# More on Selective Covering Properties in Bitopological Spaces

Authors: Ljubiša D. R. Kočinac; Selma Özçağ
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this study, we continue our investigation of selective covering properties in bitopological spaces. We discuss their behaviour under certain kinds of mappings. We also introduce selective versions of the ccc property and the star-ccc property in bitopological spaces and give few of their relations with other selective properties. Also, we consider preservation of selective covering properties of bitopological spaces under some known relations in bitopological context.


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## Body

## 1. Introduction

Throughout the study, we use the standard topological notation and terminology, mainly as in [1]. By ℕ and ℝ, we denote the set of natural numbers and the set of real numbers, respectively. By X or X,τ, we denote a topological space, while X,τ1,τ2 denotes a bitopological space [2] (called also bispace), that is a set X equipped with two (in general, unrelated) topologies. The closure and interior of a subset A of a space X,τ are denoted by ClA and IntA, respectively. When X,τ1,τ2 is a bitopological space and A⊂X, then ClτiA and IntτiA, i=1,2, denote the closure and interior in the space X,τi.IfA is a subset of a topological space X, x∈X, and ℱ is a family of subsets of X, then StA,ℱ=∪F∈ℱ:A∩F≠∅, and Stx,ℱ=Stx,ℱ denote the star of A and x with respect to ℱ.For a spaceX,τ, we also use the following notations:(i)
O is the collection of all open covers of X(ii)
OD=U⊂τ:∪Uis dense inX (U with this property is called an open weak cover of X)(iii)
O¯=U⊂τ:∪U¯:U∈U=X (U with this property is called an open almost cover of X)In case of bitopological spaces, we writeOi, ODi, O¯i to indicate that O, OD, O¯ is related to the space X,τi, i=1,2.IfA and ℬ are families of sets, then(1)
The symbolSfinA,ℬ denotes the selection hypothesis: for each sequence A1,A2,… of elements of A, there is a sequence B1,B2,…, such that for each n∈ℕ, Bn is a finite subset of An, and ∪n∈ℕBn∈ℬ (see [3]);(2)
The symbolS1A,ℬ denotes the selection hypothesis: for each sequence A1,A2,… of elements of A, there is a sequence b1,b2,…, such that bn∈An for each n∈ℕ, and bn:n∈ℕ∈ℬ (see [3]).The following covering properties are defined in this way:(i)
SfinO,O is the Menger property [4](ii)
S1O,O is the Rothberger property [5](iii)
SfinO,OD is the weak Menger property [6] (see also [7, 8])(iv)
SfinO,O¯ is the almost Menger property [9] (see also [10])(v)
S1O,OD is the weak Rothberger property [6](vi)
S1O,O¯ is the almost Rothberger property [9]A spaceX has the Hurewicz property if for each sequence U1,U2,… of open covers of X, there is a sequence V1,V2,…, such that for each n∈ℕ, Vn is a finite subset of Un, and each x∈X belongs to ∪Vn for all but finitely many n [11].For selection principles theory see [12].Recall that a family of pairwise disjoint open sets in a topological spaceX is called a cellular open family. X is said to be a ccc space if every cellular open family in it is countable.Observe that any separable space, clearly, is ccc.Selective version of the ccc space known as “selectively ccc spaces” was first introduced (under this name) by Aurichi [13].Definition 1.
(see [13]) Let X,τ be a topological space. X is selectively ccc if for every sequence An:n∈ℕ of maximal cellular open families in X, there is a sequence An:n∈ℕ, such that for each n, An∈An and ∪n∈ℕAn is dense in X.In fact, these spaces were considered in the literature under different names. In [14, 15], Scheepers (first) studied the selectively ccc spaces under the notation S1D,D, where D is the collection of open families with a dense union (i.e., D is OD in our notation). In [16], these spaces have been studied under the notation S1DO,D, where DO denotes the collection of open families with a dense union. In both studies [13, 16], it was explained that all these three definitions are equivalent.In this study, we consider selectively ccc spaces as spaces belonging to the classS1OD,OD.IfA and ℬ are collections of covers of a space X, then(1)
The symbolSS1A,ℬ denotes the selection hypotheses that for each sequence U1,U2,… of elements of A, there is a sequence x1,x2,… of elements of X such that Stxn,Un:n∈ℕ∈ℬ (see [17])(2)
The symbolSSO,1∗A,ℬ denotes the selection hypotheses: for each open cover U of X and each sequence U1,U2,… of elements of A, there is a sequence U1,U2,…, such that for each n, Un∈Un and St∪n∈ℕUn,U∈ℬ (see [18])In [19], the following spaces have been defined and studied.Definition 2.
(see [19]) A topological space X is said to be selectively star-ccc if it belongs to the class SSO,1∗OD,O.In this study, we introduce and study the selective versions of the ccc property and selectively star-ccc property in bitopological spaces. We establish some relations with other selective bitopological properties.The study is organized as follows. In Section2, we consider preservation of the bitopological almost Menger and almost Rothberger properties under certain classes of mappings. Section 3 deals with selectively ccc and selectively star-ccc bitopological spaces. Finally, in Section 4, we discuss the behaviour of several topological and bitopological covering properties under known relations between the two topologies.


