More on Selective Covering Properties in Bitopological Spaces

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.

1. Introduction

Throughout the study, we use the standard topological notation and terminology, mainly as in [1]. By and , we denote the set of natural numbers and the set of real numbers, respectively. By or , we denote a topological space, while denotes a bitopological space [2] (called also bispace), that is a set equipped with two (in general, unrelated) topologies. The closure and interior of a subset of a space are denoted by and , respectively. When is a bitopological space and , then and , , denote the closure and interior in the space .

If is a subset of a topological space , , and is a family of subsets of , then , and denote the star of and with respect to .

For a space , we also use the following notations:(i) is the collection of all open covers of (ii) ( with this property is called an open weak cover of )(iii) ( with this property is called an open almost cover of )

In case of bitopological spaces, we write , , to indicate that , , is related to the space , .

If and are families of sets, then(1)The symbol denotes the selection hypothesis: for each sequence of elements of , there is a sequence , such that for each , is a finite subset of , and (see [3]);(2)The symbol denotes the selection hypothesis: for each sequence of elements of , there is a sequence , such that for each , and (see [3]).

The following covering properties are defined in this way:(i) is the Menger property [4](ii) is the Rothberger property [5](iii) is the weak Menger property [6] (see also [7, 8])(iv) is the almost Menger property [9] (see also [10])(v) is the weak Rothberger property [6](vi) is the almost Rothberger property [9]

A space has the Hurewicz property if for each sequence of open covers of , there is a sequence , such that for each , is a finite subset of , and each belongs to for all but finitely many [11].

For selection principles theory see [12].

Recall that a family of pairwise disjoint open sets in a topological space is called a cellular open family. 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 be a topological space. is selectively ccc if for every sequence of maximal cellular open families in , there is a sequence , such that for each , and is dense in .

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 , where is the collection of open families with a dense union (i.e., is in our notation). In [16], these spaces have been studied under the notation , where 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 class .

If and are collections of covers of a space , then(1)The symbol denotes the selection hypotheses that for each sequence of elements of , there is a sequence of elements of such that (see [17])(2)The symbol denotes the selection hypotheses: for each open cover of and each sequence of elements of , there is a sequence , such that for each , and (see [18])

In [19], the following spaces have been defined and studied.

Definition 2. (see [19]) A topological space is said to be selectively star-ccc if it belongs to the class .

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 Section 2, 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 is said to be -almost Menger, , if for each sequence of -open covers of , there exists a sequence of finite families, such that for each , and .

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 mapping is called(1)--continuous [26] (respectively,-strongly -continuous [27]) if for each and each -open set containing , there exists a -open set containing , such that (respectively, ). It is obvious that each -strongly -continuous mapping is --continuous.(2)-almost continuous [28] if is the -open set in for every -regular open set in . In addition, is called -almost continuous if it is and -almost continuous. Recall that a set in a bitopological space is -regular open [29] if .

We note that -almost continuous mappings are --continuous [30], so the following result generalizes the result in [20], which states that the -almost Menger property is preserved by -almost continuous mappings.

Theorem 1 (see [21]). An -continuous image of an -almost Menger bitopological space is -almost Menger.

In [21], the definitions of -almost -sets and the -almost Hurewicz property were introduced, and the following theorem was obtained.

Theorem 2 (see [21]). Let be an -almost -set and be a bitopological space. If is an --continuous surjection, then is -almost Hurewicz.

Now, we consider the behaviour of the -almost Rothberger property under --continuous mappings.

A bitopological space is said to be -almost Rothberger, , if for each sequence of -open covers of , there is a sequence , such that for each , and [20].

Theorem 3. An --continuous image of an -almost Rothberger bitopological space is -almost Rothberger.

Proof. Let be an -almost Rothberger bitopological space. Let be the image of , where is an --continuous mapping.
Let be a sequence of -open covers of . For each and fixed , there is a set , such that . Since is --continuous, there exists a -open set containing , such that . For each , is a -open cover of . Apply the fact that is -almost Rothberger. There is a sequence , such that for each , and . For each and , we assign a set , such that . We obtain the sequence , with andThis means that is -almost Rothberger.

Corollary 1. The -almost Rothberger property is preserved by -almost continuous mappings.

