Open set recognition algorithm based on Conditional Gaussian Encoder

Abbreviations:

$P$: pairwise spaces; $KC$: closed compact spaces; $P$-$KC$: pairwise closed compact spaces; $P$-Hausdorff: pairwise Hausdorff spaces; $KC$-spaces: closed compact spaces; $P$-$KC$-spaces: pairwise closed compact spaces; $LC$-spaces: closed lindlöf spaces; $P$-$LC$-spaces: pairwise closed lindlöf spaces; $\mathbb{R}$: the set of all real numbers; $\mathbb{Q}$: the set of all rational numbers; $\mathbb{Z}$: the set of all integers; $\mathbb{N}$: the set of all natural numbers; $\tau _{u}$: the usual topology; $\tau _{s}$: the Sorgenfrey line topology; $\tau _{cof}$: the coffinite topology; $\tau _{coc}$: the cocountable topology; $\tau _{l.r}$: the left-ray topology; $\tau _{r.r}$: the right-ray topology; $\tau _{dis}$: the discrete topology; $\tau _{ind}$: the indiscrete topology; $Cl_{i}A$: the $\tau_{i}$-closure of $A$

1.   Introduction

The basic concepts of bitopological spaces began to be studied in 1963 by mathematician Kelly ^[1]. Several authors have since addressed the problem of defining compactness in bitopological spaces, as seen in Kim ^[2].

In 1969, Fletcher et al. ^[3] introduced the main definitions of $\tau _{i}\tau _{j}$-open covers and $P$-open covers in bitopological spaces. A cover $\tilde{U}$ of a bitopological space $\left(X, \tau _{i}, \tau _{j}\right)$ is called $\tau _{i}\tau _{j}$-open if

If $\tilde{U}$ contains at least one non-empty member of $\tau _{i}$ and at least one non-empty member of $\tau _{j}$, it is called $P$-open. They also defined the concepts of pairwise compact ($P$-compact) spaces. In 1972, Datta ^[4] studied the concept of semi-compact ($S$-compact) spaces in the bitopological space $\left(X, \tau _{i}, \tau _{j}\right)$. Cooke and Reilly ^[5] discussed the relationships between these previous definitions in 1975. In 1983, Fora and Hdieb ^[6] introduced the concepts of pairwise Lindelöf ($P$-Lindelöf) and semi-Lindelöf ($S$-Lindelöf) spaces. They also provided the definitions of certain types of functions as follows: function

is called $P$-continuous (or $P$-closed, respectively) if both functions

and

are continuous (or closed, respectively).

This overview introduces $KC$-bitopological spaces, emphasizing their importance in relation to compactness and closedness. The concept of $KC$-topological spaces was first developed by Hewitt in the early 1940s, with foundational definitions and illustrative examples provided, see ^[7]. Hewitt introduced minimal topological spaces, establishing that every Hausdorff compact space is a minimal $KC$-space "$mKC$-space". By 1947, it was proven that every compact $KC$-space is an $mKC$-space. In 1965, the auther Aull, shows his contribution in ^[7] developed a space thats found between $T_1$ and $T_2$, namely, $KC$-space. The relationship between them is represented by the following diagram: $T_2 \rightarrow KC \rightarrow T_1$, researchers examined space in ^[8,9,10,11]. Wilansky ^[12] explored the relationships between separation axioms between $T_{1}$ and $T_{2}$ spaces, highlighting $KC$-topological spaces. It was established that every $T_{2}$-space is a $KC$-space, and every $KC$-space is a $T_{1}$-space, see also ^[13,14]. In 2004, Valdis showed that a minimal space where every compact subset is closed is countably compact^[15], leading to the result that every minimal $KC$-space is countably compact ^[16,17,18]. Ali ^[8] in 2006 expanded on this by introducing $KC$-space, minimal $KC$-space, and minimal Hausdorff spaces, as well as minimal $LC$-topological spaces. Ali found that the direct image of a $KC$-space under a continuous function is not necessarily a $KC$-space unless certain conditions are met. Further, the relationships between minimal $KC$-topological spaces and minimal Hausdorff or $LC$-topological spaces were studied.

In 2006, Alas and Wilson ^[19] examined the minimal properties of spaces between $T_{1}$ and $T_{2}$ spaces, noting that $KC$-spaces extend Hausdorff spaces. They introduced $KC$-closed spaces and discussed their relationship with $KC$-spaces, see also ^[9,16,19]. In 2008, Bella and Costantini ^[9] showed that any minimal $KC$-space must be compact, though not necessarily Hausdorff. In 2009, Oprsal ^[20] addressed the problem of whether every $KC$-space with a weaker $KC$-topology is compact, a question resolved by Bella and Costantini ^[9]. In 2010, Zarif and Razzak ^[11] linked $KC$-topological spaces with concepts such as connected functions, closed functions, and $K$-functions, yielding significant results. In 2011, Adnan ^[7] introduced Cooke-topological spaces and analyzed their properties and relationships with $KC$-spaces. Bella and Costantini ^[21] introduced $SC$-spaces, defined by closed convergent sequences and their limits, and explored the relationships between $T_{2}$, $KC$, $SC$, and $T_{1}$ spaces as $T_{2} \longrightarrow KC \longrightarrow SC \longrightarrow T_{1}$. Jebour and Saleh ^[22] introduced $K$-$(SC)$-spaces as weaker versions of $KC$-topological spaces, developing new results, see also ^[23,24]. In 2015, Ali and Abker ^[25] introduced minimal compact closed spaces "$mKC$-spaces" and examined their relations with other spaces, including $KC$-topological spaces. A new definition for $\alpha$-$KC$-spaces was proposed, with results indicating that every $KC$-space is an $\alpha$-$KC$-space, see the study ^[26]. For more studies about this topic, see ^[13,27]. In 2018, Almohor and Hdieb ^[28] explored properties of pairwise $L$-closed spaces ($LC$-topological spaces), contributing to the field by generalizing relationships between $LC$-topological spaces and $KC$-topological spaces. Nadhim et al. introduced concepts of strong and weak forms of $KC$-topological spaces^[29].