## 2. Mappings and Almost Menger Bispaces

The first study of weak covering properties in the bitopological context began with the study [20] on the almost Menger property in bitopological spaces and continued in [21, 22] where some results on almost Menger and weak Menger properties in the bitopological spaces were obtained. Selective properties in bitopological spaces have been studied in [23–25].Let us recall the following definition.Definition 3.
(see [20]). A bitopological space X,τ1,τ2 is said to be i,j-almost Menger, i,j=1,2, if for each sequence Un:n∈ℕ of τi-open covers of X, there exists a sequence Vn:n∈ℕ of finite families, such that for each n, Vn⊂Un and X=∪n∈ℕ∪V∈VnClτjV.This section focuses on the behaviour of almost Menger, almost Rothberger, and almost Hurewicz properties under some kinds of mappings between bitopological spaces.We recall the following definitions. A mappingf:X,τ1,τ2⟶Y,σ1,σ2 is called(1)
i,j-θ-continuous [26] (respectively,i,j-strongly θ-continuous [27]) if for each x∈X and each σi-open set V⊂Y containing fx, there exists a τi-open set U⊂X containing x, such that fClτjU⊂ClσjV (respectively, fClτjU⊂V). It is obvious that each i,j-strongly θ-continuous mapping is i,j-θ-continuous.(2)
i,j-almost continuous [28] if f←B is the τi-open set in X for every i,j-regular open set B in Y. In addition, f is called p-almost continuous if it is 1,2 and 2,1-almost continuous. Recall that a set B in a bitopological space X,τ1,τ2 is i,j-regular open [29] if B=IntτiClτjB.We note thatp-almost continuous mappings are i,j-θ-continuous [30], so the following result generalizes the result in [20], which states that the i,j-almost Menger property is preserved by p-almost continuous mappings.Theorem 1 (see [21]).
Ani,j−θ-continuous image of an i,j-almost Menger bitopological space is i,j-almost Menger.In [21], the definitions of i,j-almost γ-sets and the i,j-almost Hurewicz property were introduced, and the following theorem was obtained.Theorem 2 (see [21]).
LetX,τ1,τ2 be an i,j-almost γ-set and Y,σ1,σ2 be a bitopological space. If f:X⟶Y is an i,j-θ-continuous surjection, then Y,σ1,σ2 is i,j-almost Hurewicz.Now, we consider the behaviour of thei,j-almost Rothberger property under i,j-θ-continuous mappings.A bitopological spaceX,τ1,τ2 is said to be i,j-almost Rothberger, i,j=1,2, if for each sequence Un:n∈ℕ of τi-open covers of X, there is a sequence Un:n∈ℕ, such that for each n, Un∈Un and X=∪n∈ℕClτjUn [20].Theorem 3.
Ani,j-θ-continuous image of an i,j-almost Rothberger bitopological space is i,j-almost Rothberger.Proof.
LetX,τ1,τ2 be an i,j-almost Rothberger bitopological space. Let Y=fX be the image of X, where f:X,τ1,τ2⟶Y,σ1,σ2 is an i,j-θ-continuous mapping.
LetVn:n∈ℕ be a sequence of σi-open covers of Y. For each n∈ℕ and fixed x∈X, there is a set Vx,n∈Vn, such that fx∈Vx,n. Since f is i,j-θ-continuous, there exists a τi-open set Ux,n⊂X containing x, such that fClτjUx,n⊂ClσjVx,n. For each n, Un:=Ux,n:x∈X is a τi-open cover of X. Apply the fact that X is i,j-almost Rothberger. There is a sequence xn:n∈ℕ, such that for each n∈ℕ, Uxn,n∈Un and X=∪n∈ℕClτjUxn,n. For each n and Uxn,n, we assign a set Vxn,n∈Vn, such that fClτjUxn,n⊂ClσjVxn,n. We obtain the sequence Vxn,n:n∈ℕ, with Vxn,n∈Vn and(1)Y=fX=f∪n∈ℕClτjUxn,n=∪n∈ℕfClτjUxn,n⊂∪n∈ℕClσjVxn,n.This means that Y,σ1,σ2 is i,j-almost Rothberger.Corollary 1.
Thei,j-almost Rothberger property is preserved by p-almost continuous mappings.Theorem 4.
Iff:X,τ1,τ2⟶Y,σ1,σ1 is an i,j-strongly θ-continuous mapping and X is i,j-almost Menger, then Y,σi is Menger.Proof.
LetVn:n∈ℕ be a sequence of σi-open covers of Y and x∈X. For each n∈ℕ, there exists a set Vx,n∈Vn with fx∈Vx,n. Since f is i,j-strongly θ-continuous, there is a τi-open set Ux,n⊂X with x∈Ux,n, such that fClτjUx,n⊂Vx,n. Then, Un:=Ux,n:x∈X is a τi-open cover of X. Since X is i,j-almost Menger bitopological space, there exists a sequence ℋn:n∈ℕ, such that for each n∈ℕ, ℋn is a finite subset of Un, and X=∪n∈ℕ∪ClτjH:H∈ℋn. Now, for each n∈ℕ and H∈ℋn, we can choose VH∈Vn, such that fClτjH⊆VH. Let ℱn=VH:H∈ℋn; we obtain a sequence ℱn:n∈ℕ of finite subsets of Vn, n∈ℕ. We will show ∪ℱn is a σi-cover of Y. Indeed,(2)Y=fX=f∪n∈ℕ∪ClτjH:H∈ℋn,=∪n∈ℕ∪fClτjH:H∈ℋn,⊂∪n∈ℕ∪VH:H∈ℋn.
This shows thatY,σi is Menger.Remark 1.
It is obvious that iff:X,τ1,τ2⟶Y,σ1,σ2 is an i,j-strongly θ-continuous mapping, then f is i-continuous; furthermore, if X,τ1,τ2 is the i,j-regular bitopological space, then the reverse implication is also true.Corollary 2.
Iff:X,τ1,τ2⟶Y,σ1,σ1 is i-continuous mapping and X is i,j-regular, i,j-almost Menger bitopological space, then Y,σi is Menger.Now, recall the notions of contra continuous and precontinuous mappings in bitopological spaces. A mappingf:X,τ1,τ2⟶Y,σ1,σ2 is said to be i,j-contracontinuous and i,j-precontinuous [29], if the preimage of each σi-open set V of Y is τj-closed in X (i,j-preopen in X, i.e., f←V⊂IntτiClτjf←V), where i,j=1,2.Theorem 5.
IfX,τ1,τ2 is the i,j almost Menger bitopological space and f:X,τ1,τ2⟶Y,σ1,σ1 is an i,j-contracontinuous and i,j-precontinuous mapping, then Y,σi is a Menger space.Proof.
LetVn:n∈ℕ be a sequence of covers of Y by σi-open sets. Since f is i,j-contra continuous, for each n∈ℕ and V∈Vn, the set f←V is τj-closed in X. Then, we have f←V⊂IntτiClτjf←V, since f is i,j-precontinuous. Now, we obtain f←V⊂Intτif←V, which gives f←V=Intτif←V. Then, for each n∈ℕ, we set Un:=f←V:V∈Vn. Then, Un:n∈ℕ is a sequence of τi-open covers of X. Since X is i,j-almost Menger, there exists a sequence ℋn:n∈ℕ, such that for each n∈ℕ, ℋn is a finite subset of Un and ∪n∈ℕ∪ClτjH:H∈ℋn=X. Let ℱn=fH:H∈ℋn. It is obvious that ℱn:n∈ℕ is a sequence of finite subsets of Vn for each n∈ℕ, and ∪n∈ℕℱn is an σi-open cover of Y which means that Y,σi is a Menger space.
A mappingf:X,τ1,τ2⟶Y,σ1,σ2 is said to be i,j-weakly continuous [26] if for each x∈X and each σi-open subset V⊂Y containing fx, there exist a τi-open set U⊂X with x∈U, such that fU⊂ClσjV.Theorem 6.
Iff:X,τ1,τ2⟶Y,σ1,σ2 is an i,j-weakly continuous mapping and X,τi is Menger, then Y,σ1,σ2 is an i,j-almost Menger space.Proof.
LetX,τi be a Menger space and Vn:n∈ℕ be a sequence of σi-open covers of Y. Let x∈X, then for each n∈ℕ, there exists V∈Vn with fx∈V. Since f is i,j-weakly continuous, there is a τi-open set Ux,n⊂X with x∈Ux,n and fUx,n⊂ClσjV. Now, let Un:=Ux,n:x∈X. Clearly, Un is a τi-open cover of X. Since X,τi is Menger, we find a sequence ℱn:n∈ℕ of finite sets, such that for each n∈ℕ, ℱn⊂Un and ∪n∈ℕℱn is a τi-open cover of X. For each n∈ℕ and U∈ℱn, we can choose a set VU∈Vn, such that fU⊂ClσjVU. Set ℋn=VU:U∈ℱn. We show that ∪ℋn is a cover of Y. Indeed, as X=∪n∈ℕ∪U:U∈ℱn, we have(3)Y=fX=f∪n∈ℕ∪U:U∈ℱn⊂∪n∈ℕ∪fU:U∈ℱn⊂∪n∈ℕ∪ClσjVU:U∈ℱn.
So,Y,σ1,σ2 is i,j-almost Menger.Remark 2.
Sincei,j-θ-continuous mappings are i,j-weakly continuous, Theorem 6 is also true under θ-continuous mappings.


