On almost set-Menger spaces in bitopological context

1.   Introduction

1.1.   Selection principles

Covering properties of a topological space is one of the most active research fields and has a long history which appears in papers ^[1,2,3,4]. More recently, the theory known as infinite combinatorial topology or selection principles in mathematics was introduced by M. Scheepers ^[5,6] applying selection principles to different open covers of a topological space and initiated a systematic study of selection principles. One of the most important features of the theory is to gather topology with the other fields of mathematics such as game theory, Ramsey theory, algebraic structures, etc. The theory has been widely studied and is still being studied. It also gives new viewpoints and frames for theoretical and applicable areas (see ^[7,8,9]). Several topological properties and concepts are defined and characterized by way of two classical selection principles stated in ^[6] as follows:

Let $X$ be an infinite set, and $\mathcal{A}$ and $\mathcal{B}$ are the families of subsets of it.

$S_{1}(\mathcal{A}, \mathcal{B})$ is the selection principle: For every sequence $(A_{n}: n\in\mathbb{\mathbb{N}})$ of members of $\mathcal{A}$, there exists a sequence $(b_{n}: n\in\mathbb{N})$ such that for each $n\in\mathbb{N}$, $b_{n}\in A_{n}$ and $\{b_{n}:n\in\mathbb{N}\}\in\mathcal{B}$.

$S_{fin}(\mathcal{A}, \mathcal{B})$ is the selection principle: For every sequence $(A_{n}: n\in\mathbb{N})$ of members of $\mathcal{A}$, there is a sequence $(B_{n}: n\in\mathbb{N})$ such that $B_{n}\subset A_{n}$ where $B_{n}$ is a finite set for all $n\in\mathbb{N}$ and $\bigcup_{n\in\mathbb{N}} B_{n}\in\mathcal{B}$.

In ^[3] Menger introduced the Menger basis property for metric spaces. In ^[1], Hurewicz introduced a property which is nowadays known as the Menger property (MP) and proved that the Menger basis property is equivalent to the MP. It is known that a topological space $X$ having the MP is equivalent to $X\in S_{fin}(\mathcal{O}, \mathcal{O})$ in Scheepers' notation where $\mathcal{O}$ is the family of open covers of $X$ and defined as the following:

$X$ has the MP if for all the sequence $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ where $\mathcal{D}_{n}$ is an open cover of $X$ for each $n$, there exists a finite subset $\mathcal{C}_{n}\subset\mathcal{D}_{n}$ such that $\bigcup_{n\in\mathbb{N}}\bigcup\mathcal{C}_{n} = X$.

Hurewicz showed that the statement that a metrizable space is $\sigma$-compact if and only if it has MP is equivalent to Menger conjecture. In ^[2], he introduced a property which is stronger than the MP currently referred to as the Hurewicz Property that is defined as the following.

Let $X$ be a topological space and $(\mathcal{D}_{n}: n\in\mathbb{N})$ be any sequence of open covers of $X$. If there is a sequence $(\mathcal{C}_{n}: n\in\mathbb{N})$ such that each $\mathcal{C}_{n}$ is the finite subset of $\mathcal{D}_{n}$ for each $n$ and all member $x$ of $X$, $\mid\{n\in\mathbb{N}: x\notin\bigcup\mathcal{C}_{n}\}\mid < \omega$ holds. It is well known that every $\sigma$-compact topological space has the Hurewicz property, and every topological space which has the Hurewicz property has the Menger property.

The Menger property has recently been studied extensively ^[10,11,12]. Also, general forms of the Menger property have been studied. Kočinac ^[13,14] introduced the almost Menger property. A topological space $X$ called almost Menger if every sequence $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ of open covers of $X$ there is a sequence $(\mathcal{C}_{n})_{n\in\mathbb{N}}$ where each $\mathcal{C}_{n}$ is finite subset of $\mathcal{D}_{n}$ for eacn $n\in\mathbb{N}$ and $X = \bigcup_{n\in\mathbb{N}}\overline{\bigcup\mathcal{C}_{n}}$. Kocev ^[15] studied this notion systematically. Also, the weak Menger property has been introduced in ^[16] and studied in ^[17,18]. These generalizations of the Menger property got their place in many papers ^{[19,20,21,22,23,24]}.

1.2.   Relatively topological spaces and set covering properties

In ^[25], a cardinal function $sL$ was defined by Arhangel'skii. Let $X$ be a topological space. Then the $sL(X)$ of $X$ is the minimal cardinality $\kappa$ such that for every subset $S\subset X$ and every open cover $\mathcal{D}$ of $\overline{S}$, there is a subfamily $\mathcal{D}^{*}\subset\mathcal{D}$ providing $\mid\mathcal{D^{*}}\mid\leq\kappa$ and $S\subset\overline{\bigcup\mathcal{D}^{*}}$. If $sL(X) = \omega$, then the space $X$ is called $s$-Lindelöf space.

In any topological subject, it could be important to know "how a given subspace of a topological space is located in the space". Let $X$ be a topological space, and denote with $\mathcal{P}(X)$ the power set of $X$. Let $Y\subset X$ and $P\subset \mathcal{P}(X)$. The properties of the subset $Y$ or each member of $P$ depend on how $Y$ or members of $P$ are placed in $X$. That is why it is so natural that two investigation areas arise. One of them is assigning a relative $S$ of the subset $Y$ of $X$ for given a topological property $S$, and second one is assigning a property $P$-$S$ showing how every member $P$ located in $X$. With his collaborators, A.V. Arhangel'skii first applied these investigations ^[25,26,27]. In this sense, the same investigations have been applied to selection principles, too (see ^{[21,28,29,30,31]}).

Motivated by the definition of an "$s$-Lindelöf cardinal function" of Arhangel'skii and this line of investigations, in ^[32] Kočinac and Konca defined set-Menger, set-Rothberger, set-Hurewicz spaces and their weaker forms considering the Menger, Rothberger and Hurewicz covering properties. Later, they defined in ^[33] the star versions of related spaces, and they initiated the investigation new sorts of selective covering properties known as set covering properties (see also ^[34]).

Definition 1.1. ^[32] Let $X$ be a topological space and $P\subset\mathcal{P}(X)$ such that $\emptyset\notin P$. $X$ is said to be:

(1) $P$-Menger if for all $S\in P$ and each sequence $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ of open covers of $\overline{S}$, there is a sequence $(\mathcal{D}_{n}^{*})_{n\in\mathbb{N}})$ where $\mathcal{D}_{n}^{*}$ is the finite subfamily of $\mathcal{D}_{n}$ for each $n$ and $S\subset\bigcup_{n\in\mathbb{N}}\bigcup\mathcal{D}_{n}^{*}$.

(2) Almost $P$-Menger if $S\subset\bigcup_{n\in\mathbb{N}}\overline{\bigcup\mathcal{D}_{n}^{*}}$.

If $P = \mathcal{P}\setminus\{\emptyset\}$ it is said that $X$ is set-Menger (for short SM) or almost set-Menger (ASM), respectively.

In this sense, we extend this investigation and introduce the $ij$-almost-set-Menger ($ij$-ASM) property and study it in bitopological spaces. Bitopological selection principles, which is the active research field, have been discussed in several papers ^{[35,36,37,38,39,40,41,42,43]}. The paper is generated as follows. In Section 2, we introduce the $ij$-almost-set-Menger property in bitopological context. After giving some examples for this kind of spaces, we investigate the equivalences of these spaces to other spaces like set-Menger and $ij$-almost Lindelöf. We also look at the behavior of $ij$-almost-set-Menger spaces under some different types of mappings defined in bitopological spaces. Later, in Section 3, We keep in sight the preservation of this property under union, subspaces and products. We introduce the class of $ij$-almost $P_{\gamma}$-set and investigate some properties in Section 4.