Theorem 4. If is an -strongly -continuous mapping and is -almost Menger, then is Menger.

Proof. Let be a sequence of -open covers of and . For each , there exists a set with . Since is -strongly -continuous, there is a -open set with , such that . Then, is a -open cover of . Since is -almost Menger bitopological space, there exists a sequence , such that for each , is a finite subset of , and . Now, for each and , we can choose , such that . Let ; we obtain a sequence of finite subsets of , . We will show is a -cover of . Indeed,This shows that is Menger.

Remark 1. It is obvious that if is an -strongly -continuous mapping, then is -continuous; furthermore, if is the -regular bitopological space, then the reverse implication is also true.

Corollary 2. If is -continuous mapping and is -regular, -almost Menger bitopological space, then is Menger.

Now, recall the notions of contra continuous and precontinuous mappings in bitopological spaces. A mapping is said to be -contracontinuous and -precontinuous [29], if the preimage of each -open set of is -closed in (-preopen in , i.e., ), where .

Theorem 5. If is the almost Menger bitopological space and is an -contracontinuous and -precontinuous mapping, then is a Menger space.

Proof. Let be a sequence of covers of by -open sets. Since is -contra continuous, for each and , the set is -closed in . Then, we have , since is -precontinuous. Now, we obtain , which gives . Then, for each , we set . Then, is a sequence of -open covers of . Since is -almost Menger, there exists a sequence , such that for each , is a finite subset of and . Let . It is obvious that is a sequence of finite subsets of for each , and is an -open cover of which means that is a Menger space.
A mapping is said to be -weakly continuous [26] if for each and each -open subset containing , there exist a -open set with , such that .

Theorem 6. If is an -weakly continuous mapping and is Menger, then is an -almost Menger space.

Proof. Let be a Menger space and be a sequence of -open covers of . Let , then for each , there exists with . Since is -weakly continuous, there is a -open set with and . Now, let . Clearly, is a -open cover of . Since is Menger, we find a sequence of finite sets, such that for each , and is a -open cover of . For each and , we can choose a set , such that . Set . We show that is a cover of . Indeed, as , we haveSo, is -almost Menger.

Remark 2. Since --continuous mappings are -weakly continuous, Theorem 6 is also true under -continuous mappings.

3. Selectively ccc and Selectively Star-ccc Bispaces

3.1. Selectively ccc Bispaces

Definition 4. A bitopological space is said to satisfy i-countable chain condition (for short i-ccc), , if the topological space satisfies countable chain condition. satisfies countable chain condition if it satisfies i-countable chain condition for .

Definition 5. A bitopological space is -selectively ccc if for every sequence of elements of , there is a sequence , such that , , and is dense in , i.e., if is satisfied.

Proposition 1. Let be a bitopological space such that (i.e., the topology is weaker than ).(1)If is -selectively ccc, then is selectively ccc(2)If is selectively ccc, then is -selectively ccc

Since the selectively ccc property is stronger than the ccc property, we have the following.

Corollary 3. Let be a bitopological space such that . If is -selectively ccc bispace, then has the ccc property.

Proposition 2. Let be a bitopological space and be -closed and -open subspace in . If is -selectively ccc, then is -selectively ccc.

Proof. (For i = 1 and j = 2). Let be a -closed and -open subset of and let be a sequence of members of . Thus, for each , is an element of . Since is -selectively ccc, there is a sequence , such that for every , and . If , we set ; if , then we take to be any element from . Then, it is clear that .
Recall that a subset of a bispace is called bidense (here called double dense or shorty d-dense) in if is dense in both and . is -separable if there is a countable set which is d-dense in .

Theorem 7. Let be a d-dense set in a bispace . is -selectively ccc if and only if is selectively ccc.

Proof. (For i = 1 and j = 2). Let be a sequence of -open families such that is dense in . Since is -dense, then each is dense in . As is -selectively ccc, there is a sequence , such that for each , , and is dense in . Since is dense in , then is dense in .
Let be a sequence of -open families, such that each is dense in . The restriction to , is a -open family for each . Since is -selectively ccc, there is a sequence , such that and is dense in . But is dense in , so is dense in .