## 3. Selectively ccc and Selectively Star-ccc Bispaces

### 3.1. Selectively ccc Bispaces

Definition 4.
A bitopological spaceX,τ1,τ2 is said to satisfy i-countable chain condition (for short i-ccc), i=1,2, if the topological space X,τi satisfies countable chain condition. X satisfies countable chain condition if it satisfies i-countable chain condition for i=1,2.Definition 5.
A bitopological spaceX,τ1,τ2 is i,j-selectively ccc i,j=1,2 if for every sequence Un:n∈ℕ of elements of ODi, there is a sequence Un:n∈ℕ, such that Un∈Un, n∈ℕ, and ∪n∈ℕUn is dense in X,τj, i.e., if S1ODi,ODj is satisfied.Proposition 1.
LetX,τ1,τ2 be a bitopological space such that τ1≤τ2 (i.e., the topology τ1 is weaker than τ2).(1)
IfX,τ1,τ2 is 1,2-selectively ccc, then X,τ1 is selectively ccc(2)
IfX,τ2 is selectively ccc, then X,τ1,τ2 is 2,1-selectively cccSince the selectively ccc property is stronger than the ccc property, we have the following.Corollary 3.
LetX,τ1,τ2 be a bitopological space such that τ1≤τ2. If X is 1,2-selectively ccc bispace, then X,τ1 has the ccc property.Proposition 2.
LetX,τ1,τ2 be a bitopological space and Y⊂X be τi-closed and τj-open subspace in X. If X is i,j-selectively ccc, then Y,τ1|Y,τ2|Y is i|Y,j|Y-selectively ccc.Proof.
(Fori = 1 and j = 2). Let Y be a τ1-closed and τ2-open subset of X and let Un:n∈ℕ be a sequence of members of OD1|Y. Thus, for each n∈ℕ, Vn=Un∪X∖Y is an element of OD1. Since X is 1,2-selectively ccc, there is a sequence Vn:n∈ℕ, such that for every n∈ℕ, Vn∈Vn and Cl2∪n∈ℕVn=X. If Vn∈Un, we set Wn=Vn; if Vn=X∖Y, then we take Wn to be any element from Un. Then, it is clear that Cl2|Y∪n∈ℕWn=Y.
Recall that a subsetA of a bispace X,τ1,τ2 is called bidense (here called double dense or shorty d-dense) in X if A is dense in both X,τ1 and X,τ2. X is d-separable if there is a countable set A which is d-dense in X.Theorem 7.
LetY⊂X be a d-dense set in a bispace X,τ1,τ2. X is i,j-selectively ccc if and only if Y is i|Y,j|Y selectively ccc.Proof.
(Fori = 1 and j = 2). Let Vn:n∈ℕ be a sequence of τ1|Y-open families such that ∪Vn is dense in Y,τ1|Y. Since Y is τ1-dense, then each ∪Vn is dense in X,τ1. As X is 1,2-selectively ccc, there is a sequence Vn:n∈ℕ, such that for each Vn∈Vn, n∈ℕ, and ∪n∈ℕVn is dense in X,τ2. Since Y is dense in X,τ2, then ∪n∈ℕVn is dense in Y,τ2|Y.
LetUn:n∈ℕ be a sequence of τ1-open families, such that each ∪Un is dense in X,τ1. The restriction to Y, Un|Y=U∩Y:U∈Un is a τ1|Y-open family for each n. Since Y is 1|Y,2|Y-selectively ccc, there is a sequence Vn:n∈ℕ, such that Vn∈Un|Y and ∪n∈ℕVn is dense in Y,τ2|Y. But Y is dense in X,τ2, so ∪n∈ℕVn is dense in X,τ2.Theorem 8.
LetX,τ1,τ2 and Y,σ1,σ2 be bitopological spaces and f:X⟶Y be a τi−σi-continuous, τi−σi-open and τj−σj continuous, τj−σj-closed surjection. If X is i,j-selectively ccc, then Y is i,j-selectively ccc.Proof.
(Fori = 1 and j = 2). Let Vn:n∈ℕ be a sequence of σ1-open families in Y such that ∪Vn is σ1-dense in Y. Since f is τ1−σ1 continuous and τ1−σ1 open, for each n∈ℕ, Un:=f←V:V∈Vn is τ1-open family in X and ∪Un is τ1-dense in X. Because X is 1,2-selectively ccc, we can pick Un∈Un with ∪Un:n∈ℕ dense in X,τ2. For the sequence Vn:n∈ℕ, such that f←V=Un, we have ∪n∈ℕVn dense in Y,σ2, since f is a τ2−σ2 continuous and τ2−σ2 closed mapping. Thus, Y,σ1,σ2 is 1,2 selectively ccc.
BitopologicalR-separability has been studied by Kočinac and Özçağ in [31].
Denote byD1 and D2 the collection of all dense subsets of X,τ1 and X,τ2, respectively. We say that X is Rτi,τj-separable i,j=1,2, if for each sequence Dn:n∈ℕ of elements of Di, there are dn∈Dn, n∈ℕ, such that dn:n∈ℕ is an element of Dj, i.e., if S1Di,Dj holds.
We have the following fact.Proposition 3.
EveryRτi,τj-separable bispace is i,j-selectively ccc.Corollary 4.
If the bitopological spaceX,τ1,τ2 has a τi-closed and τj-open Rτi,τj-separable subspace, then X is τi,τj selectively ccc.Proof.
It follows by Proposition2 and the previous result.
LetK denote the collection of all k-covers of a space X (open covers such that every compact subset of X is contained in a member of the cover), and let Ω denote the collection of all ω-covers of X (open covers such that every finite subset of X is contained in a member of the cover and X itself is not a member of the cover).Example 1.
If a Tychonoff spaceX with a countable base belongs to the class S1K,Ω, then ([31], Theorem 4) the function bispace CX,τp,τk is Rτk,τp-separable. Therefore, this function bispace is selectively τk,τp-star-ccc, where τp denotes the pointwise topology, and τk denotes the compact-open topology on the set CX of all continuous real-valued functions on a Tychonoff space X.