In this paper, we first introduce the concept of closed compactness, referred to as "$KC$-spaces" in bitopological spaces. We provide basic definitions for $KC$-bitopological spaces and pairwise $KC$-topological spaces, deriving many related results. Illustrative examples and theories of these two main concepts are discussed. We also explore how these concepts relate to other topological concepts within bitopological spaces.

Next, we study the effect of various types of functions on $KC$-bitopological spaces and pairwise $KC$-topological spaces. We examine the necessary conditions for the direct image of a $KC$-bitopological space and a pairwise $KC$-topological space to remain a $KC$-bitopological space and a pairwise $KC$-topological space, respectively. Additional conditions are established for the inverse image of these spaces to retain their properties.

Finally, we analyze the relationship between $KC$-bitopological spaces, pairwise $KC$-topological spaces, and other bitopological spaces, such as $P$-compact spaces. This involves studying the impact of different functions on $KC$-bitopological spaces and pairwise $KC$-topological spaces. For instance, a bitopological space $\left(X, \tau _{i}, \tau _{j}\right)$ is considered compact if both $\left(X, \tau _{i}\right)$ and $\left(X, \tau _{j}\right)$ are compact spaces.

2.   $KC$-bitopological spaces

In this section, we introduce the concepts of $KC$-bitopological spaces and pairwise $KC$-topological spaces.

We will examine their properties and explore their relationships with other bitopological spaces.

Definition 1. (1) A bitopological space $\left(X, \tau _{1}, \tau _{2}\right)$ is said to be a $P$-$KC$-space if each $\tau _{i}$-compact subset of $X$ is $\tau _{j}$-closed for all $\left(i\neq j, \; i, j = 1, 2\right)$.

(2) A bitopological space $\left(X, \tau _{1}, \tau _{2}\right)$ is said to be a $KC$-space if each $\tau _{i}$-compact subset of $X$ is $\tau _{i}$-closed for all $\left(\; i = 1\; ,\; 2\right)$.

Remark 1. Part (2) in the above definition is equivalent to that both $\left(X, \tau _{1}\right)$ and $\left(X, \tau _{2}\right)$ are $KC$-spaces.

In general: If both $(X, \tau _{i})$ and $\left(X, \tau _{j}\right)$ have the same topological property $P$, then the bitopological space $\left(X, \tau _{i}, \tau _{j}\right)$ has property $P$.

So, we say that a bitopological space $\left(X, \tau _{i}, \tau _{j}\right)$ is a $KC$-space, if both $\left(X, \tau _{i}\right)$ and $\left(X, \tau _{j}\right)$ are $KC$-spaces.

Example 1. (1) The bitopological space $\left(\mathbb{R}, \tau _{ind}, \tau _{dis}\right)$ is not a $KC$-space, since any compact subset of $\left(\mathbb{R} \; ,\tau _{dis}\right)$ is closed but (1, 2) is a compact subset in $\left(\mathbb{R} \; ,\tau _{ind}\right)$, which is not closed.

(2) The bitopological space $\left(\mathbb{R}, \tau _{dis}, \tau _{u}\right)$ is $KC$-space since any compact subset of both $\left(\mathbb{R} \; ,\tau _{dis}\right)$ and $\left(\mathbb{R} \; ,\tau _{u}\right)$ is closed.

(3) The bitopological space $\left(\mathbb{R}, \tau _{cof}, \tau _{coc}\right)$ is a $KC$-space.

Example 2. (1) The bitopological space $\left(\mathbb{R}, \tau _{ind}, \tau _{dis}\right)$ is not a $P$-$KC$-space.

(2) The bitopological space $\left(\mathbb{R}, \tau _{cof}, \tau _{dis}\right)$ is a $P$-$KC$-space.

(3) The bitopological space $\left(\mathbb{R}, \tau _{coc}, \tau _{dis}\right)$ is a $P$-$KC$-space.

Example 3. (1) The bitopological space $\left(\mathbb{R}, \tau _{u}, \tau _{s}\right)$ is a $KC$-space, but not a $P$-$KC$-space.

(2) The bitopological space $\left(\mathbb{R}, \tau _{l.r}, \tau _{r.r}\right)$ is never a $KC$-space nor a $P$-$KC$-space.

(3) The bitopological space $\left(\mathbb{R}, \tau _{ind}, \tau _{l.r}\right)$ is never a $KC$-space nor a $P$-$KC$-space.

Definition 2. (1) A bitopological space $(X, \tau_1, \tau_2)$ is said to be a $P$-$LC$-space. Each $\tau_i$-lindlöf subset of $X$ is $\tau_i$-closed for all $(i\neq j, i, j = 1, 2)$.