2.   Definitions, equivalences and examples

We use usual notations and terminology for topological spaces as in ^[44]. Our notation and terminology will follow ^[45] for bitopological spaces. By $\mathbb{N}, \mathbb{P}$ and $\mathbb{R}$, we denote the sets of natural, irrational and real numbers, respectively. During the paper, by $(X, \sigma)$ we denote a topological space while $(X, \sigma_{1}, \sigma_{2})$ (sometimes $X$) denotes a bitopological space (or shortly, bispace) which is a set $X$ equipped with two topologies, in general unrelated, $\sigma_{1}$ and $\sigma_{2}$ (see ^[46]). For any $A\subset X$, $\sigma_{i} - cl(A)$ denotes closure of $A$, and $\sigma_{i} - int(A)$ denotes the interior of $A$ with respect to the topology $\sigma_{i}$ $(i = 1, 2)$.

Definition 2.1. A bitopological space ${(X, \sigma_{1}, \sigma_{2})}$ is said to be $ij$-almost-set-Menger ($ij$-ASM, for short) ($i, j = 1, 2$) if, for all nonempty $A\subset X$ and for each $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ sequence of $\sigma_{i}$-open covers of ${\sigma_{i} - cl}(A)$, there exists a sequence $(\mathcal{C}_{n})_{n\in\mathbb{N}}$ of finite families such that $\mathcal{C}_{n}\subset\mathcal{D}_{n}$ for each $n\in\mathbb{N}$ and $A\subset\bigcup_{n\in\mathbb{N}}\bigcup_{V\in\mathcal{C}_{n}}{\sigma_{j} - cl}(V)$

In the light of this definition, we can give the following proposition.

Proposition 2.1. Let ${(X, \sigma_{1}, \sigma_{2})}$ be a bitopological space,

(1) If $(X, \sigma_{1})$ is a Menger space (or SM), then ${(X, \sigma_{1}, \sigma_{2})}$ is $12$-ASM.

(2) If $(X, \sigma_{1})$ is ASM and $\sigma_{2}\leq\sigma_{1}$ ($\sigma_{2}$ is coarser than $\sigma_{1}$), then ${(X, \sigma_{1}, \sigma_{2})}$ is $12$-ASM.

Proof. (1) Obvious from the corresponding definitions.

(2) Let $A\subset X$, and $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ be any sequence of $\sigma_{1}$-open covers of $\sigma_{1} - cl(A)$. Since $(X, \sigma_{1})$ is ASM, then there exists a sequence $(\mathcal{C}_{n})_{n\in\mathbb{N}}$ where $\mathcal{C}_{n}\subset\mathcal{D}_{n}$ is finite for each $n\in\mathbb{N}$ and $A\subset\bigcup_{n\in\mathbb{N}}\bigcup_{V\in\mathcal{C}_{n}}\sigma_{1} - cl(V)$ holds. Since $\sigma_{2}\leq\sigma_{1}$, then $\sigma_{1} - cl(V)\subset\sigma_{2} - cl(V)$ for all $V\in\mathcal{C}_{n}$ and $n\in\mathbb{N}$. This shows that ${(X, \sigma_{1}, \sigma_{2})}$ is $12$-ASM.

Example 2.1. Assume $\mathbb{R}$ equipped with cocountable topology $\sigma_{1}$ and the Sorgenfrey topology $\sigma_{2}$. Since $(\mathbb{R}, \sigma_{1})$ is SM, $(\mathbb{R}, \sigma_{1}, \sigma_{2})$ is $12$-ASM.

Example 2.2. Assume the bispace $(\mathbb{R}, \sigma_{1}, \sigma_{2})$ where $\sigma_{1}$ and $\sigma_{2}$ are the Smirnov's deleted topology and usual topology, respectively (see ^[47]). The topological space $(\mathbb{R}, \sigma_{1})$ is Menger (and SM) and $\sigma_{2}\leq\sigma_{1}$. So the bispace $(\mathbb{R}, \sigma_{1}, \sigma_{2})$ is $12$-ASM.

On the other hand, the assertion converse in Proposition $2.1(1)$ does not hold in general as the following example illustrates.

Example 2.3. Let $\mathbb{P}$ be the irrational numbers set and $a\in\mathbb{P}$ fixed point. Assume particular point topology $\sigma_{1} = \{U\subseteq\mathbb{R}: a\in U\}\cup\{\emptyset\}$ and $\sigma_{2} = \{U\cap\mathbb{P}: U\in\sigma_{1}\}$. Then, the followings are obtained.

(1) Since $a\in U$ for all $U\in\sigma_{1}$ it is obvious that $\sigma_{2}\leq\sigma_{1}$.

(2) $\mathcal{D} = \{\{x, a\}:x\in\mathbb{R}\}$ is an open cover for $(\mathbb{R}, \sigma_{1})$. Choose $\mathcal{D}_{n} = \mathcal{D}$ for all $n\in\mathbb{N}$, then $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ is the sequence of open covers of $(\mathbb{R}, \sigma_{1})$. This sequence assures that $(\mathbb{R}, \sigma_{1})$ is not Menger (so not SM).

(3) Since each nonempty open subset is dense in $(\mathbb{R}, \sigma_{1})$, then $(\mathbb{R}, \sigma_{1})$ is ASM.

(4) $(\mathbb{R}, \sigma_{1}, \sigma_{2})$ is $12$-ASM with the statement $1$.

That is why it is so natural to discuss under what conditions the converse assertion in Proposition $2.1(1)$ holds. In this sense, we firstly give the following definition.

Definition 2.2. ^[48] A bispace ${(X, \sigma_{1}, \sigma_{2})}$ is said to be an $ij$-regular space $(i, j = 1, 2, \, i\neq j)$ if, for each element $x\in X$ and each $\sigma_{i}$-closed set $F$ with $x\notin F$, there are a $\sigma_{i}$-open set $U$ and $\sigma_{j}$-open set $V$ such that $x\in U$, $V\supset F$ and $U\cap V = \emptyset$.

Theorem 2.1. If ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-ASM and an $ij$-regular bispace, then $(X, \sigma_{i})$ is SM.

Proof. Let $A\subset X$ and $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ be a sequence of $\sigma_{i}$-open covers of ${\sigma_{i} - cl}(A)$. Then, for each $n\in\mathbb{N}$ and $x\in{\sigma_{i} - cl}(A)$, we can choose $U_{x}^{n}\in\mathcal{D}_{n}$ such that $U_{x}^{n}$ contains $x$. Since ${(X, \sigma_{1}, \sigma_{2})}$ is an $ij$-regular bispace, there exist $V_{x}^{n}\in\sigma_{i}$ for each $x\in{\sigma_{i} - cl}(A)$ and $U_{x}^{n}$ such that $x\in V_{x}^{n}\subset{\sigma_{j} - cl}(V_{x}^{n})\subset U_{x}^{n}$ (see ^[48]). Now let $\mathcal{G}_{n} = \{V_{x}^{n}: x\in{\sigma_{i} - cl}(A)\}$ for all $n\in\mathbb{N}$. $(\mathcal{G}_{n})$ is a $\sigma_{i}$-open cover of ${\sigma_{i} - cl}(A)$ for all $n\in\mathbb{N}$ and

is a refinement of $\mathcal{D}_{n}$. On the other side, since ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-ASM, there is a sequence $(\mathcal{C}_{n})_{n\in\mathbb{N}}$ such that $\mathcal{C}_{n}$ is the finite subset of $\mathcal{G}_{n}$ and $A\subset\bigcup_{n\in\mathbb{N}}\bigcup_{V\in\mathcal{C}_{n}}{\sigma_{j} - cl}(V)$. For each $n\in\mathbb{N}$ and $V\in\mathcal{C}_{n}$ we can choose $U_{V}\in\mathcal{D}_{n}$ such that ${\sigma_{j} - cl}(V)\subset U_{V}$ due to that ${\sigma_{j} - cl}(\mathcal{G}_{n})$ refines $\mathcal{D}_{n}$. Let $\mathcal{G}_{n}^{*} = \{U_{V}: V\in\mathcal{C}_{n}\}$. Then each $\mathcal{G}_{n}^{*}\subset\mathcal{D}_{n}$ is finite and $\bigcup_{n\in\mathbb{N}}\bigcup\mathcal{G}_{n}^{*}\supset A$ which completes the proof.

