Abstract

A space is said to be set selectively star-ccc if for each nonempty subset of , for each collection of open sets in such that , and for each sequence of maximal cellular open families in , there is a sequence such that, for each , and . In this paper, we introduce set selectively star-ccc spaces and investigate the relationship between set selectively star-ccc and other related spaces. We also study the topological properties of set selectively star-ccc spaces. Some open problems are posed.

1. Introduction

In 1996, Scheepers [1] initiated the systematic study of selection principle in topology and their relations to game theory and Ramsey theory (also see [2]). After this study, it becomes one of the most active areas in set theoretic topology. Kočinac [3, 4] applied the star operator to these selection principles and introduced and studied the new selection principles called star selection principles. It should be noted that classical selection principles have been used to define and characterize various covering properties such as Rothberger [5], Menger [6], and star-Menger [3]. In this paper, we use the following selection principle of the Scheepers type from [1]. Let and be families of sets.

Then, denotes, for each sequence of elements of , there is a sequence such that, for each , and is an element of .

If is the family of all open covers of a space , then is the Rothberger covering property.

On the contrary, Arhangel’skii [7] defined a cardinal function and spaces such that ; we call -Lindelöf: a space is -Lindelöf if, for each nonempty subset of and each open cover of by sets open in , there is a countable set such that . Following this idea and modifying it, Kočinac and Konca [8] considered new types of selective covering properties called set-covering properties. A space is said to have the set-Menger property [8, 9] if, for each nonempty subset of and each sequence of collections of sets open in such that , there is a sequence such that, for each , is a finite subset of and . Recently, Kočinac, Konca, and Singh initiated a study of star versions of the set-Menger covering property.

In [10], Aurichi introduced the class of selectively ccc spaces. Bal and Kočinac [11] introduced and studied the star version of selectively ccc spaces called selectively star-ccc spaces.

The purpose of this paper is to introduce the set selectively star-ccc spaces, the class which lies between Lindelöf spaces and selectively star-ccc spaces. We investigate the relationship between set selectively star-ccc and other related spaces and study the topological properties of set selectively star-ccc spaces. Some open problems are posed.

2. Preliminaries

By “a space” we mean “a topological space.” Throughout the paper, an open cover of a subset means elements of are open in such that , unless otherwise stated.

If is a subset of a space and is a collection of subsets of , then

We usually write . One defines , and for , .

A family of pairwise disjoint open sets in a topological space is called a cellular open family. A space is said to be a ccc space if every cellular open family in is countable.

Definition 1 (see [10]). A space is said to be selectively ccc space if for each sequence of maximal cellular open families in , there is a sequence such that, for each , and is dense in .

Definition 2 (see [12–14]). A space is said to be -separable if, for each sequence of dense subsets of , there is a sequence such that, for each , and is dense in .
Every -separable space is a selectively ccc, and every selectively ccc space is ccc.

Definition 3. (see [11]). A space is said to be selectively star-ccc if, for each open cover of X and for each sequence of maximal cellular open families in , there is a sequence such that, for each , and .

Definition 4 (see [15]). A space is said to be strongly star-Lindelöf if, for each open cover of , there is a countable subset of such that .
Note that strongly star-Lindelöf spaces are also called star countable in [16].
In a similar way, Kočinac, Konca, and Singh defined the following.

Definition 5. Let . A space is said to be set strongly -starcompact (resp., set strongly -star-Lindelöf) if, for each nonempty subset of and for each collection of open sets in such that , there is a finite (resp., countable) subset of such that .
We say set strongly starcompact and set strongly star-Lindelöf instead of set strongly 1-starcompact and set strongly 1-star-Lindelöf, respectively. It is clear, by the definitions, that every set strongly star-Lindelöf space is strongly star-Lindelöf and every set strongly -starcompact space is set strongly -star-Lindelöf, .

Definition 6. (see [17, 18]). A space is said to be absolutely countably compact (shortly, ) if, for each open cover of and for each dense subset of , there is a finite subset of such that .
Clearly, a compact space is , and an Hausdroff space is countably compact (see [17]). In a similar way, we define the following.