(2) A bitopological space $(X, \tau_1, \tau_2)$ is said to be a $LC$-space. Each $\tau_i$-lindlöf subset of $X$ is $\tau_i$-closed for all $(i = 1, 2)$.

Since every compact ($\tau_{i}$-compact, respectively) is lindelöf ($\tau_{i}$-lindelöf, respectively) the prove of the following two theorems is clear.

Theorem 1. Every $P$-$LC$-bitopological space is a $P$-$KC$-space.

Theorem 2. Every $LC$-bitopological space is a $KC$-space.

Example 4. $(\mathcal{R}, \tau_{dis}, \tau_{coc})$ is a $P$-$LC$-space.

Example 5. $(\mathcal{R}, \tau_{dis}, \tau_{ind})$ is not a $LC$-space.

Example 6. $(\mathcal{R}, \tau_{r.r}, \tau_{l.r})$ is never a $LC$-space nor a $P$-$LC$-space.

It is easy to prove the following theorems:

Theorem 3. Every discrete space is a $KC$-space.

Theorem 4. Every discrete bitopological space $\left(X, \tau _{dis}, \tau _{dis}\right) \;$is a $KC$-space.

Definition 3. Let $\left(X, \overset{1}{\tau _{X}}\right)$ and $\left(X, \overset{2}{ \tau _{X}}\right)$ be two topological spaces defined on the same set $X$ and $A\subseteq X$. If $\left(A, \overset{1}{\tau _{A}}\right)$ and $\left(A, \overset{2}{\tau _{A}}\right)$ are two subspaces of $\left(X, \overset{1}{\tau _{X}}\right)$ and $\left(X, \overset{2}{\tau _{X}} \right)$, respectively, then $\left(A, \overset{1}{\tau _{A}}, \overset{2}{ \tau _{A}}\right)$ is a subspace of $\left(X, \overset{1}{\tau _{X}}, \overset{2}{\tau _{X}}\right)$.

Theorem 5. Every subspace of a $P$-$KC$-space is a $P$-$KC$-space.

Proof. Assume that $\left(X, \overset{1}{\tau _{X}}, \overset{2}{\tau _{X}}\right)$ a $P$-$KC$-space and $\left(A, \overset{1}{\tau _{A}}, \overset{2}{ \tau _{A}}\right)$ a subspace of $\left(X, \overset{1}{\tau _{X}}, \overset{2}{\tau _{X}}\right)$. Let $K$ be a $\overset{1}{\tau _{A}}$ compact subset of $\left(A, \overset{1}{\tau _{A}}\right)$, $K$ is a$\; \overset{1}{\tau _{X}}$ compact subset of $\left(X, \overset{1}{\tau _{X}} \right)$ but $\left(X, \overset{1}{\tau _{X}}, \overset{2}{\tau _{X}} \right)$ is a $P$-$KC$space, so $K$ is $\overset{2}{\tau _{X}}$ closed. Hence,

is $\overset{2}{\tau _{A}}$ closed in $\left(A, \overset {2}{\tau _{A}}\right).$

Therefore, any $\overset{1}{\tau _{A}}$-compact subset of $A$ is $\overset{2}{\tau _{A}}$-closed. In the same way, we prove that any $\overset{2}{\tau _{A}}$-compact subset of $A$ is $\overset{1}{\tau _{A}}$-closed. Thus, $\left(A, \overset{1}{\tau _{A}}, \overset{2}{\tau _{A}} \right)$ is a $P$-$KC$-space. □

Theorem 6. Every subspace of a $KC$-bitopological space $\left(X, \overset{1}{\tau _{X}}, \overset{2}{\tau _{X}}\right) \;$is a $KC$-space.

Proof. Assume that $\left(X, \overset{1}{\tau _{X }}, \overset{2}{\tau _{X}}\right)$ is a $KC$-space and $\left(A, \overset{1}{\tau _{A}}, \overset{2}{\tau _{A}}\right)$ is a subspace of $\left(X, \overset{1}{\tau _{X }}, \overset{2 }{\tau _{X}}\right)$. Let $K$ be a $\overset{1}{\tau _{A}}$ compact subset of $\left(A, \overset{1}{\tau _{A}}\right)$, $K$ is a $\overset{1}{ \tau _{A}}$compact subset of $\left(X, \overset{1}{\tau _{X}}\right)$, but $\left(X, \overset{1}{\tau _{X}}, \overset{2}{\tau _{X}}\right)$ is a $KC$-space, so $K$ is $\overset{1}{\tau _{X}}$ closed. Hence,

is $\overset{1}{\tau _{A}}$ closed in $\left(A, \overset{1}{\tau _{A}} \right).$

Therefore, any $\overset{1}{\tau _{A}}$-compact subset of $A$ is $\overset{1}{\tau _{A}}$-closed. In the same way we prove that any $\overset{2 }{\tau _{A}}$-compact subset of $A$ is $\overset{2}{\tau _{A}}$-closed. Thus $\left(A, \overset{1}{\tau _{A}}, \overset{2}{\tau _{A}}\right)$ is $KC$- space. □

Theorem 7. The intersection of any two $KC$-spaces is a $KC$-space.