The class of $ij$-ASM bispaces can be characterized in terms of $ij$-regular open sets. We now give the characterization such this spaces with $ij$-regular open sets.

Definition 2.3. ^[45,49] Let ${(X, \sigma_{1}, \sigma_{2})}$ be a bispace and $A\subset X$. A subset $A$ of $X$ is $ij$-regular open set (respectively $ij$-regular closed set) if $A = {\sigma_{i} - int}({\sigma_{j} - cl}(A) \big({\rm{respectively}} A = {\sigma_{i} - cl}({\sigma_{j} - int}(A)\big)$.

It can easily be seen that every $ij$-regular open set in ${(X, \sigma_{1}, \sigma_{2})}$ is $\sigma_{i}$-open.

Theorem 2.2. A bispace ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-ASM if and only if for each $A\subset X$ and every sequence $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ of covers of ${\sigma_{i} - cl}(A)$ by $ij$-regular open sets in $X$, there is a sequence $(\mathcal{C}_{n})_{n\in\mathbb{N}}$ where each $\mathcal{C}_{n}$ is finite subset of $\mathcal{D}_{n}$ for all $n$, and $A\subset\bigcup_{n\in\mathbb{N}}\bigcup_{V\in\mathcal{C}_{n}}{\sigma_{j} - cl}(V)$.

Proof. $(\Rightarrow)$ Obvious.

$(\Leftarrow)$ Let $A\subset X$ and $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ be the sequence of $\sigma_{i}$-open covers of ${\sigma_{i} - cl}(A)$. If we put $\mathcal{C}_{n} = \{{\sigma_{i} - int}({\sigma_{j} - cl}(U)): U\in\mathcal{D}_{n}\}$ for all $n$, we obtain a sequence $(\mathcal{C}_{n})_{n\in\mathbb{N}}$ of covers of ${\sigma_{i} - cl}(A)$ by $ij$-regular open sets in $X$. Then, by assumption, there is $\mathcal{C}_{n}^{*}\subset\mathcal{C}_{n}$ for all $n$ such that $\mathcal{C}_{n}^{*}$ is the finite subset and $A\subset\bigcup_{n\in\mathbb{N}}\bigcup{\sigma_{j} - cl}(\mathcal{C}_{n}^{*})$ where ${\sigma_{j} - cl}(\mathcal{C}_{n}^{*}) = \{{\sigma_{j} - cl}(V): V\in\mathcal{C}_{n}^{*}\}$. For every $n\in\mathbb{N}$ and every $V\in\mathcal{C}_{n}$, there is $U_{V}\in\mathcal{D}_{n}$ such that $V = {\sigma_{i} - int}({\sigma_{j} - cl}(U_{V}))$. Then, the sequence $(\mathcal{D}_{n}^{*})_{n\in\mathbb{N}}$ is the desired one where $\mathcal{D}_{n}^{*} = \{U_{V}: V\in\mathcal{C}_{n}^{*}\}$. For seeing this, one can observe that ${\sigma_{j} - cl}(U_{V})$ is a $ji$-regular closed subset of $X$, and

Thus, $A\subset\bigcup_{n\in\mathbb{N}}\bigcup{\sigma_{j} - cl}(\mathcal{D}_{n}^{*})$.

We now give the relations between $ij$-ASM and $ij$-Almost Lindelöf bispaces.

Definition 2.4. ^[50] A bipsace ${(X, \sigma_{1}, \sigma_{2})}$ is said to be $ij$-almost Lindelöf (for short $ij$-AL) if for every $\sigma_{i}$-open cover $\mathcal{D}$ of $X$, there is a countable subset $\{V_{n}:n\in\mathbb{N}\}$ of $\mathcal{D}$ such that $X = \bigcup_{n\in\mathbb{N}}\sigma_{j} - cl(V_{n})$.

In the following theorem, we see that every $ij$-ASM bispace is $ij$-AL.

Theorem 2.3. Every $ij$-ASM bispace is $ij$-AL.

Proof. Let ${(X, \sigma_{1}, \sigma_{2})}$ be a bispace and $\mathcal{D}$ be any $\sigma_{i}$-open cover of $X$. Let $\mathcal{D}_{n} = \mathcal{D}$ for each $n\in\mathbb{N}$ and $A\subset X$. Put $B = X\setminus A$. Since $\mathcal{D}$ is a $\sigma_{i}$-open cover of $X$, then $\mathcal{D}_{n}$ is a $\sigma_{i}$-open cover of both ${\sigma_{i} - cl}(A)$ and ${\sigma_{i} - cl}(B)$. Then, we clearly obtain a sequence $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ of $\sigma_{i}$-open covers of ${\sigma_{i} - cl}(A)$ and ${\sigma_{i} - cl}(B)$. Since ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-ASM, there exist $\mathcal{C}_{n}^{A}, \mathcal{C}_{n}^{B}\subset\mathcal{D}_{n}$ where $C_{n}^{A}$ and $C_{n}^{B}$ are finite for all $n\in\mathbb{N}$ with $A\subset \bigcup_{n\in\mathbb{N}}\bigcup_{U\in\mathcal{C}_{n}^{A}}\sigma_{j} - cl(V)$ and $B\subset\bigcup_{n\in\mathbb{N}}\bigcup_{U\in\mathcal{C}_{n}^{B}}\sigma_{j} - cl(U)$. Since $\mathcal{C}_{n}^{A}$ and $\mathcal{C}_{n}^{B}$ are finite, then the family $\mathcal{W}_{n} = \mathcal{C}_{n}^{A}\cup\mathcal{C}_{n}^{B}$ is a finite family for each $n\in\mathbb{N}$. Then $\mathcal{W} = \bigcup_{n\in\mathbb{N}}\mathcal{W}_{n}$ is a countable subfamily of $\mathcal{D}$, which is clearly providing that $X = A\cup B\subset\bigcup_{n\in\mathbb{N}}\bigcup_{W\in\mathcal{W}_{n}}\sigma_{j} - cl(W)$. So, $X$ is $ij$-AL.

The following example shows the inverse implication, in general not true.

Example 2.4. Consider $\mathbb{R}$ endowed with the two topologies; $\sigma_{1}$ is the Sorgenfrey topology, and $\sigma_{2}$ is the family of sets $U\setminus C$, where $U\in\sigma_{1}$ and $C\subset\mathbb{R}$ and $\mid C\mid\leq\omega$. The bispace $(\mathbb{R}, \sigma_{1}, \sigma_{2})$ is $12$-AL, since $(R, \sigma_{1})$ is Lindelöf. But it fails to be $12$-ASM since $(\mathbb{R}, \sigma_{1})$ is not almost Menger so not ASM (see ^[17]) and $\sigma_{1} - cl(U) = \sigma_{2} - cl(U)$ for every $\sigma_{1}$-open set $U$.

It is a quite natural question under what conditions these properties are equivalent. Let us give the definition of $P$-space.

Definition 2.5. ^[51] A space $X$ is called $P$-space if every intersection of countably many open sets is open.

Theorem 2.4. Let ${(X, \sigma_{1}, \sigma_{2})}$ be $ij$-AL. If $(X, \sigma_{i})$ is a $P$-space, then ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-ASM.