### 3.2. Selectively Star-ccc Bispaces

We now present a bitopological version of selectively star-ccc spaces introduced and studied in [19] (see also [32–34]).Definition 6.
A bispaceX,τ1,τ2 is said to be i,j-selectively star-ccc, i,j=1,2, if for every sequence Un:n∈ℕ of τi-open weak covers and every open cover C of X,τj, there is a sequence Un:n∈ℕ, such that for every n, Un∈Un and St∪n∈ℕUn,C=X.The following result shows that the class of selectively star-ccc spaces is big enough.Theorem 9.
LetX,τ1,τ2 be a bispace, such that τ1≥τ2. If X,τ2 is a Lindelöf space, then X,τ1,τ2 is a 1,2-selectively star-ccc space.Proof.
LetUn:n∈ℕ be a sequence of τ1-open weak covers of X, and let C be an open cover of X,τ2. There exists a countable set V=C1,C2,…⊂C, such that X=∪V. Since each Ci, i∈ℕ, is τ1-open, for every n∈ℕ, there exists Un∈Un, such that Un∩Cn≠∅. It follows that St∪n∈ℕUn,V=∪V=X. This proves the theorem.Corollary 5.
The bispaceℝ,τS,τ, where ℝ is the set of real numbers, τ is the usual metric topology, and τS is the Sorgenfrey topology, is τS,τ-selectively star-ccc.Recall that a spaceX is said to be strongly star Lindelöf, if for each open cover U of X, there is a countable set C⊂X, such that StC,U=X.Call a bispaceX,τ1,τ2i,j-selectively 2-star-ccc if for every τj-open cover U of X and every sequence Vn:n∈ℕ of τi-open weak covers of X, there is a sequence Vn:n∈ℕ, such that Vn∈Vn, n∈ℕ, and St2∪n∈ℕVn,U=X, where St2V,U=StStV,U,U.Theorem 10.
LetX,τ1,τ2 be a bispace such that τ1≥τ2. If X,τ2 is a strongly star Lindelöf space, then X,τ1,τ2 is an 1,2-selectively 2-star-ccc space.Proof.
LetU be an arbitrary τ2-open cover of X and Vn:n∈ℕ be a sequence of τ1-open weak covers of X. Since X,τ2 is strongly star Lindelöf, there exists a countable set A=a1,a2,…⊂X, such that StA,U=X. Therefore, Stan,U:n∈ℕ is a τ2-open cover of X. For each n∈ℕ, there exists Un∈U, such that an∈Un⊂Stan,U. So, Un⊂Stan,U⊂StUn,U. For every n∈ℕ, the set ∪n∈ℕ∪Vn is dense in X,τ1 and hence in X,τ2, so that Un∩∪n∈ℕVn≠∅. This implies that for each n∈ℕ, there exists Vn∈Vn, such that Un∩Vn≠∅. Therefore, for each n∈ℕ,(4)StUn,U⊂StStVn,U,U=St2Vn,U,which implies(5)X=StA,U⊂St2∪n∈ℕVn,U.
This shows thatX,τ1,τ2 is (1, 2)-selectively 2-star-ccc.Theorem 11.
Everyi,j-selectively ccc bispace X,τ1,τ2 is i,j-selectively star-ccc.Proof.
(Fori = 1, j = 2). Let C be a τ2-open cover of X and Un:n∈ℕ be a sequence of τ1-open weak covers of X. Since X is 1,2-selectively ccc, there exists a sequence Un:n∈ℕ, such that for every n, Un∈Un, and M=∪n∈ℕUn is dense in X,τ2. So, M intersects each element of C, and thus,(6)St∪n∈ℕUn,C=X,whereX is a 1,2-selectively star-ccc space.
From Proposition3 and Theorem 11, we have the following corollary.Corollary 6.
EveryRτi,τj-separable bispace is i,j-selectively star-ccc.We show now that the converse in this corollary is not always true.Example 2.
LetX,τ1,τ2 be a bispace with τ1≥τ2, such that X,τ2 is a compact and nonseparable space. By Theorem 9, the bitopological space X,τ1,τ2 is 1,2-selectively star-ccc. But it is not Rτ1,τ2-separable by the following fact: Rτ1,τ2-separability implies separability of X,τ2.