Definition 7. A space is said to be set absolutely countably compact (shortly, set-) if, for each nonempty subset of , for each collection of open sets in such that , and for every dense subset of , there is a finite subset of such that .

Lemma 1. Every compact space is set-.

Proof. Let be any nonempty subset of , be any collection of open sets in such that , and be any dense subset of . Since a closed subset of a compact space is compact, is compact; hence, there exists a finite subset of such that . For each , . Take . Put . Then, is a finite subset of and . Thus, is set-.
It is clear from the definitions that every set- space is and thus we have the following corollary.

Corollary 1. Every Hausdorff set- space is countably compact.

The following is an open question.

Problem 1. Does an space which is not set- exist?
Recall that a subspace of a space is -dense in if, for every , there is a countable such that .

Lemma 2. If is countably compact and every dense subspace of is -dense in , then is set-.

Proof. Let be any nonempty subset of , be any collection of open sets in such that , and be any dense subset of . Since a closed subset of a countably compact space is countably compact, is countably compact, and so there exists a finite subset such that . Since is -dense in , for every , pick a countable such that . Set . Then, is countable, and for each , . So, is a countable cover of . By countable compactness of , there is a finite subset such that .
By Lemma 2, we have the following result.

Theorem 1. If a countably compact space has a countable tightness, then is set-.

Recall that a collection of infinite subsets of is said to be almost disjoint if the sets are finite for all distinct elements . For an almost disjoint family , put and topologize as follows: for each element and each finite set , is a basic open neighborhood of ; each is isolated. The spaces of this type are called Isbell-Mrówka-spaces [19] or spaces.

Throughout the paper, the cardinality of a set is denoted by . Let denote the first infinite cardinal, the first uncountable cardinal, and the cardinality of the set of all real numbers. For a cardinal , let be the smallest cardinal greater than . As usual, a cardinal is an initial ordinal and an ordinal is the set of smaller ordinals. A cardinal is often viewed as a space with the usual order topology. Other notation and terminology follow [20].

A sub set of is said to be regular open if . A subset is said to be regular closed if its complement is regular open or, equivalently, if .

In this section, we give some examples showing the relationships between set selectively star-ccc spaces and other related spaces.

Definition 8. A space is said to be set selectively star-ccc if, for each nonempty subset of , for each collection of open sets in such that , and for each sequence of maximal cellular open families in , there is a sequence such that, for each , and .
It is clear by the definitions that every set selectively star-ccc space is selectively star-ccc.

Theorem 2. Every Lindelöf space is set selectively star-ccc.

Proof. Let be any nonempty subset of a Lindelöf space , be any collection of open sets in such that , and be a sequence of maximal cellular open families in . Since space is Lindelöf, closed subset is also Lindelöf. Thus, there exists a countable subset of such that . For each , is maximal cellular open family in , and thus there exists such that . Thus,Therefore, is a set selectively star-ccc space.
The following example shows that the converse of Theorem 2 need not be true.

Example 1. (1)Let the ordinal space be equipped with the usual order topology. Then, is a countably compact space of countable tightness, and by Theorem 1, is set-. By Theorem 5, is a set selectively star-ccc space. However, is not Lindelöf.(2)Let be any Tychonoff space such that the function space is not Lindelöf. For a Tychonoff space the function space is selectively ccc; thus, by Theorem 3, is set selectively star-ccc.

Theorem 3. Every selectively ccc space is set selectively star-ccc.

Proof. Let be any nonempty subset of a selectively ccc space , be any collection of open sets in such that , and be any sequence of maximal cellular open families in . Since the space is selectively ccc, there exists a sequence such that, for each , and is dense in . Therefore, , which shows that is a set selectively star-ccc space.
The following example shows that converse of Theorem 3 is not true.

Example 2. Let be the discrete space of cardinality and let , where , be the one-point Lindelöfication of . The topology on is defined as follows: for each , is isolated and a set containing is open if and only if is countable. Then, is a Tychonoff Lindelöf space, and thus it is a set selectively star-ccc space. However, the collection is a maximal cellular open family in which is not countable. Thus, is not a ccc space; hence, not selectively ccc (since every selectively ccc space is ccc).