Proof. Let $A\subset X$ and $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ be a sequence of $\sigma_{i}$-open covers of ${\sigma_{i} - cl}(A)$. We may suppose that every $\mathcal{D}_{n}$ is closed under finite unions without loss of generality. Now, if we put $\mathcal{G} = \{\bigcap_{n\in\mathbb{N}}U_{n}: U_{n}\in\mathcal{D}_{n}\}$, then since $(X, \sigma_{i})$ is $P$-space, we obtain an $\sigma_{i}$-open over of ${\sigma_{i} - cl}(A)$. On the other hand, since ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-AL, ${\sigma_{i} - cl}(A)$ is $ij$-AL. Then there is a countable subset $\mathcal{G}^{*} = \{G_{n}:n\in\mathbb{N}\}$ of $\mathcal{G}$ providing ${\sigma_{i} - cl}(A)\subset\bigcup_{n\in\mathbb{N}}{\sigma_{j} - cl}(G_{n})$. Let $G_{n} = \bigcap_{m\in\mathbb{N}} U_{m}^{n}$ where $U_{m}^{n}\in\mathcal{D}_{m}$. Since $G_{n}\subset U_{n}^{n}$ for each $n\in\mathbb{N}$, we clearly obtain $A\subset\bigcup_{n\in\mathbb{N}}\sigma_{j} - cl(U_{n}^{n})$. So ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-ASM.

Corollary 2.1. Let ${(X, \sigma_{1}, \sigma_{2})}$ be an $ij$-regular bispace, and $(X, \sigma_{1})$ is $P$-space. Then, the following expressions are equivalent:

(1) $(X, \sigma_{1})$ is Menger,

(2) $(X, \sigma_{1})$ is ASM,

(3) ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-ASM,

(4) ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-AL,

(5) $(X, \sigma_{1})$ is Lindelöf.

In what follows, we study some behaviors of $ij$-ASM bispaces under some types of mappings.

Definition 2.6. ^[45] Let ${(X, \sigma_{1}, \sigma_{2})}$ and $(Y, \rho_{1}, \rho_{2})$ be bispaces and $f:X\to Y$ be a mapping. $f$ is said to be $d$-continuous (pairwise continuous) if the mappings $f_{i}:(X, \sigma_{i})\to(Y, \rho_{i})$ are continuous ($i$-continuous) for $i = 1, 2$.

Theorem 2.5. Let ${(X, \sigma_{1}, \sigma_{2})}$ be $ij$-ASM bispace, and let $(Y, \rho_{1}, \rho_{2})$ be a bispace. If $f:X\to Y$ is a $d$-continuous surjection, then $(Y, \rho_{1}, \rho_{2})$ is $ij$-ASM

Proof. Let $B\subset Y$, $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ be a sequence of open sets of $Y$ providing $\rho_{i} - cl(B)\subset\bigcup\mathcal{D}_{n}$ for all $n\in\mathbb{N}$ and $A = f^{-1}(B)$. Since $f$ is $d$-continuous, $f^{-1}(U)\in\sigma_{i}$ for all $n\in\mathbb{N}$ and $U\in\mathcal{D}_{n}$. Moreover, by the $d$-continuity of $f$, we obtain ${\sigma_{i} - cl}(A)\subset f^{-1}(\rho_{i} - cl(B))\subset f^{-1}(\bigcup \mathcal{D}_{n})$ for all $n\in\mathbb{N}$. Then, $(\mathcal{D}_{n}^{A})_{n\in\mathbb{N}}$ is the sequence of $\sigma_{i}$-open covers of ${\sigma_{i} - cl}(A)$ where $\mathcal{D}_{n}^{A} = \{f^{-1}(U): U\in\mathcal{D}_{n}\}$ for each $n\in\mathbb{N}$. Since ${(X, \sigma_{1}, \sigma_{2})}$ is an $ij$-ASM bispace, then there is a $\mathcal{C}_{n}^{A}\subset\mathcal{D}_{n}^{A}$ such that $\mathcal{C}_{n}^{A}$ is finite for all $n\in\mathbb{N}$, and $A\subset\bigcup_{n\in\mathbb{N}}\bigcup_{V'\in\mathcal{C}_{n}^{A}} {\sigma_{j} - cl}(V')$ holds. We can choose a $U_{V'}\in\mathcal{D}_{n}$ such that $V' = f^{-1}(U_{V'})$ for all $V'\in\mathcal{C}_{n}^{A}$ and $n\in\mathbb{N}$. Let $\mathcal{C}_{n} = \{U_{V'}:V'\in\mathcal{C}_{n}^{A}\}$. Then each $\mathcal{C}_{n}$ is the finite subset of $\mathcal{D}_{n}$ for all $n$, and

which concludes that $(Y, \rho_{1}, \rho_{2})$ is $ij$-ASM.

Definition 2.7. ^[52] A mapping $f:{(X, \sigma_{1}, \sigma_{2})}\to(Y, \rho_{1}, \rho_{2})$ is $12$-continuous if $f^{*}:(X, \sigma_{1})\to(Y, \rho_{2})$ is continuous.

Proposition 2.2. Let ${(X, \sigma_{1}, \sigma_{2})}$ be $12$-ASM bispace and $f:{(X, \sigma_{1}, \sigma_{2})}\to(Y, \rho_{1}, \rho_{2})$ is $21$-continuous. If $\sigma_{2}\leq\sigma_{1}$, then $(Y, \rho_{1}, \rho_{2})$ is $12$-ASM.

Definition 2.8. ^[53] Let ${(X, \sigma_{1}, \sigma_{2})}$ and $(Y, \rho_{1}, \rho_{2})$ be bispaces and $f:{(X, \sigma_{1}, \sigma_{2})}\to (Y, \rho_{1}, \rho_{2})$ be a mapping. $f$ is $ij$-strongly-$\theta$-continuous if each $x\in X$ and every $U\in\rho_{i}$ such that $f(x)\in U$, there exists an open set $V\in\sigma_{i}$ such that $x\in V$ and $f({\sigma_{j} - cl}(V))\subset U$.

Clearly, if $f$ is an $ij$-strongly-$\theta$-continuous mapping, then $f$ is $i$-continuous.

Theorem 2.6. Let ${(X, \sigma_{1}, \sigma_{2})}$ and $(Y, \rho_{1}, \rho_{2})$ be bispaces, and $f:{(X, \sigma_{1}, \sigma_{2})}\to(Y, \rho_{1}, \rho_{2})$ is $ij$-strongly-$\theta$-continuous and surjective. Then, $(Y, \rho_{i})$ is SM.

Proof. Let $B\subset Y$ and $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ be a sequence of $\rho_{i}$-open covers of $\rho_{i} - cl(B)$. Let $A = f^{-1}(B)$ and $x\in{\sigma_{i} - cl}(A)$. We obtain $f(x)\in f({\sigma_{i} - cl}(f^{-1}(B))) = \rho_{i} - cl(B)$ for all $x\in{\sigma_{i} - cl}(A)$, since $f$ is $i$-continuous. Then, we can choose a $U_{x}^{n}\in\mathcal{D}_{n}$ such that $f(x)\in U_{x}^{n}$ for each $n\in\mathbb{N}$. Since $f$ is $ij$-strongly-$\theta$-continuous, there is a $\sigma_{i}$-open $V_{x}^{n}$ such that $x\in V_{x}^{n}$ and $f({\sigma_{j} - cl}(V_{x}^{n}))\subset U_{x}^{n}$. Then, $(\mathcal{D}_{n}^{A})_{n\in\mathbb{N}}$ is a sequence of $\sigma_{i}$-open covers of ${\sigma_{i} - cl}(A)$ where $\mathcal{D}_{n}^{A} = \{V_{x}^{n}: x\in{\sigma_{i} - cl}(A)\}$. Since ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-ASM, there is $\mathcal{C}_{n}^{A}\subset\mathcal{D}_{n}^{A}$ such that $\mathcal{C}_{n}^{A}$ is a finite subset for each $n\in\mathbb{N}$ providing that $A\subset\bigcup_{n\in\mathbb{N}}\bigcup\mathcal{C}_{n}^{A}$. Let $F_{n}$ be a finite subset of ${\sigma_{i} - cl}(A)$ for each $n\in\mathbb{N}$ and let $\mathcal{C}_{n}^{A} = \{V_{x}^{n}: x\in F_{n}\}$. Then, $\mathcal{C}_{n} = \{U_{x}^{n}: x\in F_{n}\}$ is the finite subset of $\mathcal{D}_{n}$ for each $n\in\mathbb{N}$. Indeed, we have

So, $(Y, \rho_{i})$ is SM.