## 3.1. Selectively ccc Bispaces

Definition 4.
A bitopological spaceX,τ1,τ2 is said to satisfy i-countable chain condition (for short i-ccc), i=1,2, if the topological space X,τi satisfies countable chain condition. X satisfies countable chain condition if it satisfies i-countable chain condition for i=1,2.Definition 5.
A bitopological spaceX,τ1,τ2 is i,j-selectively ccc i,j=1,2 if for every sequence Un:n∈ℕ of elements of ODi, there is a sequence Un:n∈ℕ, such that Un∈Un, n∈ℕ, and ∪n∈ℕUn is dense in X,τj, i.e., if S1ODi,ODj is satisfied.Proposition 1.
LetX,τ1,τ2 be a bitopological space such that τ1≤τ2 (i.e., the topology τ1 is weaker than τ2).(1)
IfX,τ1,τ2 is 1,2-selectively ccc, then X,τ1 is selectively ccc(2)
IfX,τ2 is selectively ccc, then X,τ1,τ2 is 2,1-selectively cccSince the selectively ccc property is stronger than the ccc property, we have the following.Corollary 3.
LetX,τ1,τ2 be a bitopological space such that τ1≤τ2. If X is 1,2-selectively ccc bispace, then X,τ1 has the ccc property.Proposition 2.
LetX,τ1,τ2 be a bitopological space and Y⊂X be τi-closed and τj-open subspace in X. If X is i,j-selectively ccc, then Y,τ1|Y,τ2|Y is i|Y,j|Y-selectively ccc.Proof.
(Fori = 1 and j = 2). Let Y be a τ1-closed and τ2-open subset of X and let Un:n∈ℕ be a sequence of members of OD1|Y. Thus, for each n∈ℕ, Vn=Un∪X∖Y is an element of OD1. Since X is 1,2-selectively ccc, there is a sequence Vn:n∈ℕ, such that for every n∈ℕ, Vn∈Vn and Cl2∪n∈ℕVn=X. If Vn∈Un, we set Wn=Vn; if Vn=X∖Y, then we take Wn to be any element from Un. Then, it is clear that Cl2|Y∪n∈ℕWn=Y.
Recall that a subsetA of a bispace X,τ1,τ2 is called bidense (here called double dense or shorty d-dense) in X if A is dense in both X,τ1 and X,τ2. X is d-separable if there is a countable set A which is d-dense in X.Theorem 7.
LetY⊂X be a d-dense set in a bispace X,τ1,τ2. X is i,j-selectively ccc if and only if Y is i|Y,j|Y selectively ccc.Proof.
(Fori = 1 and j = 2). Let Vn:n∈ℕ be a sequence of τ1|Y-open families such that ∪Vn is dense in Y,τ1|Y. Since Y is τ1-dense, then each ∪Vn is dense in X,τ1. As X is 1,2-selectively ccc, there is a sequence Vn:n∈ℕ, such that for each Vn∈Vn, n∈ℕ, and ∪n∈ℕVn is dense in X,τ2. Since Y is dense in X,τ2, then ∪n∈ℕVn is dense in Y,τ2|Y.
LetUn:n∈ℕ be a sequence of τ1-open families, such that each ∪Un is dense in X,τ1. The restriction to Y, Un|Y=U∩Y:U∈Un is a τ1|Y-open family for each n. Since Y is 1|Y,2|Y-selectively ccc, there is a sequence Vn:n∈ℕ, such that Vn∈Un|Y and ∪n∈ℕVn is dense in Y,τ2|Y. But Y is dense in X,τ2, so ∪n∈ℕVn is dense in X,τ2.Theorem 8.
LetX,τ1,τ2 and Y,σ1,σ2 be bitopological spaces and f:X⟶Y be a τi−σi-continuous, τi−σi-open and τj−σj continuous, τj−σj-closed surjection. If X is i,j-selectively ccc, then Y is i,j-selectively ccc.Proof.
(Fori = 1 and j = 2). Let Vn:n∈ℕ be a sequence of σ1-open families in Y such that ∪Vn is σ1-dense in Y. Since f is τ1−σ1 continuous and τ1−σ1 open, for each n∈ℕ, Un:=f←V:V∈Vn is τ1-open family in X and ∪Un is τ1-dense in X. Because X is 1,2-selectively ccc, we can pick Un∈Un with ∪Un:n∈ℕ dense in X,τ2. For the sequence Vn:n∈ℕ, such that f←V=Un, we have ∪n∈ℕVn dense in Y,σ2, since f is a τ2−σ2 continuous and τ2−σ2 closed mapping. Thus, Y,σ1,σ2 is 1,2 selectively ccc.
BitopologicalR-separability has been studied by Kočinac and Özçağ in [31].
Denote byD1 and D2 the collection of all dense subsets of X,τ1 and X,τ2, respectively. We say that X is Rτi,τj-separable i,j=1,2, if for each sequence Dn:n∈ℕ of elements of Di, there are dn∈Dn, n∈ℕ, such that dn:n∈ℕ is an element of Dj, i.e., if S1Di,Dj holds.
We have the following fact.Proposition 3.
EveryRτi,τj-separable bispace is i,j-selectively ccc.Corollary 4.
If the bitopological spaceX,τ1,τ2 has a τi-closed and τj-open Rτi,τj-separable subspace, then X is τi,τj selectively ccc.Proof.
It follows by Proposition2 and the previous result.
LetK denote the collection of all k-covers of a space X (open covers such that every compact subset of X is contained in a member of the cover), and let Ω denote the collection of all ω-covers of X (open covers such that every finite subset of X is contained in a member of the cover and X itself is not a member of the cover).Example 1.
If a Tychonoff spaceX with a countable base belongs to the class S1K,Ω, then ([31], Theorem 4) the function bispace CX,τp,τk is Rτk,τp-separable. Therefore, this function bispace is selectively τk,τp-star-ccc, where τp denotes the pointwise topology, and τk denotes the compact-open topology on the set CX of all continuous real-valued functions on a Tychonoff space X.


## 3.2. Selectively Star-ccc Bispaces

We now present a bitopological version of selectively star-ccc spaces introduced and studied in [19] (see also [32–34]).Definition 6.
A bispaceX,τ1,τ2 is said to be i,j-selectively star-ccc, i,j=1,2, if for every sequence Un:n∈ℕ of τi-open weak covers and every open cover C of X,τj, there is a sequence Un:n∈ℕ, such that for every n, Un∈Un and St∪n∈ℕUn,C=X.The following result shows that the class of selectively star-ccc spaces is big enough.Theorem 9.
LetX,τ1,τ2 be a bispace, such that τ1≥τ2. If X,τ2 is a Lindelöf space, then X,τ1,τ2 is a 1,2-selectively star-ccc space.Proof.
LetUn:n∈ℕ be a sequence of τ1-open weak covers of X, and let C be an open cover of X,τ2. There exists a countable set V=C1,C2,…⊂C, such that X=∪V. Since each Ci, i∈ℕ, is τ1-open, for every n∈ℕ, there exists Un∈Un, such that Un∩Cn≠∅. It follows that St∪n∈ℕUn,V=∪V=X. This proves the theorem.Corollary 5.
The bispaceℝ,τS,τ, where ℝ is the set of real numbers, τ is the usual metric topology, and τS is the Sorgenfrey topology, is τS,τ-selectively star-ccc.Recall that a spaceX is said to be strongly star Lindelöf, if for each open cover U of X, there is a countable set C⊂X, such that StC,U=X.Call a bispaceX,τ1,τ2i,j-selectively 2-star-ccc if for every τj-open cover U of X and every sequence Vn:n∈ℕ of τi-open weak covers of X, there is a sequence Vn:n∈ℕ, such that Vn∈Vn, n∈ℕ, and St2∪n∈ℕVn,U=X, where St2V,U=StStV,U,U.Theorem 10.
LetX,τ1,τ2 be a bispace such that τ1≥τ2. If X,τ2 is a strongly star Lindelöf space, then X,τ1,τ2 is an 1,2-selectively 2-star-ccc space.Proof.
LetU be an arbitrary τ2-open cover of X and Vn:n∈ℕ be a sequence of τ1-open weak covers of X. Since X,τ2 is strongly star Lindelöf, there exists a countable set A=a1,a2,…⊂X, such that StA,U=X. Therefore, Stan,U:n∈ℕ is a τ2-open cover of X. For each n∈ℕ, there exists Un∈U, such that an∈Un⊂Stan,U. So, Un⊂Stan,U⊂StUn,U. For every n∈ℕ, the set ∪n∈ℕ∪Vn is dense in X,τ1 and hence in X,τ2, so that Un∩∪n∈ℕVn≠∅. This implies that for each n∈ℕ, there exists Vn∈Vn, such that Un∩Vn≠∅. Therefore, for each n∈ℕ,(4)StUn,U⊂StStVn,U,U=St2Vn,U,which implies(5)X=StA,U⊂St2∪n∈ℕVn,U.
This shows thatX,τ1,τ2 is (1, 2)-selectively 2-star-ccc.Theorem 11.
Everyi,j-selectively ccc bispace X,τ1,τ2 is i,j-selectively star-ccc.Proof.
(Fori = 1, j = 2). Let C be a τ2-open cover of X and Un:n∈ℕ be a sequence of τ1-open weak covers of X. Since X is 1,2-selectively ccc, there exists a sequence Un:n∈ℕ, such that for every n, Un∈Un, and M=∪n∈ℕUn is dense in X,τ2. So, M intersects each element of C, and thus,(6)St∪n∈ℕUn,C=X,whereX is a 1,2-selectively star-ccc space.
From Proposition3 and Theorem 11, we have the following corollary.Corollary 6.
EveryRτi,τj-separable bispace is i,j-selectively star-ccc.We show now that the converse in this corollary is not always true.Example 2.
LetX,τ1,τ2 be a bispace with τ1≥τ2, such that X,τ2 is a compact and nonseparable space. By Theorem 9, the bitopological space X,τ1,τ2 is 1,2-selectively star-ccc. But it is not Rτ1,τ2-separable by the following fact: Rτ1,τ2-separability implies separability of X,τ2.