Proof. Let $\left(X, \overset{1}{\tau _{X}}\right)$ and $\left(X, \overset{2}{ \tau _{X }}\right)$ be two $KC$-spaces. Let

and $K$ be a compact subset of $\left(X, \overset{3}{\tau _{X }}\right)$, $K$ is a compact subset of both $\left(X, \overset{1}{\tau _{X}}\right)$ and $\left(X, \overset{2}{\tau _{X }}\right).$ Thus, $K$ is closed in both $\overset{1}{ \tau _{X }}$ and $\overset{2}{\tau _{X }}$, so $K$ is closed in $\overset{3} {\tau _{X }}$. Therefore, $\left(X, \overset{3}{\tau _{X }}\right)$ is a $KC$-space. □

Definition 4. The intersection of the bitopological spaces $\left(X, \tau _{i}, \tau _{j}\right)$ and $\left(X, \overset{\ast }{\tau _{i}}, \overset{\ast }{ \tau _{j}}\right)$ is the bitopological space $\left(X, \tau _{i}\cap \overset{\ast }{\tau _{i}}, \tau _{j}\cap \overset{\ast }{\tau _{j}}\right)$.

Theorem 8. The intersection of any two $KC$-bitopological spaces defined on the same set is a $KC$-bitopological space.

Proof. Let $\left(X, \tau _{i}, \tau _{j}\right)$ and $\left(X, \overset{\ast }{ \tau _{i}}, \overset{\ast }{\tau _{j}}\right)$ be two $KC$-bitopological spaces, and

be the intersection of $\left(X, \tau _{i}, \tau _{j}\right)$ and $\left(X, \overset{\ast }{\tau _{i}}, \overset{\ast }{\tau _{j}}\right),$ where

and

Let $K$ be a $\sigma _{i}$-compact subset of $\left(X, \sigma _{i}\right)$. By above theorem, $K$ is $\sigma _{i}$-closed subset of $\left(X, \sigma _{i}\right)$. Hence, $\left(X, \sigma _{i}\right)$ is a $KC$-space. Similarly, $\left(X, \sigma _{j}\right)$ is $KC$-space. Therefore, $\left(X, \sigma _{i}, \sigma _{j}\right)$ is a $KC$-space. □

Theorem 9. The intersection of any two $P$-$KC$-topological spaces defined on the same set is a $P$-$KC$-space.

Proof. Let $\left(X, \tau _{i}, \tau _{j}\right)$ and $\left(X, \overset{\ast }{ \tau _{i}}, \overset{\ast }{\tau _{j}}\right)$ be two $P$-$KC$-topological spaces, and

be the intersection of $\left(X, \tau _{i}, \tau _{j}\right)$ and $\left(X, \overset{\ast }{\tau _{i}}, \overset{\ast }{\tau _{j}}\right),$ where

and

Let $K$ be a $\sigma _{i}$- compact subset of $\left(X, \sigma _{i}\right)$. $K$ is $\tau _{i}$- compact and $\tau _{i}^{*}$-compact, but both $\left(X, \tau _{i}, \tau _{j}\right)$ and $\left(X, \overset{\ast }{\tau _{i}}, \overset{\ast }{ \tau _{j}}\right)$ are $P$-$KC$-spaces, so $K$ is closed in both $\tau _{j}$ and $\overset{\ast }{\tau _{j}}.$ Hence, $K$ is $\sigma _{j}$- closed. Therefore, $\left(X, \sigma _{i}, \sigma _{j}\right)$ is a $P$-$KC$- space. □

Definition 5. A bitopological space $\left(X, \tau _{i}, \tau _{j}\right)$ is called $P$-Hausdorff if for distinct points $a$ and $b$, there is a $\tau_{i}$-open set $U$ and a $\tau _{j}$-open set $V$ such that $a\in U, b\in V$, and

Definition 6. A bitopological space $\left(X, \tau _{i}, \tau _{j}\right)$ is called Hausdorff (resp. compact) if both $\left(X, \tau _{i}\right)$ and $\left(X, \tau_{j}\right)$ are Hausdorff (resp. compact) spaces.

Theorem 10. Every Hausdorff space is a $KC$-space.

Proof. Let $A$ be a $\tau _{i}$-compact subset of $X$. Every compact subspace of a Hausdorff space is closed, then $A$ is $\tau _{i}$-closed. Similarly, we can show that if $A$ is a $\tau _{j}$-compact subset of $X$, then $A$ is $\tau _{j}$-closed. □

Example 7. The bitopological space $\left(\mathbb{R}, \tau _{dis}, \tau _{u}\right)$ is a Hausdorff space; so it is a $KC$-space.

Remark 2. The converse of above theorem is not true; see the following example.

Example 8. The bitopological space $\left(\mathbb{R}, \tau _{cof}, \tau _{coc}\right)$ is a $KC$-space but not a Hausdorff space.

Recall that: In a topological space $\left(X, \tau _{X}\right)$, if every countable intersection of any collection of open sets is open, then $X$ is called a $p$-space.

Theorem 11. Every Hausdorff $p$-space is a $LC$-space.

Proof. Assume that $\left(X, \tau _{i}, \tau _{j} \right)$ is a Hausdroff $P$-space. Let $D$ be a $\tau_{i}$-Lindelöf subset of $X$. $D$ is a $\tau_{i}$-Lindelöf subset of Hausdroff $P$-space $\left(X, \tau _{i} \right)$. Therefore, $D$ is $\tau_{i}$-closed.

Similarly, we can get if $D$ is a $\tau_{j}$-Lindelöf, then $D$ is $\tau_{j}$-closed. □

Example 9. A space $\left(\mathbb{N}, \tau_{dis}, \tau_{coc} \right)$ is a Hausdorff $P$-space, so it is a $LC$-space.

