- Email: [email protected]
More on selectively star-ccc spaces
Topology and its Applications 268 (2019) 106905
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Topology and its Applications www.elsevier.com/locate/topol
More on selectively star-ccc spaces ✩ Yan-Kui Song a , Wei-Feng Xuan b,∗ a
Institute of Mathematics, School of Mathematical Science, Nanjing Normal University, Nanjing, 210046, China b School of statistics and Mathematics, Nanjing Audit University, Nanjing, 211815, China
a r t i c l e
i n f o
Article history: Received 26 June 2019 Received in revised form 28 September 2019 Accepted 29 September 2019 Available online 1 October 2019 MSC: primary 54D20 secondary 54E35 Keywords: Selectively star-ccc ccc Selectively 2-star-ccc Pseudocompact Strongly star-Lindelöf Regular closed Gδ -subspace
a b s t r a c t Bal and Kočinac in [2] introduced and studied the class of selectively star-ccc spaces. A space X is called selectively star-ccc if for every open cover U of X and for every sequence (An : n ∈ ω) of maximal pairwise disjoint open families in X there exists a sequence (An : n ∈ ω) such that An ∈ An for every n ∈ ω and St( n∈ω An , U ) = X. In this paper, we first provide some sufficient conditions for ccc spaces to be selectively 2-star-ccc, which partially answer Problem 4.4 of Bal and Kočinac [2]. We give a Tychonoff example of a pseudocompact selectively 2-star-ccc which is not strongly star-Lindelöf, which gives a positive answer to Problem 4.8 from [2] and Question 3.11 from [16]. We also show that a regular closed Gδ -subspace of a Tychonoff pseudocompact selectively star-ccc space may not be selectively star-ccc. We finally prove that the product of a selectively star-ccc space and a Lindelöf space may not be selectively star-ccc. © 2019 Elsevier B.V. All rights reserved.
1. Introduction In 1996, Scheepers [15] started a systematic study of classical selection principles, which attracted a huge number of mathematicians to this field. Every selection principle give rise of some particular type of topological spaces, such as Rothberger spaces [14], Menger spaces [12], and star-Menger spaces [5–7]. In this paper, we use the following selection principle of the Rotheberger type. Let A and B be families of sets. Then S1 (A, B) is the selection principle: for every sequence (An : n ∈ ω) of elements of A one can select bn ∈ An , n ∈ ω, so that {bn : n ∈ ω} ∈ B. If O is the family of all open covers of a space X, then S1 (O, O) is the Rothberger covering property. If A is a subset of a space X and U is a family of subsets of X, then the star of a subset A ⊂ X with respect to U is the set ✩
The first author is supported by NSFC project 11771029. The second author is supported by NSFC project 11801271.
* Corresponding author. E-mail addresses: [email protected] (Y.-K. Song), [email protected] (W.-F. Xuan). https://doi.org/10.1016/j.topol.2019.106905 0166-8641/© 2019 Elsevier B.V. All rights reserved.
2
Y.-K. Song, W.-F. Xuan / Topology and its Applications 268 (2019) 106905
St(A, U) =
{U ∈ U : U ∩ A = ∅}.
We also put St2 (A, U) = St(St(A, U), U). If A = {x} for some x ∈ X, then we write St(x, U) instead of St({x}, U) for simplicity. Kočinac applied this star operator in the field of selection principles and introduced and studied many selection principles using this operator (see [5–9]), which are termed as “star-selection principles theory” nowadays. Aurichi in [1] introduced and studied the class of selectively ccc spaces. Recall that a space X is a selectively ccc space if for every sequence (An : n ∈ ω) of maximal pairwise disjoint open families in X there is a sequence (An : n ∈ ω) such that An ∈ An for every n ∈ ω and n∈ω An is dense in X. Recently, based on the notion of Aurichi’s selectively ccc spaces, Bal and Kočinac [2] introduced a star version of selectively ccc spaces. Definition 1.1. ([2]) A space X is called selectively star-ccc if for every open cover U of Xand for every sequence (An : n ∈ ω) of maximal pairwise disjoint open families in X, there is a sequence (An : n ∈ ω) such that An ∈ An for every n ∈ ω and St( n∈ω An , U) = X. In [2], Bal and Kočinac discussed the relations between selectively star-ccc spaces and some other existing selective properties. Several examples were also given which makes selectively star-ccc spaces different from other spaces. In [16], Song and Xuan showed that there exists a Tychonoff selectively 2-star-ccc space which is neither strongly