## 4. Covering Properties and Relations on Bispaces

In this section, we will investigate the behaviour and preservation of covering properties studied in this study under various relations (as<C and <N) between two topologies on a bitopological space X,τ1,τ2.In 1957, Weston [35] introduced the C-relation, while in [30], the N-relation was defined. The abovementioned relations and their combinations with the set-theoretic operation inclusion are considered in [30].LetX,τ1,τ2 be a bitopological space. Obviously, τ2<τ1 implies Clτ1A⊂Clτ2A for each A⊂X.Definition 7 (see [30, 35]).
The topologyτ1 is coupled to the topology τ2 on a set X, denoted by τ1Cτ2, if Clτ1A⊂Clτ2A for each A∈τ1.Remark 3.
(see [30]). If the open sets and closed sets are equal for a topology τ1 on a set X, then τ1 is coupled to every topology on X. For example, the discrete and indiscrete topologies are coupled to each topology on X.TheC-relation defines a partial order ≤ (we use the notation <C) on the family of all topologies on X by(7)τ1<Cτ2⇔τ1Cτ2∧τ1<τ2.It is easy to see thatτ1<Cτ2 implies τ1Cτ2∧τ2Cτ1, but the reverse implication is not correct.Example 3.
LetX=a,b,c,d,e,τ1=∅,e,a,b,c,d,X, and τ2=∅,c,a,b,d,e,X. It is clear that τ1<Cτ2 and τ2<Cτ1, but these topologies are not comparable by inclusion.A setX together with the topologies τ1Cτ2τ1<Cτ2 is denoted by X,τ1Cτ2X,τ1<Cτ2.Now, we consider theN-relation between two topologies on a set X.Definition 8.
(see [30]). A topology τ1 is near to a topology τ2 on a set X (briefly τ1Nτ2) if Clτ1A⊂Clτ2A for each A∈τ2.The discrete topology onX is near any topology on X, and every topology on X is near the indiscrete topology on X.Similar to theC-relation, the N-relation defines a partial order on the family of all topologies on X by(8)τ1<Nτ2⇔τ1Nτ2∧τ1<τ2.We mention that the notions of coupling and nearness of topologies are independent of each other; however, we have the following implication:(9)τ1<Nτ2⇒τ1<Cτ2.A setX together with the topologies τ1Nτ2τ1<Nτ2 is denoted by X,τ1Nτ2 (X,τ1<Nτ2).We omit the proof of the following proposition.Proposition 4.
LetX,τ1Cτ2 be a bitopological space. If X,τ1 is almost Menger, then X,τ1Cτ2 is 1,2-almost Menger.However, in the following example, we show that the reverse implication is not true in general.Example 4.
LetX be the Euclidean plane ℝ2, τ1 be the Sorgenfrey topology, and τ1 be the ordinary metric topology on X. Clearly, X,τ1Cτ2 is 1,2-almost Menger; however, X,τ1 does not have the almost Menger property, since X,τ1 is not almost Lindelöf.On the other hand, we have the following result.Theorem 12.
A bispaceX,τ1<Cτ2 is 1,2-almost Menger if and only if X,τ1 is almost Menger.Proof.
LetUn:n∈ℕ be a sequence of τ1-open covers of X. Since X,τ1<Cτ2 is 1,2-almost Menger, there exists a sequence Vn:n∈ℕ of finite sets with Vn⊂Un and(10)∪n∈ℕ∪Clτ2V:V∈Vn=X.
Byτ1<Cτ2, we have τ1Cτ2 and τ1<τ2. For V∈τ1<τ2, we have Clτ2V⊂Clτ1V. Therefore, ∪n∈ℕ∪Clτ1V:V∈Vn=X gives that X,τ1 is almost Menger.
LetUn:n∈ℕ be a sequence of τ1-open covers of X. By the almost Mengerness of X,τ1, one can find finite sets Vn⊂Un, n∈ℕ, such that ∪n∈ℕ∪Clτ1V:V∈Vn=X. Because τ1<τ2, each Vn is a family of τ2-open sets. So, for each n and each V∈Vn, τ1<Cτ2 implies Clτ1V⊂Clτ2V. It follows from here that(11)X=∪n∈ℕ∪Clτ1V:V∈Vn⊂∪n∈ℕ∪Clτ2V:V∈Vn=X.
Therefore,X,τ1,τ2 is τ1,τ2-almost Menger.
It should be noted that the Proposition4 extends and generalizes the following result obtained by Özçağ and Eysen in [20].Corollary 7.
IfX,τ1 is almost Menger and τ2<τ1, then the bitopological space X,τ1,τ2 is 1,2-almost Menger.Similar to the proof of the first part of Theorem12, we prove the following.Theorem 13.
IfX,τ1<Nτ2 is 1,2-almost Menger, then X,τ1 is almost Menger.Since our main aim was, as mentioned in the introduction, to investigate the behaviour of some covering properties under several functions, we turn to mappings between bitopological spaces equipped with some relations.The classes of alli,j-θ-continuous functions of X to Y are denoted by i,j−θCX,Y.Corollary 8 (see [30]).
LetX,τ1,τ2 and Y,σ1,σ2 be bitopological spaces. For X,τ1<Cτ2 and Y,σ1<Cσ2, we have(12)1,2−θCX,Y=1−θCX,Y.Theorem 14.
Letf:X,τ1<Cτ2⟶Y,σ1<Cσ2 be 1,2-θ-continuous mapping. If X,τ1<Cτ2 is 1,2-almost Menger, thenProof.
LetX,τ1<Cτ2 be an 1,2-almost Menger bitopological space. By Theorem 12, X,τ1 is almost Menger, and the mapping fi:X,τ1⟶Y,σ1 is θ-continuous by Corollary 8. Since θ-continuous image of an almost Menger space is almost Menger ([36], Theorem 2.1), then Y,σ1 is almost Menger. Thus, Y,σ1<Cσ2 is 1,2-almost Menger by Theorem 12.Corollary 9 (see [30]).
The following conditions are satisfied for a bitopological spaceX,τ1Cτ2.(1)
IfX,τ1,τ2 is 1,2-regular, then X,τ1,τ2 is 1-regularMoreover, for a bitopological spaceX,τ1<Cτ2, we have(2)
X,τ1,τ2 is 1,2-regular ⇔X,τ1,τ2 is 1-regularTheorem 15.
LetX,τ1<Cτ2 be 1,2-regular and 1,2-almost Menger bitopological space. If f:X,τ1,τ2⟶Y,σ1,σ2 is 1-continuous mapping, then Y,σ1<Cσ2 is 1,2-almost Menger.Proof.
LetX,τ1<Cτ2 be 1,2-almost Menger and 1,2-regular bitopological space, then X,τ1 is almost Menger by Theorem 12 and X,τ1 is regular by Corollary 9; thus, we obtain that X,τ1 is Menger ([10], Theorem 2.1). Since f:X,τ1⟶Y,σ1 is continuous and continuous image of a Menger space is Menger, then Y,σ1 is Menger and Y,σ1 is almost Menger. By using Theorem 12, Y,σ1<Cσ2 is 1,2-almost Menger.Proposition 5.
Iff:X,τ1Cτ2⟶Y,σ1,σ2 is 1,2-strongly θ-continuous mapping, then f is 1-strongly θ-continuous.Proof.
Letf:X,τ1Cτ2⟶Y,σ1,σ2 be 1,2-strongly θ-continuous mapping. Let x∈X and V be σ1-open set in Y containing fx. Since f is 1,2-strongly θ-continuous mapping, there is a τ1-open set U in X containing x, such that fClτ2U⊆V. Since τ1Cτ2, we have fClτ1U⊆fClτ2U⊆V; thus, f is 1-strongly θ-continuous.Proposition 6.
f:X,τ1<Cτ2⟶Y,σ1,σ2 is 1,2-strongly θ-continuous mapping if and only if f is 1-strongly θ-continuous.Proof.
Immediate.Corollary 10.
Letf:X,τ1<Cτ2⟶Y,σ1,σ1 be i,j-strongly θ-continuous mapping. If X,τ1<Cτ2 is 1,2-almost Menger, then Y,σ1 is Menger.Proof.
IfX,τ1<Cτ2 is i,j-almost Menger, then by Proposition 4, X,τ1 is almost Menger. Since f:X,τ1⟶Y,σ1 is strongly θ-continuous by the Proposition 5, we obtain that Y,σ1 is Menger by ([36], Theorem 2.7).