Definition 7. If $\left(X, \overset{1}{\tau _{X}}\right)$ and $\left(X, \overset{2}{ \tau _{X}}\right)$ are two metric spaces, then $\left(X, \overset{1}{\tau _{X}}, \overset{2}{\tau _{X}}\right)$ is called the bitopological metric space.

Corollary 1. Every metric space is a $KC$-space.

Proof. Since every metric space is a Hausdorff then it is a $KC$-space. □

Corollary 2. Every bitopological metric space is a $KC$-space.

The following theorem can be found in ^[6].

Theorem 12. If a bitopological space $\left(X, \tau _{i}, \tau _{j}\right)$ is a $P$- Hausdorff space, then for each $x$ in $X$ we have:

a)

where $Cl_{i}\; V_{\alpha}$ is a $\tau_{i}$-closure of $V_{\alpha}$.

b)

where $Cl_{j}\; U_{\beta}$ is a $\tau_{j}$-closure of $U_{\beta}$.

Proof. Let $y\in X$ such that $x\neq y$. There exists a $\tau _{i}$-open set $U_{i}$ and a $\tau _{j}$-open set $U_{j}$ such that $y\in U_{i}$, $x\in U_{j}$, and

Since

then

Hence, $x\in Cl_{i}U_{j}$, $\forall \tau _{j}$-open sets $U_{j}$ and

This proves part (a). The proof of part (b) is similar to (a).

□

Theorem 13. Let $\left(X, \tau _{i}, \tau _{j}\right)$ be a $P$-Hausdorff space. Then, every $\tau _{i}$-compact subset of $X$ is $\tau _{j}$-closed $\left(i\neq j, i, \; j = 1, 2\right)$.

Proof. Let $B$ be a $\tau _{i}$-compact subset of $X$ and $x\in X-B$. By above theorem,

Since

$\left\{ X-Cl_{i}\; U_{\alpha }:\alpha \in \Delta \right\}$ is a $\tau _{i}$-open cover of a $\tau _{i}$-compact set $B$. So, there exists a finite subset $\overset{\ast }{\Delta } \subseteq \Delta$ such that $\left\{ X-Cl_{i}\; U_{\alpha }:\alpha \in \overset{\ast }{\Delta }\right\}$ is a cover of $B$.

Hence,

Letting

then $U$ is a $\tau _{j}$-open set such that

Hence, $B$ is $\tau _{j}$-closed. □

Theorem 14. Every $P$-Hausdorff space $\left(X, \tau _{i}, \tau _{j}\right)$ is a $p$-$KC$-space.

Proof. Let $A$ be a $\tau _{i}$-compact subset of a $P$-Hausdorff space $X$. By above theorem, $A$ is $\tau _{j}$-closed where $\left(i\neq j, \; i\, \, ,\; j = 1, 2\right)$. Hence, $\left(X, \tau _{i}, \tau _{j}\right)$ is a $P$-$KC$-space. □

Example 10. The bitopological space $\left(\mathbb{R}, \tau _{cof}, \tau _{dis}\right)$ is a $P$-Hausdorff, so it is a $P$-$KC$-space.

Remark 3. The converse of above theorem is not true, see the following example:

Example 11. The bitopological space $\left(\mathbb{R}, \tau _{cof}, \tau _{coc}\right)$ is a $P$-$KC$-space but not a $P$-Hausdorff.

Theorem 15. Every $P$-Hausdorff $P$-space is a $P$-$LC$-space.

Proof. Let $B$ be a $\tau _{i}$-lindelöf subset of $X$ and $x\in X-B$.

Since

$\left\{ X-Cl_{i}\; U_{\alpha }:\alpha \in \Delta \right\}$ is a $\tau _{i}$-open cover of a $\tau _{i}$-lindelöf set $B$. So, there exists a countable subset $\overset{\ast }{\Delta }\subseteq \Delta$ such that $\left\{ X-Cl_{i}\; U_{\alpha }:\alpha \in \overset{\ast }{\Delta }\right\}$ is a cover of $B$. Hence,

Letting

then $U$ is a $\tau _{j}$-open set such that $x\in U\subseteq X-B$. Hence, $B$ is $\tau _{j}$-closed. □

Example 12. $\left(\mathbb{Z}, \tau _{coc}, \tau _{dis}\right)$ is a $P$-Hausdorff $P$-space, so it is a $P$-$LC$-space.

Example 13. $\left(\mathbb{N}, \tau _{coc}, \tau _{cof}\right)$ is a $P$-$LC$-space but not a $P$-Hausdorff and $P$-space.

Theorem 16. Let $\left(X, \tau_{i}, \tau_{j}\right)$ be a compact $P$-$KC$-space. Then, $\tau_{i} = \tau_{j}$.

Proof. Let $\phi \neq W$ and $w\in \tau_{i}$. Then, $X-W$ is a $\tau_{i}$-closed subset of a compact space $\left(X, \tau_{i}\right)$. So, $X-W$ is $\tau_{i}$-compact subset of $P$-$KC$-space $X$. Therefore, $X-W$ is a $\tau_{j}$-closed. Hence, $W$ is a $\tau_{j}$-open and $\tau_{i}\subseteq \tau_{j}$. Similarly, we can get $\tau_{j}\subseteq \tau_{i}$. Consequently, $\tau_{i} = \tau_{j}$. □

Corollary 3. If a space $\left(X, \tau_{i}, \tau_{j}\right)$ is a compact $P$-Hausdorff, then $\tau_{i} = \tau_{j}$.