star Lindelöf nor selectively star-ccc, which gives a positive answer to a question of Bal and Kočinac [2]. Under 2ℵ0 = 2ℵ1 , Song and Xuan [16] even provide a normal example of a selectively 2-star-ccc space which is neither strongly star Lindelöf nor selectively star-ccc. It was also shown in the same paper that every open Fσ -subset of a selectively star-ccc space is selectively star-ccc. In this paper, we first provide some sufficient conditions for ccc spaces to be selectively 2-star-ccc, which partially answer Problem 4.4 of Bal and Kočinac [2]. We also give a Tychonoff example of a pseudocompact selectively 2-star-ccc which is not strongly star-Lindelöf, which gives a positive answer to Problem 4.8 from [2] and Question 3.11 from [16]. Moreover, we show that a regular closed Gδ -subspace of a Tychonoff pseudocompact selectively star-ccc space may not be selectively star-ccc. Finally, we prove that the product of a selectively star-ccc space and a Lindelöf space may not be selectively star-ccc. 2. Notation and terminology All topological spaces in this paper are assumed to be Hausdorff unless otherwise is stated. Throughout the paper, the cardinality of a set A is denoted by |A|. Let c denote the cardinality of the continuum, ω1 the first uncountable cardinal and ω the first infinite cardinal. For a pair of ordinals α, β with α < β, we write (α, β) = {γ : α < γ < β}, (α, β] = {γ : α < γ ≤ β} and [α, β] = {γ : α ≤ γ ≤ β}. As usual, a cardinal is an initial ordinal and an ordinal is the set of smaller ordinals. Every cardinal is often viewed as a space with the usual order topology. Definition 2.1. A space X satisfies the countable chain condition (abbreviated as ccc) if any disjoint family of open sets in X is countable, that is, the Souslin number (or cellularity) of X is at most ω. We note that the following property here called strongly star-Lindelöf, has different names in articles such as [10] and [18]. Definition 2.2. We say that a space X is strongly star-Lindelöf if for every open cover U of X, there exists a countable subset A of X such that St(A, U) = X.
Y.-K. Song, W.-F. Xuan / Topology and its Applications 268 (2019) 106905
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Definition 2.3. ([11]) A space X is absolutely countably compact if for every open cover U of X and every dense subset D of X, there exists a finite subset F of D such that St(F, U) = X. Clearly, a compact space is absolutely countably compact, and an absolutely countably compact Hausdorff space is countably compact. Definition 2.4. ([2]) A space X is called selectively 2-star-ccc if for every open cover U of Xand for every sequence (An : n ∈ ω) of maximal pairwise disjoint open families in X, there is a sequence (An : n ∈ ω) such that An ∈ An for every n ∈ ω and St2 ( n∈ω An , U) = X. All notations and terminology not explained in the paper are given in [3]. 3. General results and examples In this section, we begin with some easy observations which look interesting. Theorem 3.1. Let X be a space which has a dense subset Y of isolated points. If for any open cover U of X there is a countable set D ⊂ Y such that St(D, U) = X, then X is selectively star-ccc. Proof. Let U be an open cover of X and let (An : n ∈ ω) be a sequence of maximal pairwise disjoint open families in X. It follows from our hypothesis that there is a countable set D = {dn : n ∈ ω} ⊂ Y such that St(D, U) = X. For each n ∈ ω, since the disjoint family An is maximal and dn is an isolated point of X, there exists An ∈ An such that dn ∈ An , and thus St(dn , U) ⊂ St(An , U). One can easily verify that X = St(D, U) = St(
An , U).
n∈ω
This proves that X is selectively star-ccc.
2
The following result is an immediate consequence of Theorem 3.1. Corollary 3.2. If X is a space with a countable dense subset of isolated points, then X is selectively star-ccc. The next group of results give partial answers to Problem 4.4 of [2]. Theorem 3.3. If X is a Baire ccc space, then X is selectively 2-star-ccc. Proof. Let U be an open cover of X and let (An : n ∈ ω) be a sequence of maximal pairwise disjoint open families in X. Clearly, An is dense in X for each n ∈ ω. Since X is Baire, it follows that D = { An : n ∈ ω} is dense in X. Since X is ccc, D is also ccc. Thus, for the open cover U of Y , there is a countable subfamily W = {Un : n ∈ ω} ⊂ U such that D ⊂ W. Clearly, Un ∩ D = ∅ for each n ∈ ω. For each Un ∈ W, we may choose some dn ∈ D and An ∈ An such that dn ∈ Un and dn ∈ An . It follows that Un ⊂ St(An , U), and so D ⊂ W ⊂ St( n∈ω An , U). Since D is dense in X, the set St( n∈ω An , U) is also dense in X, which implies that St2 ( n∈ω An , U) = X. The proof is complete. 2 Theorem 3.4. If X is a ccc space which has a dense paracompact subspace, then X is selectively 2-star-ccc. Proof. Let Y be a dense paracompact subspace of X. Since X is ccc, the subspace Y has to be ccc. We claim that Y being ccc paracompact is Lindelöf. It is well known that every paracompact space with countable extent is Lindelöf, so if Y is not Lindelöf then Y must have an uncountable closed discrete subset D. Using