Theorem 8. Let and be bitopological spaces and be a -continuous, -open and continuous, -closed surjection. If is -selectively ccc, then is -selectively ccc.

Proof. (For i = 1 and j = 2). Let be a sequence of -open families in such that is -dense in . Since is continuous and open, for each , is -open family in and is -dense in . Because is -selectively ccc, we can pick with dense in . For the sequence , such that , we have dense in , since is a continuous and closed mapping. Thus, is selectively ccc.
Bitopological -separability has been studied by Kočinac and Özçağ in [31].
Denote by and the collection of all dense subsets of and , respectively. We say that is -separable , if for each sequence of elements of , there are , , such that is an element of , i.e., if holds.
We have the following fact.

Proposition 3. Every -separable bispace is -selectively ccc.

Corollary 4. If the bitopological space has a -closed and -open -separable subspace, then is selectively ccc.

Proof. It follows by Proposition 2 and the previous result.
Let denote the collection of all -covers of a space (open covers such that every compact subset of is contained in a member of the cover), and let denote the collection of all -covers of (open covers such that every finite subset of is contained in a member of the cover and itself is not a member of the cover).

Example 1. If a Tychonoff space with a countable base belongs to the class , then ([31], Theorem 4) the function bispace is -separable. Therefore, this function bispace is selectively -star-ccc, where denotes the pointwise topology, and denotes the compact-open topology on the set of all continuous real-valued functions on a Tychonoff space .

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 bispace is said to be -selectively star-ccc, , if for every sequence of -open weak covers and every open cover of , there is a sequence , such that for every , and .

The following result shows that the class of selectively star-ccc spaces is big enough.

Theorem 9. Let be a bispace, such that . If is a Lindelöf space, then is a -selectively star-ccc space.

Proof. Let be a sequence of -open weak covers of , and let be an open cover of . There exists a countable set , such that . Since each , , is -open, for every , there exists , such that . It follows that . This proves the theorem.

Corollary 5. The bispace , where is the set of real numbers, is the usual metric topology, and is the Sorgenfrey topology, is -selectively star-ccc.

Recall that a space is said to be strongly star Lindelöf, if for each open cover of , there is a countable set , such that .

Call a bispace -selectively 2-star-ccc if for every -open cover of and every sequence of -open weak covers of , there is a sequence , such that , , and , where .

Theorem 10. Let be a bispace such that . If is a strongly star Lindelöf space, then is an -selectively 2-star-ccc space.

Proof. Let be an arbitrary -open cover of and be a sequence of -open weak covers of . Since is strongly star Lindelöf, there exists a countable set , such that . Therefore, is a -open cover of . For each , there exists , such that . So, . For every , the set is dense in and hence in , so that . This implies that for each , there exists , such that . Therefore, for each ,which impliesThis shows that is (1, 2)-selectively 2-star-ccc.

Theorem 11. Every -selectively ccc bispace is -selectively star-ccc.

Proof. (For i = 1, j = 2). Let be a -open cover of and be a sequence of -open weak covers of . Since is -selectively ccc, there exists a sequence , such that for every , , and is dense in . So, intersects each element of , and thus,where is a -selectively star-ccc space.
From Proposition 3 and Theorem 11, we have the following corollary.

Corollary 6. Every -separable bispace is -selectively star-ccc.

We show now that the converse in this corollary is not always true.

Example 2. Let be a bispace with , such that is a compact and nonseparable space. By Theorem 9, the bitopological space is -selectively star-ccc. But it is not -separable by the following fact: -separability implies separability of .

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 and ) between two topologies on a bitopological space .

In 1957, Weston [35] introduced the -relation, while in [30], the -relation was defined. The abovementioned relations and their combinations with the set-theoretic operation inclusion are considered in [30].

Let be a bitopological space. Obviously, implies for each .

Definition 7 (see [30, 35]). The topology is coupled to the topology on a set , denoted by , if for each .

Remark 3. (see [30]). If the open sets and closed sets are equal for a topology on a set , then is coupled to every topology on . For example, the discrete and indiscrete topologies are coupled to each topology on .

The -relation defines a partial order (we use the notation ) on the family of all topologies on by

It is easy to see that implies , but the reverse implication is not correct.

Example 3. Let , and . It is clear that and , but these topologies are not comparable by inclusion.