Proof. Let $H\in \tau_{i}$. $X-H$ is a $\tau_{i}$-closed subset of a compact space $\left(X, \tau_{i}\right)$. Therefore, $X-H$ is a $\tau_{i}$-compact subset of $P$-Hausdorff space $X$. Hence, $X-H$ is $\tau_{j}$-closed, so $H$ is $\tau_{j}$-open. So, $H\in \tau_{j}$, and then $\tau_{i}\subseteq \tau_{j}$. In the same way, we get $\tau_{j}\subseteq \tau_{i}$. Consequently, $\tau_{i} = \tau_{j}$. □

Theorem 17. Let $\left(X, \tau _{i}, \tau _{j}\right)$ be a Lindelöf $P$-$LC$-space $P$-space. Then, $\tau _{i} = \tau _{j}$.

Proof. Let $\phi \neq O\in \tau_{i}$. Then, $X-O$ is $\tau_{i}$-closed subset of a Lindelöf $P$-space $\left(X, \tau _{i}\right)$. So, $X-O$ is $\tau _{i}$-Lindelöf. But, $X$ is a $P$-$LC$-space, so $X-O$ is $\tau _{j}$-closed and then $O\in \tau _{j}$. Therefore, $\tau _{i}\subseteq \tau _{j}$. By the same technique, we obtained $\tau _{j}\subseteq \tau _{i}$. Consequently, $\tau _{i} = \tau _{j}$. □

Corollary 4. If the space $\left(X, \tau _{i}, \tau _{j}\right)$ is a Lindelöf $P$-Hausdorff $P$-space, then $\tau _{i} = \tau _{j}$.

The proof comes directly from the fact that "every $P$-Hausdorff $P$-space is a $P$-$LC$-space".

3.   Mappings on $KC$-bitopological spaces

In this section, we discuss the effects of various types of functions on $KC$-bitopological spaces and pairwise $KC$-topological spaces.

Definition 8. A function

is said to be a compact function, if $\; g^{-1}\left\{ y\right\}$ is $\tau _{i}$-compact and $\tau _{j}$-compact for each $y\in Y$.

We can find the following definition in ^[11].

Definition 9. A function

is said to be a $K$-function, if $g^{-1}\left\{ B\right\}$ is a compact subset of $X$ for all compact subsets $B$ of $Y$ and $g\left(A\right)$ is a compact subset of $Y$ for all compact subsets $A$ of $X$.

Definition 10. A function

is said to be a $K$-function, if both functions

and

are $K$-functions.

Theorem 18. Let

be an onto closed $K$-function. If $X$ is a $P$-$KC$-space, then so is $Y$.

Proof. Let $B$ be a $\sigma _{i}$-compact subset of $Y$. To show that $B$ is a $\sigma _{j}$-closed subset of $Y.$ Since $g$ is a $K$-function, then $g^{-1}\left\{ B\right\}$ is a $\tau _{i}$-compact subset of $X$.

But, $X$ is a $P$-$KC$-space, so $g^{-1}\left\{ B\right\}$ is a $\tau _{j}$-closed. Since $g$ is an onto closed, function then

is $\sigma _{j}$-closed in $Y$. So, $Y$ is a $P$-$KC$-space. □

We presented some definitions that will be used later.

Definition 11. ^[6] A function

is called $P$-continuous ($P$-closed, respectively) if the functions

and

are continuous (closed, respectively).

Theorem 19. Let

be an onto $P$-closed $K$-function. If $X$ is $P$-Hausdorff, then $Y$ is a $P$-$KC$-space.

Proof. Assume that

be an onto $P$-closed $K$-function. Let $A$ be a $\sigma _{i}$-compact subset of $Y$. Since $g$ is a $K$-function, then $g^{-1}\left(A\right)$ is a $\tau _{i}$-compact subset of $X$. Since $X$ is $P$-Hausdorff, then $g^{-1}\left(A\right)$ is $\tau _{j}$-closed. Hence,

is $\sigma _{j}$-closed because

is closed. Hence, $Y$ is a $P$-$KC$-space. □

Theorem 20. Let

be an onto $P$-closed $K$-function. If $X$ is Hausdorff, then $Y$ is a $KC$-space.

Proof. Assume that

be an onto $P$-closed $K$-function. Let $A$ be a $\sigma _{i}$-compact subset of $Y$. Since $g$ is a $K$-function, then $g^{-1}\left(A\right)$ is a $\tau _{i}$-compact subset of $X$. Since $X$ is Hausdorff, then $g^{-1}\left(A\right)$ is $\tau _{i}$- closed. Hence,

is $\sigma _{i}$-closed because

is closed. Hence, $Y$ is a $KC$-space. □

Definition 12. ^[3] A cover $\widetilde{U}$ of a bitopological space $\left(X, \tau _{i}, \tau _{j}\right)$ is called a $P$-open cover if $\widetilde{U}$ contains at least one non-empty element of $\tau _{i}$ and at least one non-empty element of $\tau _{j}$.

Definition 13. ^[3] A bitopological space $\left(X, \tau _{i}, \tau _{j}\right)$ is called $P$-compact (resp. Lindelöf) if each $P$-open cover of $X$ has a finite (resp. countable) subcover.