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the collectionwise normality of Y , D has an uncountable disjoint expansion, which contradicts the fact that Y is ccc. It was shown in [2] that every Lindelöf space is selectively star-ccc. Since Y is dense in X, we can deduce easily that X is selectively 2-star-ccc. The proof is complete. 2 Corollary 3.5. If X is a Čech-complete ccc space, then X is selectively 2-star-ccc. Proof. By a well known result of Šapirovskij [17], X contains a dense paracompact Čech-complete subspace. Theorem 3.4 does the rest. 2 Corollary 3.6. If X is a ccc space which has a dense metrizable subspace, then X is selectively 2-star-ccc. Proof. We only note that every metrizable space is paracompact. 2 Theorem 3.7. If X is a ccc space which has a dense monotonically normal subspace (hence, a dense GOspace), then X is selectively 2-star-ccc. Proof. The proof is similar to the one which was given in Theorem 3.4. We only note that every ccc monotonically normal space is (hereditarily) Lindelöf (see [4]). 2 The following example shows that a Tychonoff selectively 2-star-ccc pseudocompact space may not be strongly star Lindelöf, which answers the Problem 4.8 posed by Bal and Kočinac [2] and Question 3.11 by Song and Xuan [16]. We will use Alexandroff duplicate. Recall that A(X) is the Alexandroff duplicate of a space X if the underlying set of A(X) is X × {0, 1}; each point of X × {1} is isolated and a basic neighborhood of a point x, 0 ∈ X × {0} is of the form (U × {0}) ∪ ((U × {1}) \ {x, 1 }), where U is a neighborhood of x in X. It is evident that A(X) is countably compact if and only if X is countably compact. We also need the following lemma of Vaughan (see [19]). Lemma 3.8. If X is countably compact, then A(X) is absolutely countably compact, i.e., if for every open cover U of A(X) and every dense subset D of A(X), there exists a finite subset F of D such that St(F, U) = A(X). Example 3.9. There exists a Tychonoff pseudocompact selectively 2-star-ccc space which is not strongly star Lindelöf. Proof. Let D = {dα : α < c} be a discrete space of cardinality c and let βD be the Čech-stone compactification of D. Let Y = (βD × (c + 1)) \ ((βD \ D) × {c}) as a subspace of βD × (c + 1). Let X = A(Y ) \ ((D × {c}) × {1}). Clearly, the space X is a Tychonoff pseudocompact space, since A(βD × c) is a dense countably compact subspace of X. Next, we show that X is not strongly star Lindelöf. For each α < c, let Uα = A(({dα } × (c + 1)) \ {dα , c , 1 }.
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Then Uα is open in X and Uα ∩ Uα = ∅ whenever α = α . Let U = {Uα : α < c} ∪ {A(βD × c)}. Clearly, U is an open cover of X. Let B be a countable subset of X; then there exists β < c such that B ∩ Uβ = ∅, and thus dβ , c , 0 ∈ / St(B, U), since Uβ is the only element of U containing dβ , c , 0 and Uβ ∩ B = ∅, which shows that X is not strongly star Lindelöf. Finally, we show that X is selectively 2-star-ccc. We only need to show that A(βD × c) is selectively 1-star-ccc, since it is dense in X. To this end, let U be an open cover of X and (An : n ∈ ω) be a sequence of maximal pairwise disjoint open families in X. Let I be the set of all isolated points of c and let A = ((D × I) × {0}) ∪ {(βD × c) × {1}}. Then A be a dense subset of X and every point of A is isolated. Since βD×c is countably compact, A(βD×c) is absolutely countably compact by Lemma 3.8, there exists a finite subset F ⊂ A such that A(βD × c) ⊂ St(F, U). We can enumerate F as {xn : n ∈ ω}. For each n ∈ ω, since An is a maximal pairwise disjoint open family in X and xn is an isolated point of X, we may find some An ∈ An such that xn ∈ An , thus St(xn , U) ⊂ St(An , U). Therefore, we have A(βD × c) ⊂ St(F, U) ⊂ St(
An , U).
n∈ω
This shows that A(βD × c) is selectively star-ccc. The proof is complete.