Since every $ij$-strogly-$\theta$-continuous mapping is $i$-continuous, we can give the following result.

Corollary 2.2. If $f:{(X, \sigma_{1}, \sigma_{2})}\to(Y, \rho_{1}, \rho_{2})$ is an $i$-continuous mapping, and ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-regular and an $ij$-ASM bispace, then $(Y, \rho_{i})$ is SM.

What about the pre-images of $ij$-ASM bispaces? We need some definitions for looking at the behavior.

Definition 2.9. ^[45] Let ${(X, \sigma_{1}, \sigma_{2})}$ and $(Y, \rho_{1}, \rho_{2})$ be bitopological spaces. A mapping $f:X\to Y$ is called $d$-closed if induced mappings $f_{i}:(X, \sigma_{i})\to (Y, \rho_{i})$ are closed for $i = 1, 2$.

Definition 2.10. A bispace ${(X, \sigma_{1}, \sigma_{2})}$ is called $d$-compact if the spaces $(X, \sigma_{i})$ are compact for $i = 1, 2$.

Definition 2.11 ^[54] Let ${(X, \sigma_{1}, \sigma_{2})}$ and $(Y, \rho_{1}, \rho_{2})$ be bispaces and $f:X\to Y$ is $d$-closed and $d$-continuous mapping. $f$ is called perfect if for all $y\in Y$, the set $f^{-1}(y)$ is $d$-compact in $X$.

Definition 2.12 ^[55] Let ${(X, \sigma_{1}, \sigma_{2})}$ and $(Y, \rho_{1}, \rho_{2})$ be bispaces and $f:X\to Y$ be a mapping. $f$ is called $ij$-preopen if $f(V)\subset \rho_{i}-int(\rho_{j}-cl(f(V)))$ for all $V\in\sigma_{i}$.

Proposition 2.3. ^[56] $f:{(X, \sigma_{1}, \sigma_{2})}\to (Y, \rho_{1}, \rho_{2})$ is an $ij$-preopen mapping if and only if $f^{-1}(\rho_{i}-cl(U))\subset\sigma_{i}-cl(f^{-1}(U))$ for all $U\in\rho_{i}$.

Theorem 2.7. Let $(Y, \rho_{1}, \rho_{2})$ be $ij$-ASM, and $f:{(X, \sigma_{1}, \sigma_{2})}\to (Y, \rho_{1}, \rho_{2})$ is a perfect $ji$-preopen mapping. Then, ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-ASM.

Proof. Let $A\subset X$ and $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ be the sequence of open covers of ${\sigma_{i} - cl}(A)$. Let $B = f(A)\subset Y$ and $y\in\rho_{i} - cl(f(A))$. Then, there is a finite subset $\mathcal{C}_{y}^{n}$ of $\mathcal{D}_{n}$ for each $n\in\mathbb{N}$ such that $f^{-1}(y)\subset\bigcup\mathcal{C}_{y}^{n}$. Let $\bigcup\mathcal{C}_{y}^{n} = V_{y}^{n}$. Since $f$ is $i$-closed, $U_{y}^{n} = Y\setminus f(X\setminus V_{y}^{n})$ is a $\rho_{i}$-open neighbourhood of $y$. For every $n\in\mathbb{N}$, let $\mathcal{H}_{n} = \{U_{y}^{n}: y\in\rho_{i} - cl(f(A))\}$. Then, $(\mathcal{H}_{n})_{n\in\mathbb{N}}$ is a sequence of $\rho_{i}$-open covers of $\rho_{i} - cl(f(A))$. Since $(Y, \rho_{i}, \rho_{j})$ is $ij$-ASM, there is finite $\mathcal{H}_{n}^{*}\subset\mathcal{H}_{n}$ for all $n$ such that $f(A)\subset\bigcup_{n\in\mathbb{N}}\bigcup_{H\in\mathcal{H}_{n}^{*}}\rho_{j} - cl(H)$. Let $F_{n}\subset f(A)$ be finite for all $n\in\mathbb{N}$ and $\mathcal{H}_{n}^{*} = \{U_{y_{i}}^{n}: i\in F_{n}\}$. Then, $\mathcal{D}_{n}^{*} = \bigcup_{i\in F_{n}}\mathcal{C}_{y_{i}}^{n}\subset\mathcal{D}_{n}$ is finite for all $n\in\mathbb{N}$. Then, since $f$ is $ji$-preopen, we have the following:

Hence, ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-ASM.

Definition 2.13. A mapping $f:{(X, \sigma_{1}, \sigma_{2})}\to(Y, \rho_{1}, \rho_{2})$ is called $k$-continuous if the inverse image of every $\rho_{i}$-open set is $ij$-regular open.

Theorem 2.8. A $k$-continuous surjection image of an $ij$-ASM bispace is $ij$-ASM.

3.   Union, subspaces and products

We consider the preservation of $ij$-ASM property under union, subspaces and products in this section.

Theorem 3.1. Being $ij$-ASM bispace is closed under countable union.

Proof. Let $\{(X_{n}, {\sigma_{1}}_{n}, {\rho_{1}}_{n}): n\in\mathbb{N}\}$ be countable family of $ij$-ASM bispaces and $X = \bigcup_{n\in\mathbb{N}} X_{n}$. Suppose that $\tau$ and $\sigma$ are the first and second topologies on $X$, respectively. Let $A\subset X$ and $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ be the sequence of $\tau$-open covers of $\tau - cl(A)$. Without loss of generality, we may assume that $A_{n}\subset X_{n}$ for each $n\in\mathbb{N}$ such that $A = \bigcup_{n\in\mathbb{N}}A_{n}$. Let $N_{n}$ be an infinite subset of $\mathbb{N}$, $N_{n}\cap N_{m} = \emptyset$ for each $n, m\in\mathbb{N}$ and $\mathbb{N} = \bigcup_{n\in\mathbb{N}} N_{n}$. Since ${\sigma_{i}}_{n} - cl(A_{n})\subset\tau - cl(A)$ for each $n\in\mathbb{N}$, $\mathcal{S}_{n} = (\mathcal{D}_{k}: k\in N_{n})$ is the sequence of ${\sigma_{i}}_{n}$-open covers of ${\sigma_{i}}_{n} - cl(A_{n})$. Since $(X_{n}, {\sigma_{1}}_{n}, {\rho_{2}}_{n})$ is $ij$-ASM, there is finite $\mathcal{C}_{k}\subset\mathcal{D}_{k}$ for each $k\in N_{n}$ and $n\in\mathbb{N}$ such that $A_{n}\subset\bigcup_{k\in N_{n}}\bigcup_{V\in\mathcal{C}_{k}}{\rho_{j}}_{n} - cl(V)$ holds. Then $\mathcal{D} = \{\bigcup_{k\in N_{n}}\sigma - cl(\mathcal{C}_{k}): n\in\mathbb{N}\}$ is the desired cover of $A$.

Theorem 3.2. Every $\sigma_{i}$-closed and $\sigma_{j}$-open subspace of an $ij$-ASM bispace is $ij$-ASM.