Example 14. The bitopological space $\left(\mathbb{R}, \tau _{dis}, \tau _{coc}\right)$ is not $P$-compact. Since

is a $P$-open cover of $X$ which has no finite subcover if $\widetilde{U}$ has a finite subcover $\acute{U}$, then

where $x_{i}\in \mathbb{Q} \; \forall \; i = 1, 2, 3, ..., n$.

This means

which is a contradiction.

Example 15. Consider $X = \mathbb{R}$. Let

and

Let $\tau _{1}$ and $\tau _{2}$ be the topologies on $X$ induced by the bases ${\rm{ß}}_{1}$ and ${\rm{ß}}_{2}$, respectively. $\left(\mathbb{R}, \tau _{1}, \tau _{2}\right)$ is a $P$-compact space since any $P$-open cover of $X$ must contain $\left\{ X\right\}$. Hence, $\left\{ X\right\}$ is a finite subcover of any $P$-open cover.

Theorem 21. Let $\left(X, \tau _{i}, \tau _{j}\right)$ be a $P$-compact space, every $\tau _{i}$-closed proper subset of $X$ is $\tau _{j}$-compact, where $\left(i\neq j, \; i, j = 1, 2\right)$.

Proof. Let $F$ be a $\tau _{i}$-closed proper subset of a $P$-compact $X$, and

be a $\tau_{j}$-open cover of $F$.

For each $x\in X-F$, there exists a $\tau _{i}$-open set $U\left(x\right)$ such that

Now,

is a $P$-open cover of the $P$-compact space $X$, so there exists a finite set $\Delta _{1}\subseteq \Delta$ and a finite set

such that

is a finite cover of $X$. Since

then

So, $\left\{ U_{\alpha }:\alpha \in \Delta _{1}\right\}$ is a finite subcover of $\widetilde{U}$ for $F$. Therefore, $F$ is $\tau _{j}$-compact. □

Remark 4. The expression (proper subset) in the previous theory can not be dispensed with or removed.

Example 16. Consider $\left(\mathbb{Z}, \tau_{coc}, \tau_{dis} \right)$. Then, $\mathbb{Z}$ is $\tau_{coc}$-closed but it is not $\tau_{dis}$-compact.

Theorem 22. Let

be $P$-continuous and $X$ be $P$-compact. If $Y$ is a $P$-$KC$-space then, $g$ is $P$-closed.

Proof. Let $A$ be a $\tau _{i}$-closed subset of a $P$-compact space $\left(X, \tau _{i}, \tau _{j}\right)$. $A$ is a $\tau _{j}$-compact, where $\left(i\neq j, i, \; j = 1, 2\right)$. Since $g$ is $P$-continuous, then $g\left(A\right)$ is a $\sigma _{j}$-compact subset of $Y$. But, $Y$ is a $P$-$KC$-space, so $g\left(A\right)$ is a $\sigma _{i}$-closed subset of $Y$, where $\left(i\neq j, \; i, j = 1, 2\right)$. Hence,

is closed. Similarly, we can show that

is closed. Therefore, $g$ is $P$-closed. □

Theorem 23. Let

be $P$-continuous and $X$ be a compact space. If $Y$ is a $KC$-space, then $g$ is $P$-closed.

Proof. Let $A$ be a $\tau _{i}$-closed subset of a compact space $\left(X, \tau _{i}, \tau _{j}\right)$. Then, $A$ is $\tau _{i}$-compact. Since $g$ is $P$-continuous, then $g\left(A\right)$ is a $\sigma _{i}$-compact subset of $Y$. But, $Y$ is a $KC$-space, so $g\left(A\right)$ is a $\sigma _{i}$-closed subset of $Y$. Hence,

is closed. Similarly, we can show that

is closed. Therefore, $g$ is $P$-closed. □

We now remind an important theory in single topological spaces.

Theorem 24. Every locally compact $KC$-space is $T_{2}$.

This previous theory can be generelized in bitopological spaces and proven in the same way as follows:

Theorem 25. Every locally compact $KC$-bitopological space is $T_{2}$.

Theorem 26. Let

be an onto $P$-continuous function. If $X$ is $T_{2}$ compact, and $Y$ is a $KC$-space, then $Y$ is $T_{2}.$

Proof. Since $g$ is an onto, $P$-continuous function, then $Y$ is compact, so $Y$ is locally compact. But, $Y$ is a $KC$-space, hence $Y$ is $T_{2}$. □

Theorem 27. Let

be a $P$-continuous bijective $K$-function. If $Y$ is a $P$-$KC$-space, then

is a $P$-$KC$-space.

Proof. Let $A$ be a $\tau _{i}$-compact subset of $X$. We prove that $A$ is $\tau _{j}$-closed subset of $X$. Since $A$ is a $\tau _{i}$-compact subset of $X$, then $g\left(A\right)$ is $\sigma _{i}$-compact subset of $Y$ because $g$ is a $K$-function. But, $Y$ is a $P$-$KC$-space, so $g\left(A\right)$ is a $\sigma _{j}$-closed subset of $Y$. Since $g$ is $P$- continuous bijective function, then

is $\tau _{j}$-closed in $X$. Similarly, if $A$ is a $\tau _{j}$-compact subset of $X$, then $A$ is $\tau _{i}$-closed. Hence,

is a $P$-$KC$-space. □

Theorem 28. Let

be a $P$-continuous bijective $K$-function. If $Y$ is a $KC$-space, then

is a $KC$-space.