Corollary 2. Every -separable space is set selectively star-ccc.

Example 3. (1)In Example 2, the space is set selectively star-ccc but not separable, and thus not -separable, since every -separable space is separable.(2)The space contains a dense countable subspace which is not -separable (see Theorem 50 in [13]]), but being countable is set selectively star-ccc.(3)The ordinal space is set selectively star-ccc but not separable, and thus not -separable.

Theorem 4. Let be a space which has a dense subset of isolated points. If, for any nonempty set and for each collection of open sets in such that , there is a countable set such that ; then, is a set selectively star-ccc space.

Proof. Let be any nonempty subset of , be any collection of open sets in such that , and be any sequence of maximal cellular open families in . It follows from hypothesis that there is a countable set such that . For each , is a maximal cellular open family in and is an isolated point of ; hence, there exists such that and thus . Therefore,which proves that is a set-selectively star-ccc space.

Corollary 3. If is a space with a countable dense subset of isolated points, then it is set selectively star-ccc.

The following theorem follows from the definition of set- spaces and from Theorem 4.

Theorem 5. Every set- space is set selectively star-ccc.

The converse of Theorem 5 is not true.

Example 4. Any Lindelöf space which is not countably compact is such an example. Such spaces are the Sorgenfrey line and the real line endowed with the open-minus-countable topology : belongs to if it is of the form , where is open in the usual metric topology on , and is a countable subset of .

Theorem 6. Every continuous open image of a set selectively star-ccc space is set selectively star-ccc.

Proof. Let be a set-selectively star-ccc space and be a continuous open mapping from onto . Let be any subset of , be a collection of open sets in such that , and be any sequence of maximal cellular open families in . Let . Then, is the collection of open sets in withand because is open, for every , is a maximal cellular open family in . Since is a set selectively star-ccc space, there is a sequence such that for each , and .
For each , let be such that . Now, we have to show that .
Let . There exists such that . Thus, . Choose such that and . Thus, for some . Let . Then, , which implies . Also, there exists a such that and . Thus, , . Since , . Therefore, is a set selectively star-ccc space.
Now, we discuss the nature of set selectively star-ccc property on subspaces. The following example shows that the set selectively star-ccc property is not preserved under open subspaces.

Example 5. There exists a Tychonoff set selectively star-ccc space having open subspace which is not set selectively star-ccc.
Let be the one-point Lindelöfication of the discrete space (Example 2). Then, is set selectively star-ccc space. However, its open (discrete) uncountable subspace is not set selectively star-ccc.
The following example shows that the set selectively star-ccc property is not preserved under closed subspaces.

Example 6. There exists a Tychonoff pseudocompact set selectively star-ccc space having closed subspace which is not set selectively star-ccc.

Proof. Let be the Isbell-Mrówka space with , where is a maximal almost disjoint family of infinite subsets of . Then, is a Tychonoff pseudocompact space. Since is a countable dense subset of containing isolated points, by Corollary 3, is set selectively star-ccc space. On the contrary, is a closed uncountable discrete subspace of ; thus, is not set-selectively star-ccc.
The following example shows that the set selectively star-ccc property is not preserved under regular closed subspaces.

Example 7. There exists a Tychonoff set selectively star-ccc space having regular closed subspace which is not set selectively star-ccc.

Proof. Let be the maximal almost disjoint family of infinite subsets of with . Defineand topologize as follows: has the usual product topology and is an open subspace of ; for , a basic neighbourhood is of the form , , a finite subset of . Let be the Isbell–Mrówka space.
Let be the quotient space of the disjoint sum by identifying the subspace of with the subspace of and let be the quotient map. Notice that is a regular closed subspace of .

Claim 1. is a set selectively star-ccc space.
Let be any nonempty subset of , be any collection of open sets in such that , and be any sequence of maximal cellular open families in . There are three cases: Case (i): . Note first that the Tychonoff space is set selectively star-ccc by Corollary 3. Since is a countable dense subset of isolated points of and is homeomorphic to , is also set selectively star-ccc. Thus, there are , , such that . Case (ii): . The space is countably compact; hence, set-, and, by Theorem 5, set selectively star-ccc. For each , is homeomorphic to and thus is set selectively star-ccc. It is easy to conclude that it follows from here that is also set selectively star-ccc. Therefore, there is a sequence such that, for each , and . Case (iii): , where and .Let be a partition of into two disjoint infinite subsets. By Case (i) and Case (ii), we have and . This implies that is a sequence witnessing that .
These three cases show that is set selectively star-ccc.

