conception, many properties and results can be then inferred and derived from such a study, which would be left to the future for further considerations.