Proof. Let $A$ be a $\tau _{i}$-compact subset of $X$. We prove that $A$ is a $\tau _{i}$-closed subset of $X$. Since $A$ be a $\tau _{i}$-compact subset of $X$, then $g\left(A\right)$ is a $\sigma _{i}$-compact subset of $Y$ because $g$ is a $K$-function. But, $Y$ is a $KC$-space, so $g\left(A\right)$ is a $\sigma _{i}$-closed subset of $Y$. Since $g$ is a $P$-continuous bijective function, then

is $\tau _{i}$-closed in $X$. Similarly, if $A$ is a $\tau _{j}$-compact subset of $X$, then $A$ is $\tau _{j}$-closed. Hence,

is a $KC$-space. □

Theorem 29. Let

be a $K$-function. If $X$ and $Y$ are $P$-compact $P$-$KC$-spaces, then $g$ is $P$-continuous and $P$-closed.

Proof. First, we show that $g$ is $P$-continuous. Let $A$ be a $\sigma _{i}$-closed subset of $Y$. Since $Y$ is $P$-compact, then $A$ is $\sigma _{j}$-compact $\left(i\neq j, \; i, j = 1, 2\right)$. Since $g$ is a $K$-function, then $g^{-1}\left(A\right)$ is a $\tau _{j}$-compact subset of $X$. Hence, $g^{-1}\left(A\right)$ is a $\tau _{i}$-closed subset of $X$ because $X$ is a $P$-$KC$-space. So,

is continuous. Similarly, we can show that

is continuous. Hence, $g$ is $P$-continuous.

Second, we show that $g$ is $P$-closed.

Let $C$ be a $\tau _{i}$-closed subset of $X$, then, $C$ is a $\tau _{j}$-compact subset of $X$. Since $g$ is a $K$-function, then $g\left(C\right)$ is a $\sigma _{j}$-compact subset of $Y$. So, $g\left(C\right)$ is $\sigma _{i}$-closed subset of $Y$ because $Y$ is $P$-$KC$-space. Hence,

is closed. Similarly, we can show that

is closed. Hence, $g$ is $P$-closed. □

Theorem 30. Let

be a $K$-function. If $X$ and $Y$ are compact $KC$-spaces, then $g$ is $P$-continuous and $P$-closed.

Proof. First, we show that $g$ is $P$-continuous.

Let $A$ be a $\sigma _{i}$-closed subset of $Y$. Since $Y$ is compact, then $A$ is $\sigma _{i}$-compact. Since $g$ is a $K$-function, then $g^{-1}\left(A\right)$ is a $\tau _{i}$-compact subset of $X$. Hence, $g^{-1}\left(A\right)$ is a $\tau _{i}$-closed subset of $X$ because $X$ is a $KC$-space. So,

is continuous. Similarly, we can show that

is continuous. Hence, $g$ is $P$-continuous.

Second, we show that $g$ is $P$-closed. Let $C$ be a $\tau _{i}$-closed subset of $X$. $C$ is a $\tau _{i}$-compact subset of $X$. Since, $g$ is a $K$-function, then $g\left(C\right)$ is a $\sigma _{i}$-compact subset of $Y$. So, $g\left(C\right)$ is a $\sigma_{i}$-closed subset of $Y$ because $Y$ is a $KC$-space. Hence,

is closed. Similarly, we can show that

is closed. Hence, $g$ is $P$-closed. □

4.   Conclusions

In the introduction to this research we noted, this research carefully reviewed previous studies on the topic, highlighting the main results and contributions of those studies. At the end of the introduction to the study, we provided an overview and summary of the results and conclusions we reached. The key findings are as follows:

(1) The study introduces and develops the fundamental definitions of $KC$-bitopological spaces and pairwise $KC$-topological spaces.

(2) A variety of illustrative examples are provided to support and reinforce the study's subject matter.

(3) The concepts of $KC$-bitopological spaces and pairwise $KC$-topological spaces are linked to other important topological concepts within bitopological spaces, clarifying their interrelationships.

(4) The study examines the effects of different types of functions on the direct and indirect images of $KC$-bitopological spaces and pairwise $KC$-topological spaces.

(5) Necessary and sufficient conditions are established for certain functions to ensure that the direct and indirect images of $KC$-bitopological spaces and pairwise $KC$-topological spaces remain within these respective categories.

Author contributions

Hamza Qoqazeh: proposed and set the main title, wrote the basic definitions of the subject of the study, established and provd the basic theories contained, general supervision of the research implementation process; Ali Atoom: wrote the introduction, added a number of theories and proved them; Maryam Alholi: wrote a summary; enriched the subject with illustrative examples; Eman ALmuhur: developed the main results and conclusions, checked the examples; Eman Hussein: carried out a scientific audit on the correctness of the formulation of the theories contained in the research and their proof; Anas Owledat: checked the grammar and linguistics; Abeer Al-Nana: reviewed the previous studies, documented the main references, examined the percentage of scientific inference. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

The researchers express their sincere thanks and appreciation to all who contributed to the preparation and enhancement of this research. We also extend our gratitude and respect to the esteemed scholars whose work in this field has been cited.

Conflict of interest

The researchers declare no personal interests in the publication of this paper. This research is original, and its primary aim is to contribute to the advancement of scientific knowledge in the field of general topology.