Claim 2. is not set selectively star-ccc.
Song and Xuan (Example 1 in [21]) proved that the Tychonoff space is not selectively star-ccc. Thus, is not set selectively star-ccc (since every set selectively star-ccc space is selectively star-ccc). Then, , which is homeomorphic to , is not set selectively star-ccc.
However, we have the following result on subspaces of set selectively star-ccc spaces.

Theorem 7. Every clopen subspace of a set selectively star-ccc space is also set selectively star-ccc.

Proof. Let be a set selectively star-ccc space and be an open and closed set. Let be any subset of , be any collection of open sets in such that , and be any sequence of maximal cellular open families in . Since is open, then is a collection of open sets in . For each set . Then, is a sequence of maximal cellular open families in . Since is closed, . Since is set-selectively star-ccc space, there is a sequence such that, for each , and . Set if , and  = an arbitrary element from if . Hence, , which shows that is a set selectively star-ccc space.
Observe that the set selectively star-ccc property is not finitely productive. The following example shows that the product of a set selectively star-ccc space and a Lindelöf (hence, set electively star-ccc) space is not set selectively star-ccc.

Example 8. The ordinal space (with the usual order topology) is set selectively star-ccc, and the one-point Lindelöfication of the discrete space is Lindelöf. Then, is not set selectively star-ccc because in Example 4 [22] it was shown that this product is not selectively star-ccc.
We now give a positive result about the set selectively star-ccc property in the product of topological spaces.

Example 9. The space is set selectively star-ccc.

Proof. Let be any nonempty subset of , be any collection of open sets in such that , and be any sequence of maximal cellular open families in . Consider a partition of into pairwise disjoint infinite subsets : . For each , is set selectively star-ccc because it is homeomorphic to , and is countably compact, hence set-, and thus, by Theorem 5, set selectively star-ccc. For each and for each , let and . Then, is a sequence of maximal cellular families of . Therefore, there is a sequence such that, for each , and ,which impliesSo, is set selectively star-ccc.

4. Set Selectively -Star-CCC Spaces

Definition 9. Let . A space is said to be set selectively -star-ccc if, for each nonempty subset of , for each collection of open sets in such that , and for each sequence of maximal cellular open families in , there is a sequence such that, for each , and .
The following lemma follows from the definitions.

Lemma 3. For a space , the following statements hold:(1)Every set selectively -star-ccc space is set selectively -star-ccc.(2)Every set selectively -star-ccc space is selectively -star-ccc.

Theorem 8. Every set strongly star-Lindelöf space is set selectively 2-star-ccc.

Proof. Let be any nonempty subset of a set strongly star-Lindelöf space , be any collection of open sets in such that , and be any sequence of maximal cellular open families in . Since is a set strongly star-Lindelöf space, there is a countable subset of such that . Then, is an open cover of . For each , there exists such that , that is, . For each , is dense in . Thus, for each , , which implies that, for each , there exists such that . Therefore, for each ,Hence, , which shows that is set selectively 2-star-ccc space.

Corollary 4. For a space , the following statements hold:(1)Every set strongly -star-Lindelöf space is set selectively -star-ccc(2)Every set strongly -starcompact space is set selectively -star-ccc

Lemma 4. If a space has a dense set selectively star-ccc subspace, then is set selectively 2-star-ccc.

Proof. The proof follows from the definitions.

Example 10. There exists a Tychonoff set selectively 2-star-ccc space which is neither set strongly star-Lindelöf nor set selectively star-ccc.