A set together with the topologies is denoted by .

Now, we consider the -relation between two topologies on a set .

Definition 8. (see [30]). A topology is near to a topology on a set (briefly ) if for each .

The discrete topology on is near any topology on , and every topology on is near the indiscrete topology on .

Similar to the -relation, the -relation defines a partial order on the family of all topologies on by

We mention that the notions of coupling and nearness of topologies are independent of each other; however, we have the following implication:

A set together with the topologies is denoted by ().

We omit the proof of the following proposition.

Proposition 4. Let be a bitopological space. If is almost Menger, then is -almost Menger.

However, in the following example, we show that the reverse implication is not true in general.

Example 4. Let be the Euclidean plane , be the Sorgenfrey topology, and be the ordinary metric topology on . Clearly, is -almost Menger; however, does not have the almost Menger property, since is not almost Lindelöf.

On the other hand, we have the following result.

Theorem 12. A bispace is -almost Menger if and only if is almost Menger.

Proof. Let be a sequence of -open covers of . Since is -almost Menger, there exists a sequence of finite sets with andBy , we have and . For , we have . Therefore, gives that is almost Menger.
Let be a sequence of -open covers of . By the almost Mengerness of , one can find finite sets , , such that . Because , each is a family of -open sets. So, for each and each , implies . It follows from here thatTherefore, is -almost Menger.
It should be noted that the Proposition 4 extends and generalizes the following result obtained by Özçağ and Eysen in [20].

Corollary 7. If is almost Menger and , then the bitopological space is -almost Menger.

Similar to the proof of the first part of Theorem 12, we prove the following.

Theorem 13. If is -almost Menger, then 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 all --continuous functions of to are denoted by .

Corollary 8 (see [30]). Let and be bitopological spaces. For and , we have

Theorem 14. Let be --continuous mapping. If is -almost Menger, then

Proof. Let be an -almost Menger bitopological space. By Theorem 12, is almost Menger, and the mapping is -continuous by Corollary 8. Since -continuous image of an almost Menger space is almost Menger ([36], Theorem 2.1), then is almost Menger. Thus, is -almost Menger by Theorem 12.

Corollary 9 (see [30]). The following conditions are satisfied for a bitopological space .(1)If is -regular, then is 1-regular Moreover, for a bitopological space , we have(2) is -regular is 1-regular

Theorem 15. Let be -regular and -almost Menger bitopological space. If is 1-continuous mapping, then is -almost Menger.

Proof. Let be -almost Menger and -regular bitopological space, then is almost Menger by Theorem 12 and is regular by Corollary 9; thus, we obtain that is Menger ([10], Theorem 2.1). Since is continuous and continuous image of a Menger space is Menger, then is Menger and is almost Menger. By using Theorem 12, is -almost Menger.

Proposition 5. If is -strongly -continuous mapping, then is 1-strongly -continuous.

Proof. Let be -strongly -continuous mapping. Let and be -open set in containing . Since is -strongly -continuous mapping, there is a -open set in containing , such that . Since , we have ; thus, is 1-strongly -continuous.

Proposition 6. is -strongly -continuous mapping if and only if is 1-strongly -continuous.

Proof. Immediate.

Corollary 10. Let be -strongly -continuous mapping. If is -almost Menger, then is Menger.

Proof. If is -almost Menger, then by Proposition 4, is almost Menger. Since is strongly -continuous by the Proposition 5, we obtain that 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 .(1)Let be the plane equipped with the following two topologies: is the usual Euclidean topology, and is the deleted radius topology ([37], example 77). Then, , and for any open set , , i.e., . is a Lindelöf spaces, while is not ([37], example 77).(2)Another such example is the set with the Euclidean topology and the Niemytzki plane topology

Example 6. -compactness is not preserved by the order relation
Let be the plane equipped with the following two topologies: is the usual Euclidean topology, and is the deleted radius topology ([37], example 77). It is clear that . is -compact, but is not -compact.

Remark 4. Being -compact, the space in the previous example is Alster [8]. However, 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 .

Let . Let . Let be the usual metric topology on , and , the half-disk topology ([37], example 78), is defined as follows: any point has the Euclidean local base, while the point has basic neighbourhoods of the form