Proof. Let $(A, {\sigma_{1}}_{A}, {\sigma_{2}}_{A})$ be a $\sigma_{i}$-closed and $\sigma_{j}$-open subspace of $ij$-ASM ${(X, \sigma_{1}, \sigma_{2})}$. Let $B$ any subset of $A$ and $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ be sequence of ${\sigma_{i}}_{A}$-open covers of ${\sigma_{i}}_{A} - cl(B)$. Then we can choose $V_{U}$ for each $U\in\mathcal{D}_{n}$ and $n\in\mathbb{N}$ such that $U = A\cap V_{U}$. Let $\mathcal{D}_{n}^{*} = \{V_{U}: U\in\mathcal{D}_{n}\}$. Since $A$ is $\sigma_{i}$-closed, we have

Then, $(\mathcal{D}_{n}^{*})_{n\in\mathbb{N}}$ is sequence of $\sigma_{i}$-open covers of ${\sigma_{i} - cl}(B)$. Since ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-ASM, there is a finite subset $\mathcal{C}_{n}^{*}$ of $\mathcal{D}_{n}^{*}$ for each $n\in\mathbb{N}$ such that $\bigcup_{V_{U}\in\mathcal{C}_{n}^{*}}\sigma_{j} - cl(V_{U})$ is a cover of $B$. Let $\mathcal{C}_{n} = \{U = A\cap V_{U}: V_{U}\in\mathcal{C}_{n}^{*}\}$ for each $n\in\mathbb{N}$. As $A\in\sigma_{j}$ and for all $n\in\mathbb{N}$ and $U\in\mathcal{C}_{n}$, we have

holds and thus $B\subset\bigcup_{n\in\mathbb{N}}\bigcup_{U\in\mathcal{C}_{n}}{\sigma_{j}}_{A} - cl(U)$. Hence $(A, {\sigma_{1}}_{A}, {\sigma_{2}}_{A})$ is $ij$-ASM.

In this manner, being $ij$-ASM bispace is not hereditary property as the following example illustrates.

Example 3.1. An $ij$-ASM bispace whose a subspace is not $ij$-ASM.

Consider the set $X = [0, \Omega]$ is the set of ordinals such that $\alpha\leq\Omega$ for all $\alpha\in X$ where $\Omega$ denotes the first uncountable ordinal together with the order topology $\sigma_{1}$ and $\sigma_{2}$ is the discrete topology on $X$. Then the bispace $(X, \sigma_{1}, \sigma_{2})$ is $12$-ASM, since $(X, \sigma_{1})$ is compact so it is ASM (see ^[47]). If we consider the subset $Y = X\setminus\{\Omega\}$ with its corresponding topologies ${\sigma_{1}}_{Y}$ and ${\sigma_{2}}_{Y}$, the bispace $(Y, {\sigma_{1}}_{Y}, {\sigma_{2}}_{Y})$ is not $12$-AL (see ^[57]), so by the Theorem 2.3, it is not $12$-ASM.

Theorem 3.3. Let ${(X, \sigma_{1}, \sigma_{2})}$ be $ij$-ASM bispace and $(Y, \rho_{1}, \rho_{2})$ be a $d$-compact bispace. Then $(X\times Y, \sigma_{1}\times\rho_{1}, \sigma_{2}\times\rho_{2})$ is $ij$-ASM.

Proof. Let $A$ and $B$ be any subsets of $X$ and $Y$, respectively, and $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ be any sequence of $\sigma_{i}\times\rho_{i}$-open covers of ${\sigma_{i} - cl}(A)\times\rho_{i} - cl(B) = \sigma_{i}\times\rho_{i} - cl(A\times B)$. Without loss of generality, we can assume that $\mathcal{D}_{n} = \mathcal{A}_{n}\times\mathcal{B}_{n}$ where $\mathcal{D}_{n}$ is a $\sigma_{i}$-open cover of ${\sigma_{i} - cl}(A)$, and $\mathcal{B}_{n}$ is a $\rho_{i}$-open cover of $\rho_{i} - cl(B)$ for each $n\in\mathbb{N}$. Let $x\in{\sigma_{i} - cl}(A)$. Since $Y$ is $\rho_{i}$-compact, $\rho_{i} - cl(B)$ is $\rho_{i}$-compact. Then, we can choose a finite subset $\mathcal{C}_{n}$ of $\mathcal{D}_{n}$ for each $n\in\mathbb{N}$ such that $\{x\}\times\rho_{i} - cl(B)\subset\bigcup\mathcal{C}_{n}$. Say $\mathcal{C}_{n} = \mathcal{A}_{x}^{(n)}\times\mathcal{B}_{x}^{(n)}$ for all $n$. If $U_{x}^{(n)} = \bigcap\mathcal{A}_{x}^{(n)}$, one can observe that

for each $n\in\mathbb{N}$. Let $\mathcal{G}_{n} = \{U_{x}^{(n)}: x\in{\sigma_{i} - cl}(A)\}$ for each $n\in\mathbb{N}$. Then, $(\mathcal{G}_{n})_{n\in\mathbb{N}}$ is a sequence of $\sigma_{i}$-open covers of ${\sigma_{i} - cl}(A)$. Since ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-ASM, there is a finite subset $\mathcal{H}_{n}$ of $\mathcal{G}_{n}$ such that $\mathcal{H}_{n} = \{U_{x_{k}^{n}}^{(n)}: k\in F_{n}\}$ where $F_{n}$ is the finite subset of ${\sigma_{i} - cl}(A)$ for each $n\in\mathbb{N}$, and

holds. If we choose $\mathcal{D}_{n}^{*} = \bigcup\Big(\mathcal{A}_{x_{k}^{n}}^{(n)}\times\mathcal{B}_{x_{k}^{n}}^{(n)}\Big)$, then $\mathcal{D}_{n}^{*}$ is the finite subset of $\mathcal{D}_{n}$ and we have

So, $(X\times Y, \sigma_{1}\times\rho_{1}, \sigma_{2}\times\rho_{2})$ is $ij$-ASM.

Definition 3.1. ^[6] An open cover $\mathcal{D}$ of a topological space $(X, \sigma)$ is an $\omega$-cover if $X\notin\mathcal{D}$ and each finite subset of $X$ is contained in some element of $\mathcal{D}$.

Theorem 3.4. Let ${(X, \sigma_{1}, \sigma_{2})}$ be a bispace. The power bitopological space $(X^{n}, \sigma_{1}^{n}, \sigma_{2}^{n})$ (see ^[58]) is $ij$-ASM if and only if for every $A\subset X$ and for every sequence $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ of $\sigma_{i}$-$\omega$-covers of ${\sigma_{i} - cl}(A)$, there is a sequence $(\mathcal{C}_{n})_{n\in\mathbb{N}}$ where $\mathcal{C}_{n}\subset\mathcal{D}_{n}$ is finite for each $n\in\mathbb{N}$ and for all finite subset $F$ of $A$, there is at least $n\in\mathbb{N}$ and $V\in\mathcal{C}_{n}$ such that $F\subset{\sigma_{j} - cl}(V)$.