### 4.1. More on Relations and Covering Properties

Example 5.
Lindelöfness is not preserved by the order relation<C.(1)
LetX,τ1,τ2 be the plane ℝ2 equipped with the following two topologies: τ1 is the usual Euclidean topology, and τ2 is the deleted radius topology ([37], example 77). Then, τ1<τ2, and for any open set U∈τ1, Clτ1U=Clτ2U, i.e., τ1<Cτ2. X,τ1 is a Lindelöf spaces, while X,τ2 is not ([37], example 77).(2)
Another such example is the setX=x,y∈ℝ2:y≥0 with the Euclidean topology τ1 and the Niemytzki plane topology τ2Example 6.
σ-compactness is not preserved by the order relation <C
LetX,τ1,τ2 be the plane ℝ2 equipped with the following two topologies: τ1 is the usual Euclidean topology, and τ2 is the deleted radius topology ([37], example 77). It is clear that τ1<Cτ2. X,τ1 is σ-compact, but X,τ2 is not σ-compact.Remark 4.
Beingσ-compact, the space X,τ1 in the previous example is Alster [8]. However, X,τ2 is not Alster because it is not Menger (Alster spaces are Hurewicz; hence, Menger in all finite powers).Example 7.
Mengerness is not preserved by the order relation<N.LetX=x,y∈ℝ2:y≥0. Let Y=x,y∈ℝ2:y>0. Let τ1 be the usual metric topology on X, and τ2, the half-disk topology ([37], example 78), is defined as follows: any point p∈Y has the Euclidean local base, while the point p=x,0∈X∖Y has basic neighbourhoods of the form Bp,ε∩Y∪p. Then, τ1<τ2, and for any τ2-open set U, Clτ2U=Clτ1U, so that τ1<Nτ2. The space X,τ1 is Menger, while X,τ2 is not Menger because it is not Lindelöf.Example 8.
Hurewicz property is not preserved by the order relation<C.
LetX,τ1,τ2 be the plane ℝ2 equipped with the following two topologies: τ1 is the usual Euclidean topology, and τ2 is the deleted diameter topology ([37], Example 77). It is clear that τ1<Cτ2. X,τ1 is the Hurewicz, but X,τ2 is not Hurewicz because it is not Lindelöf.
Another such example is the closed upper half-plane with the Euclidean topologyτ1 and the half-disc topology τ2 ([37], example 78).Example 9.
LetX be any uncountable scattered compact space (for instance, the product 0,ω1×0,ω of ordinal spaces) with the topology τ2, and let τ1 be the discrete topology on X. Then, τ1Nτ2, X,τ2 is a Rothberger space, and X,τ1 is not Rothberger.Theorem 16.
Ifτ1<Nτ2 and X,τ2 is almost Menger, then X,τ1 is almost Menger.Proof.
LetUn:n∈ℕ be a sequence of open covers of X,τ1. As τ1<τ2, it is a sequence of open covers of X,τ2. By assumption, there are finite sets Vn⊂Un, n∈ℕ, so that X=∪n∈ℕClτ2∪Vn. From τ1<Nτ2 it follows Clτ2∪Vn=Clτ1∪Vn for each n∈ℕ, which means that X,τ1 is almost Menger.
Similarly, one obtains.Theorem 17.
Ifτ1<Nτ2 and X,τ2 is weakly Menger, then X,τ1 is weakly Menger.Theorem 18.
LetX,τ1,τ2 be a bitopological space. If τ1<Cτ2 and X,τ2 is selectively ccc, then X,τ1 is selectively ccc.Proof.
LetUn:n∈ℕ be a sequence of τ1-open weak cover of X. Then, ∪Un∈τ1, τ1<Cτ2 and Clτ1∪Un=X, n∈ℕ, imply that for each n, Un∈τ2, and Clτ2∪Un=X. As X,τ2 is selectively ccc, there are Un∈Un, n∈ℕ, such that Clτ2∪n∈ℕUn=X. From τ1<τ2, it follows Clτ1∪n∈ℕUn⊃Clτ2∪n∈ℕUn=X, which means that X,τ1 is selectively ccc.Theorem 19.
LetX,τ1,τ2 be a bitopological space. If τ1<Cτ2 and X,τ2 is selectively star-ccc, then X,τ1 is selectively star-ccc.Proof.
LetUnn∈ℕ be a sequence of open weak covers of X,τ1 and U a τ1-open cover of X. Since ∪Un∈τ1, n∈ℕ, and τ1<Cτ2, we have Clτ1∪Un⊂Clτ2∪Un=X. Also, U is a τ2-open cover of X. Using the fact that X,τ2 is selectively star-ccc, we find Un∈Un, n∈ℕ, such that St∪n∈ℕUn,U=X, i.e., X,τ1 is selectively star-ccc.
Table1 summarizes the above results. Empty fields in the table may serve as open questions.Table 1
The preservation of some covering properties under various relations.
Propertyτ1 C τ2τ1 N τ2τ1≤Cτ2τ1≤Nτ21τ1 Lτ2¬L2τ1σ-comτ2¬σ-com3τ1 Mτ2¬M4τ2 aMτ1 aM5τ2 wMτ1 wM6τ1 Hτ2¬H7τ2 Rτ1¬R8τ1 Alsterτ2¬Alster9τ2 sel cccτ1 sel ccc