Proof. Let be the space from Example 7. Song and Xuan (Example 1 in [21]) showed that the is a Tychonoff space which is neither strongly star-Lindelöf nor selectively star-ccc. Thus, is neither set strongly star-Lindelöf nor set selectively star-ccc (since every set selectively star-ccc space is selectively star-ccc and every set strongly star-Lindelöf space is strongly star-Lindelöf).
Now, we prove that is a set selectively 2-star-ccc space. By Example 9, the space is set selectively star-ccc. On the other side, is a dense subset of . By Lemma 4, is set selectively 2-star-ccc.
Example 2 shows that there exists a Tychonoff set selectively 2-star-ccc space which is not ccc. Now, a natural question arises: is a ccc space set selectively 2-star-ccc? Song and Xuan [22] gave some sufficient conditions under which a ccc space is selectively 2-star-ccc. We use these results and give some conditions under which a ccc space is set selectively 2-star-ccc. Our results improve the corresponding results in [22].

Theorem 9. If is a ccc space which has a dense paracompact subspace , then is set selectively 2-star-ccc.

Proof. Since is a ccc space, thus has to be ccc. If we prove that is Lindelöf, then by Theorem 2 and Lemma 4, is set selectively 2-star-ccc. Since every paracompact space with countable extent is Lindelöf, so if is not Lindelöf, then must have an uncountable closed discrete subset . Using the collection-wise normality of , has an uncountable disjoint expansion, which contradicts the fact that is ccc. Thus, is Lindelöf, and hence is set selectively 2-star-ccc.

Corollary 5. If X is Čech-complete ccc space, then is set selectively 2-star-ccc.

Proof. By a well-known result of Šapirovskij [23], contains a dense paracompact Čech-complete subspace. Thus, by Theorem 9, is set selectively 2-star-ccc.

Corollary 6. If is ccc space which has a dense metrizable subspace, then is set selectively 2-star-ccc.

Proof. Since metrizable subspace is paracompact, thus by Theorem 9, is set selectively 2-star-ccc.

Theorem 10. If X is a ccc space which has a monotonically normal dense subspace (hence, a dense GO-space), then is selectively 2-star-ccc.

Proof. Since every ccc monotonically normal space is (hereditary) Lindelöf, rest of the proof is similar to the proof of Theorem 9.

5. Open Problems

We finish the paper by the following questions which we could not answer while working on this paper.

Problem 2. Does there exist a Tychonoff selectively star-ccc space which is not set selectively star-ccc? Do there exist similar examples for ?
Since every set selectively 2-star-ccc spaces is selectively 2-star-ccc, the following problem is an improved version of the Problem 4.9 in [11].

Problem 3. Are ccc spaces set selectively 2-star-ccc?

Problem 4. Does there exist in ZFC a normal set selectively 2-star-ccc space which is neither set strongly star-Lindelöf nor set selectively star-ccc?
Let us notice that, under assumption , there is such an example. It is the space , , , in (Example 2.2 in[24]). This space is set selectively 2-star-ccc because it contains the set selectively star-ccc space as a dense subspace. On the contrary, this space is not set strongly star-Lindelöf (because it is not star-Lindelöf), and it is not set selectively star-ccc as it was shown in (Example 3.4 in [21]).
In [25], Scheepers gave a game-theoretic characterization of selectively ccc spaces.

Problem 5. Do there exist game-theoretic characterizations of set selectively star-ccc and set selectively k-star-ccc spaces?
Song and Xuan (Theorem 3.6 in [21]) showed that an open -subset of selectively star-ccc space is selectively star-ccc.

Problem 6. Is open -subset of a set selectively star-ccc space also set selectively star-ccc?

6. Conclusion

Set-selective properties of topological spaces show how the subsets of a space are located in the space. We used this idea and the method of stars to study the set version of an important class of selectively star-ccc spaces. It is proved that the class of set selectively star-ccc spaces contains Lindelöf spaces, countably compact spaces of countable tightness, and Rothberger separable spaces. On the contrary, the class of set selectively star-ccc spaces is different from the class of Lindelöf spaces and some other classes of spaces which are set selectively star-ccc. A few open problems are posed to suggest a further research in this field. In particular, it would be interesting to investigate set selectively -star-ccc spaces, , defined by the iteration of the star operator.

Data Availability

The data used to support the findings of this study are cited at relevant places within the text as references and are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.