Proof. $(\Rightarrow)$ Let $A\subset X$ and $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ be a sequence of $\sigma_{i}$-$\omega$-covers of ${\sigma_{i} - cl}(A)$. Let $K_{t}$ be infinite subset of $\mathbb{N}$ with $K_{t}\cap K_{n} = \emptyset$ for all $t, n\in\mathbb{N}$ and $\mathbb{N} = \bigcup_{t\in\mathbb{N}} K_{t}$. For every $n\in\mathbb{N}$ and $k\in K_{t}$, let $\mathcal{D}_{k}^{t} = \{U^{t}: U\in\mathcal{D}_{k}\}$. Then $(\mathcal{D}_{k}^{t})_{k\in K_{t}}$ is a sequence of $\sigma_{i}^{t}$-open covers of $({\sigma_{i} - cl}(A))^{t} = \sigma_{i}^{t} - cl(A^{t})$. Since $(X^{t}, \sigma_{1}^{t}, \sigma_{2}^{t})$ is $ij$-ASM, there is a finite subset $\mathcal{C}_{k}^{t}\subset\mathcal{D}_{k}^{t}$ for each $k\in K_{t}$ and $A^{t}\subset\bigcup_{k\in K_{t}}\bigcup_{V\in\mathcal{C}_{k}^{t}}\sigma_{j}^{t} - cl(V)$ holds. For every $k\in K_{t}$ and $V\in\mathcal{C}_{k}^{t}$, we can choose $U_{V}\in\mathcal{D}_{k}$ such that $V = U_{V}^{t}$. Now say $\mathcal{C}_{k} = \{U_{V}: V\in\mathcal{C}_{k}^{t}\}$ for each $k\in K_{t}$. Then, the sequence $(\mathcal{C}_{k})_{k\in K_{t}}$ is the desired sequence. It obviously is that each $\mathcal{C}_{k}$ is finite subset of $\mathcal{D}_{k}$ and if $F = \{x_{1}, x_{2}, ..., x_{p}\}\subset A$, then there is an at least $k\in K_{p}$ and $V\in\mathcal{C}_{k}^{p}$ such that $(x_{1}, x_{2}, ..., x_{p})\in\sigma_{j}^{p} - cl(V)$. On the other hand, $V = U_{V}^{p}$ for an $U_{V}\in\mathcal{D}_{k}$. Then, we have

and hence $F\subset\sigma_{j} - cl(U_{V})$.

$(\Leftarrow)$ Let $A\subset X^{t}$ and $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ be a sequence of $\sigma_{i}^{t}$-open covers of $\sigma_{i}^{t} - cl(A)$. Let $\mathcal{D}_{n} = \{U_{k}^{(n)}: k\in S_{n}\}$ for each $n\in\mathbb{N}$ and $A = A_{1}\times A_{2}\times...\times A_{t}$. Let $F_{p}\subset{\sigma_{i} - cl}(A_{p})$ be finite subset for each $p\in\{1, 2, ..., t\}$. Then, $F_{1}\times F_{2}\times...\times F_{t}$ is a finite subset of $X^{t}$. Then, there is a finite subset $S_{n}^{F_{1}}\subset S_{n}$ such that $F_{1}\times F_{2}\times...\times F_{t}\subset\bigcup_{k\in S_{n}^{F_{1}}} U_{k}^{(n)}$. On the other hand, there is a $\sigma_{i}$-open set $V_{F_{p}}$ for each $p\in\{1, 2, ..., t\}$ such that $F_{p}\subset V_{F_{p}}$ and $V_{F_{1}}\times V_{F_{2}}\times...\times V_{F_{p}}\subset\bigcup_{k\in S_{n}^{F_{1}}} U_{k}^{(n)}$ (see^[47]). Then, for all finite subsets $F_{A_{p}}$ of ${\sigma_{i} - cl}(A_{p})$ for each $p\in\{1, 2, ..., t\}$, $\mathcal{C}_{n}^{(p)} = \{V_{F_{A_{p}}}: F_{A_{p}}\subset\sigma_{i} - cl(A_{p}) \rm{ is finite }\}$ is a $\sigma_{i}$-$\omega$-cover of $\sigma_{i} - cl(A_{p})$ for each $n\in\mathbb{N}$. By assumption, there is finite subset $\mathcal{G}_{n}^{(p)}\subset\mathcal{C}_{n}^{(p)}$ for each $n\in\mathbb{N}$ and $p\in\{1, 2, ..., t\}$, and for every finite subset $P$ of $A_{p}$, one can find a $n\in\mathbb{N}$ and $G\in\mathcal{G}_{n}^{(p)}$ such that $P\subset\sigma_{j} - cl(G)$. Let $R_{n}^{(p)}$ be a finite index set for each $n\in\mathbb{N}$ and $p\in\{1, 2, ..., t\}$. Assume that, $\mathcal{G}_{n}^{(p)} = \{V_{F_{A_{p}}^{r}}: r\in R_{n}^{(p)}\}$. In this sense, if $K_{n} = \{k\in S_{n}^{F_{A_{p}}^{r}}: p\in\{1, 2, ...t\} \rm{ and } r\in R_{n}^{(p)}\}$, then

holds. To see this, let $x = (x_{1}, x_{2}, ..., x_{t})\in A$. Then, $\{x_{p}\}\subset A_{p}$ for each $p\in\{1, 2, ..., t\}$. Thus, there is $n_{x_{p}}\in\mathbb{N}$ and $G_{x_{p}}\in\mathcal{G}_{n_{x_{p}}}^{(p)}$ such that $\{x_{p}\}\subset{\sigma_{j} - cl}(G_{x_{p}})$. Let $G_{x_{p}} = V_{F_{A_{p}}^{r_{x_{p}}}}$ for some $r_{x_{p}}\in R_{n_{x_{p}}}^{(p)}$. Then, we have

Hence, there is $k\in K_{n}$ such that $x\in\sigma_{j}^{t}-cl(U_{k}^{(n)})$. So, $(X^{t}, \sigma_{1}^{t}, \sigma_{j}^{t})$ is $ij$-ASM.

Question 3.1. An $ij$-ASM bispace $(X, \sigma_{1}, \sigma_{2})$ such that $(X^{2}, \sigma_{1}^{2}, \sigma_{2}^{2})$ is not $ij$-ASM?

4.   $ij$-Almost $P_{\gamma}$-sets

The concept of a $\gamma$-set was introduced by Gerlits and Nagy in ^[59]. Later, in ^[15] Kocev introduced the concept of an almost $\gamma$-set and studied. In this section, we will give the definition of $ij$-almost $P_{\gamma}$-set based upon the definitions of $\gamma$-set and almost $\gamma$-set. We will investigate the characterization of this class of $ij$-almost $P_{\gamma}$-sets with $ij$-regular open sets and their preservation under $d$-continuous surjection.

Definition 4.1. Let ${(X, \sigma_{1}, \sigma_{2})}$ be a bitopological space and $A\subset X$ and let $\mathcal{D}$ be an infinite $\sigma_{i}$-open cover of $A$. If the set $\{U\in\mathcal{D}: x\notin{\sigma_{j} - cl}(U)\}$ is finite for all $x\in A$, then we say that $\mathcal{D}$ is an $ij$-almost $P_{\gamma}$-cover of $A$.

Definition 4.2. A bispace ${(X, \sigma_{1}, \sigma_{2})}$ is called $ij$-almost $P_{\gamma}$-set (shortly $ij$-A$P_{\gamma}S$ if for all $A\subset X$ and for any sequence $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ of $\sigma_{i}$-$\omega$-covers of ${\sigma_{i} - cl}(A)$, there is a sequence $(U_{n})_{n\in\mathbb{N}}$ such that $U_{n}\in\mathcal{D}_{n}$ for each $n\in\mathbb{N}$ and the set $\{U_{n}:n\in\mathbb{N}\}$ is an $ij$-almost $P_{\gamma}$-cover of $A$.

Based upon this definiton, We can give following proposition,

Proposition 4.1. Let ${(X, \sigma_{1}, \sigma_{2})}$ be a bispace. If $(X, \sigma_{1})$ is $\gamma$-set (see ^[59]), then ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-A$P_{\gamma}S$.

Remark 4.1. Statement converse in Proposition $4$ is not true in general.

Example 4.1. Endow the real line by the two topologies: $\sigma_{1}$ is the particular point topology (see Example 2.3), and $\sigma_{2}$ is the indiscrete topology. Then, the bispace $(\mathbb{R}, \sigma_{1}, \sigma_{2})$ is clearly $12$-A$P_{\gamma}S$, while $(X, \sigma_{1})$ is not a $\gamma$-set.