## 4.1. More on Relations and Covering Properties

Example 5.
Lindelöfness is not preserved by the order relation<C.(1)
LetX,τ1,τ2 be the plane ℝ2 equipped with the following two topologies: τ1 is the usual Euclidean topology, and τ2 is the deleted radius topology ([37], example 77). Then, τ1<τ2, and for any open set U∈τ1, Clτ1U=Clτ2U, i.e., τ1<Cτ2. X,τ1 is a Lindelöf spaces, while X,τ2 is not ([37], example 77).(2)
Another such example is the setX=x,y∈ℝ2:y≥0 with the Euclidean topology τ1 and the Niemytzki plane topology τ2Example 6.
σ-compactness is not preserved by the order relation <C
LetX,τ1,τ2 be the plane ℝ2 equipped with the following two topologies: τ1 is the usual Euclidean topology, and τ2 is the deleted radius topology ([37], example 77). It is clear that τ1<Cτ2. X,τ1 is σ-compact, but X,τ2 is not σ-compact.Remark 4.
Beingσ-compact, the space X,τ1 in the previous example is Alster [8]. However, X,τ2 is not Alster because it is not Menger (Alster spaces are Hurewicz; hence, Menger in all finite powers).Example 7.
Mengerness is not preserved by the order relation<N.LetX=x,y∈ℝ2:y≥0. Let Y=x,y∈ℝ2:y>0. Let τ1 be the usual metric topology on X, and τ2, the half-disk topology ([37], example 78), is defined as follows: any point p∈Y has the Euclidean local base, while the point p=x,0∈X∖Y has basic neighbourhoods of the form Bp,ε∩Y∪p. Then, τ1<τ2, and for any τ2-open set U, Clτ2U=Clτ1U, so that τ1<Nτ2. The space X,τ1 is Menger, while X,τ2 is not Menger because it is not Lindelöf.Example 8.
Hurewicz property is not preserved by the order relation<C.
LetX,τ1,τ2 be the plane ℝ2 equipped with the following two topologies: τ1 is the usual Euclidean topology, and τ2 is the deleted diameter topology ([37], Example 77). It is clear that τ1<Cτ2. X,τ1 is the Hurewicz, but X,τ2 is not Hurewicz because it is not Lindelöf.
Another such example is the closed upper half-plane with the Euclidean topologyτ1 and the half-disc topology τ2 ([37], example 78).Example 9.
LetX be any uncountable scattered compact space (for instance, the product 0,ω1×0,ω of ordinal spaces) with the topology τ2, and let τ1 be the discrete topology on X. Then, τ1Nτ2, X,τ2 is a Rothberger space, and X,τ1 is not Rothberger.Theorem 16.
Ifτ1<Nτ2 and X,τ2 is almost Menger, then X,τ1 is almost Menger.Proof.
LetUn:n∈ℕ be a sequence of open covers of X,τ1. As τ1<τ2, it is a sequence of open covers of X,τ2. By assumption, there are finite sets Vn⊂Un, n∈ℕ, so that X=∪n∈ℕClτ2∪Vn. From τ1<Nτ2 it follows Clτ2∪Vn=Clτ1∪Vn for each n∈ℕ, which means that X,τ1 is almost Menger.
Similarly, one obtains.Theorem 17.
Ifτ1<Nτ2 and X,τ2 is weakly Menger, then X,τ1 is weakly Menger.Theorem 18.
LetX,τ1,τ2 be a bitopological space. If τ1<Cτ2 and X,τ2 is selectively ccc, then X,τ1 is selectively ccc.Proof.
LetUn:n∈ℕ be a sequence of τ1-open weak cover of X. Then, ∪Un∈τ1, τ1<Cτ2 and Clτ1∪Un=X, n∈ℕ, imply that for each n, Un∈τ2, and Clτ2∪Un=X. As X,τ2 is selectively ccc, there are Un∈Un, n∈ℕ, such that Clτ2∪n∈ℕUn=X. From τ1<τ2, it follows Clτ1∪n∈ℕUn⊃Clτ2∪n∈ℕUn=X, which means that X,τ1 is selectively ccc.Theorem 19.
LetX,τ1,τ2 be a bitopological space. If τ1<Cτ2 and X,τ2 is selectively star-ccc, then X,τ1 is selectively star-ccc.Proof.
LetUnn∈ℕ be a sequence of open weak covers of X,τ1 and U a τ1-open cover of X. Since ∪Un∈τ1, n∈ℕ, and τ1<Cτ2, we have Clτ1∪Un⊂Clτ2∪Un=X. Also, U is a τ2-open cover of X. Using the fact that X,τ2 is selectively star-ccc, we find Un∈Un, n∈ℕ, such that St∪n∈ℕUn,U=X, i.e., X,τ1 is selectively star-ccc.
Table1 summarizes the above results. Empty fields in the table may serve as open questions.Table 1
The preservation of some covering properties under various relations.
Propertyτ1 C τ2τ1 N τ2τ1≤Cτ2τ1≤Nτ21τ1 Lτ2¬L2τ1σ-comτ2¬σ-com3τ1 Mτ2¬M4τ2 aMτ1 aM5τ2 wMτ1 wM6τ1 Hτ2¬H7τ2 Rτ1¬R8τ1 Alsterτ2¬Alster9τ2 sel cccτ1 sel ccc


## 5. Conclusion

This study is a continuation and extension of our research in the field of selective covering properties in bitopological spaces. Complementing some earlier studies on almost Menger and almost Rothberger bitopological spaces, we investigated preservation of these properties under various classes of mappings between bitopological spaces. Next, we introduced and studied in detail the selective bitopological versions of the topological ccc property called selectively ccc and selectively star-ccc property. Relations of these bitopological properties with other known properties are established. In the last part of the study, we discussed (non) preservation of a number of topological and bitopological covering properties under known relations between the two topologies. We believe that a natural continuation of our study may be findings of relations of the properties studied in this study with game theory, function spaces, and hyperspaces.



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