Theorem 4.1. A bispace ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-A$P_{\gamma}S$ if and only if for every $A\subset X$ and every sequence $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ of $\sigma_{i}$-$\omega$-open covers of ${\sigma_{i} - cl}(A)$ by $ij$-regular open subsets of $X$, there is a sequence $(U_{n})_{n\in\mathbb{N}}$ such that $U_{n}\in\mathcal{D}_{n}$ for each $n\in\mathbb{N}$ and the set $\{U_{n}:n\in\mathbb{N}\}$ is an $ij$-almost $P_{\gamma}$-cover of $A$.

Proof. $(\Rightarrow)$ It is an obvious consequence from the fact that every $ij$-regular open set is $\sigma_{i}$-open.

$(\Leftarrow)$ Let $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ be a sequence of $\sigma_{i}$-$\omega$ covers of ${\sigma_{i} - cl}(A)$. Then, $(\mathcal{D}_{n}^{*})_{n\in\mathbb{N}}$ is the sequence of $\sigma_{i}$-$\omega$ covers of ${\sigma_{i} - cl}(A)$ by $ij$-regular open sets of $X$ where $\mathcal{D}_{n}^{*} = \{\sigma_{i} - int(\sigma_{j} - cl(U): U\in\mathcal{D}_{n}\}$ for eacn $n\in\mathbb{N}$. There exists a sequence $(U_{n}^{*})_{n\in\mathbb{N}}$ with $U_{n}^{*}\in\mathcal{D}_{n}^{*}$ for every $n\in\mathbb{N}$ and $\mathcal{D}^{*} = \{U_{n}^{*}: n\in\mathbb{N}\}$ is $P_{\gamma}$-cover of the set $A$. On the other hand, we can choose an $U_{n}\in\mathcal{D}_{n}$ such that $U_{n}^{*} = \sigma_{i} - int(\sigma_{j} - cl(U_{n})$. Then, one can easily see that $\mathcal{D} = \{U_{n}:n\in\mathbb{N}\}$ is the desired cover of $A$. Hence, ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-A$P_{\gamma}S$.

Theorem 4.2. $d$-continuous surjection of an $ij$-A$P_{\gamma}S$ bispace is $ij$-A$P_{\gamma}S$.

Proof. Let ${(X, \sigma_{1}, \sigma_{2})}$ be $ij$-A$P_{\gamma}S$ and $f:{(X, \sigma_{1}, \sigma_{2})}\to (Y, \rho_{1}, \rho_{2})$ be a $d$-continuous surjection. Let $B\subset Y$, $f^{-1}(B) = A$, and $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ be a sequence of $\rho_{i}$-$\omega$-open covers of $\rho_{i} - cl(B)$ by $ij$-regular open subsets of $Y$. Since $f$ is $i$-continuous, ${\sigma_{i} - cl}(A)\subset f^{-1}(\rho_{i} - cl(B))$ holds, and $(\mathcal{C}_{n})_{n\in\mathbb{N}}$ is the sequence of $\sigma_{i}$-$\omega$ covers of ${\sigma_{i} - cl}(A)$ where $\mathcal{C}_{n} = \{f^{-1}(U): U\in\mathcal{D}_{n}\}$ for each $n\in\mathbb{N}$. Since ${(X, \sigma_{1}, \sigma_{2})}$ is $ij$-A$P_{\gamma}S$, there is a sequence $(V_{n})_{n\in\mathbb{N}}$ providing that $V_{n}\in\mathcal{C}_{n}$ for each $n\in\mathbb{N}$ and $\mathcal{C} = \{V_{n}:n\in\mathbb{N}\}$ is $ij$-almost $P_{\gamma}$-cover of $A$. On the other side, there is a $U_{n}\in\mathcal{D}_{n}$ such that $V_{n} = f^{-1}(U_{n})$ for each $n\in\mathbb{N}$. Then, $\mathcal{D}^{*} = \{U_{n}:n\in\mathbb{N}\}$ is an $ij$-almost $P_{\gamma}$-cover of $B$. To see this, let $y\in B$ and $f(x) = y$ for some $x\in A$. Since $\mathcal{C}$ is an $ij$-almost $P_{\gamma}$-cover of $A$, the set $F_{x} = \{V_{n}\in\mathcal{C}: x\notin \sigma_{j} - cl(V_{n})\}$ is finite. Then, there is $n_{0}\in\mathbb{N}$ such that $x\in\sigma_{j} - cl(V_{n})$ for all $n > n_{0}$. Then, $f(x) = y\in f(\sigma_{j} - cl(V_{n}))$, and thus $y\in\rho_{j} - cl(U_{n})$ for all $n > n_{0}$. Therefore, we conclude that the $F^{y} = \{U_{n}\in\mathcal{D}^{*}: y\notin\rho_{j} - cl(U_{n})\}$ is finite, and since $\mathcal{C}$ is infinite, $\mathcal{D}^{*}$ is infinite. So, $(Y, \rho_{1}, \rho_{2})$ is $ij$-A$P_{\gamma}S$.

5.   Conclusions

In this paper, we dealed with the almost-set-Mengerness in bitopological spaces. Further investigations may be the similar properties of almost-set-Hurewicz and almost-set-Rothberger property (we began to investigate) in bitopological spaces. we now give the related definitions as the followings.

Definition 5.1. A bitopological space ${(X, \sigma_{1}, \sigma_{2})}$ is called:

(1) $ij$-almost-set-Hurewicz (for short, $ij$-ASH) if for all $A\subset X$ and for every sequence $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ of $\sigma_{i}$-open covers of $\sigma_{i} - cl(A)$ there is a sequence $(\mathcal{C}_{n})_{n\in\mathbb{N}}$ such that $\mathcal{C}_{n}$ is a finite subset of $\mathcal{D}_{n}$ and each $x\in A$ belongs to all but finitely many sets $\sigma_{j} - cl(\cup\mathcal{C}_{n})$. (in other words, the set $\{\cup\mathcal{C}_{n}:n\in\mathbb{N}\}$ is $ij$-almost $P_{\gamma}$-cover of $A$.)

(2) $ij$-almost-set-Rothberger ($ij$-ASR) if for all $A\subset X$ and for every sequence $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ of $\sigma_{i}$-open covers of $\sigma_{i} - cl(A)$ there is a sequence $(U_{n})_{n\in\mathbb{N}}$ such that $U_{n}\in\mathcal{D}_{n}$ and $A\subset\bigcup_{n\in\mathbb{N}}\sigma_{j} - cl(U_{n})$.

Also, we give the definition of an $ij$-weakly-set-Menger bispace as follows:

Definition 5.2. ${(X, \sigma_{1}, \sigma_{2})}$ is said to be $ij$-weakly-set-Menger (for short, $ij$-WSM) if for all $A\subset X$ and for every sequence $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ of $\sigma_{i}$-open covers of ${\sigma_{i} - cl}(A)$, there is a sequence $(\mathcal{C}_{n})_{n\in\mathbb{N}}$ such that $\mathcal{C}_{n}\subset\mathcal{D}_{n}$ is a finite subset for each $n\in\mathbb{N}$ and $A\subset\sigma_{j} - cl\Big(\bigcup_{n\in\mathbb{N}}\bigcup\mathcal{C}_{n}\Big)$

It would be interesting to study and investigate the properties of $ij$-WSM bispaces in bitopological context as well as the relations between those kind of bispaces and $ij$-ASM bispaces. Furthermore, if these properties have game-theoretic characterization can be scrutinized in bitopological context and those ones can open a way to applicable area. The possible applications of statistical convergence to the open covers of topological spaces and selection properties were given in ^[60,61]. This notions also can be studied and extended under the convergence in binary metric spaces and their induced topologies ^[62]. Also, the results obtained may be generalized to fuzzy bitopological spaces and associated with the fixed point theory ^[63,64].

We also define $ij$-weakly-set-Hurewicz ($ij$-WSH) and $ij$-weakly-set-Rothberger ($ij$-WSR) bitopological spaces in a similar way.

Conflict of interest

The authors declare that they have no competing interests.