2
Bal and Kočinac in [2] pointed out that selectively star-ccc is not hereditary with respect to closed sets. We now give a Tychonoff pseudocompact example showing that selectively star-ccc even is not hereditary with respect to regular closed Gδ -subsets. Example 3.10. There exists a Tychonoff pseudocompact selectively star-ccc space having a regular closed Gδ -subset that is not selectively star-ccc. Proof. Let S1 be the space X in Example 3.9. Then S1 is Tychonoff pseudocompact. Let us first show that S1 is not selectively star-ccc. For each α < c, let Uα = A({dα } × (c + 1)) \ {dα , c , 1 }. Then Uα is open in S1 and Uα ∩ Uα = ∅ for α = α . Let U = {Uα : α < c} ∪ {A(βD × c)}. Clearly, U is an open cover of S1 . Let I be the set of all isolated points of c and let A = ((D × I) × {0}) ∪ {(βD × c) × {1}}. Then A is a dense subset of S1 and every point of A is isolated. For each n ∈ ω, let An = {a} : a ∈ A ; then (An : n ∈ ω) is a sequence of maximal pairwise disjoint open families in S1 . Now let (An : n ∈ ω) be a sequence such that for every n ∈ N, An ∈ An . Let B = n∈ω An ; clearly, B is countable. Hence, there
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exists β < c such that B ∩ Uβ = ∅, thus dβ , c , 0 ∈ / St(B, U), since Uβ is the only element of U containing dβ , c , 0 and Uβ ∩ B = ∅, which shows that S1 is not selectively star-ccc. Let S2 = ω ∪ R be the Isbell-Mrówka space (see [13]), where R is a maximal almost disjoint family of infinite subsets of ω with |R| = c. It is well-known that S2 is Tychonoff pseudocompact. Since ω is a countable dense subset of isolated points of S2 , then S2 is selectively star-ccc by Corollary 3.2. We assume that S1 ∩ S2 = ∅. Since |R| = c, we can enumerate R as {rα : α < c}. Let Z be the quotient space of the discrete sum S1 ⊕ S2 by identifying dα , c , 0 with rα for each α < c. Now we show that the space Z is as required. It is evident that the space Z is Tychonoff pseudocompact by the construction of the topology of Z. Let ϕ : S1 ⊕ S2 → Z be the quotient map. Then ϕ(S1 ) is a regular closed subspace of Z. For each n ∈ ω, let Un = ϕ(S1 ∪ {m : m ≥ n}). Then Un is open in Z and ϕ(S1 ) = n∈ω Un . Thus ϕ(S1 ) is a regular-closed Gδ -subset of Z. However ϕ(S1 ) is not selectively star-ccc, since it is homeomorphic to S1 . Finally, we prove that Z is selectively star-ccc. Let U be an open cover of Z. Let E = ϕ(A ∪ ω) (note that the set A has been defined before); then E is a dense subset of isolated points of Z. Thus we only need to show that there exists a countable subset C of E such that St(C, U) = Z by Theorem 3.1. Since ϕ(S2 ) is homeomorphic to S2 , then π(S2 ) ⊂ St(ϕ(ω), U). On the other hand, since ϕ(A(βD × c)) is homeomorphic to A(βD × c) and βD × c is countably compact, thus ϕ(A(βD × c)) is absolutely countably compact by Lemma 3.8, we may find a finite subset F ⊂ ϕ(A) such that ϕ(A(βD × c)) ⊂ St(F, U). Let C = ϕ(ω) ∪ F . Then C is a countable subset of E such that St(C, U) = Z. The proof is complete.
2
Note that the product of a selectively star-ccc space and a Lindelöf space may not be selectively star-ccc, as can be seen in the next example. Example 3.11. There exists a selectively star-ccc space X and a Lindelöf space Y such that the product X × Y is not selectively star-ccc. Proof. Let X = ω1 with the usual order topology. Being countably compact, the space X is absolutely countably compact. Thus X is selectively star-ccc by Theorem 3.1, since the set of all the isolated points of X is dense. Let D = {dα : α < ω1 } be a discrete space of cardinality ω1 and let Y = D ∪ {d∗ } be one-point Lindelöfication of D , where d∗ ∈ / D . Now we show that X × Y is not selectively star-ccc. For each α < ω1 , let Uα = (α, ω1 ) × {dα }. Then Uα is open in X × Y and Uα ∩ Uα = ∅ for α = α . Let V = (X × Y ) \ (
Uα ).
α<ω1
It is not difficult to see that V is open in X × Y . Let U = {Uα : α < ω1 } ∪ {V }. Then U is an open cover of X × Y . Let S be the set of all isolated points of ω1 and let D = S × D . Then D is a dense subset of X × Y and every point of D is isolated. For each n ∈ ω, let An = {d} : d ∈ D . Then, one can easily see that (An : n ∈ ω) is a sequence of maximal pairwise disjoint open families in X × Y .
Y.-K. Song, W.-F. Xuan / Topology and its Applications 268 (2019) 106905
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Let (An : n ∈ ω) be a sequence such that for every n ∈ ω, An ∈ An . Let A = n∈ω An ; then A is countable. Thus f (A) is a countable subset of X, where f : X × Y → X is the projection. Therefore, there exists β < ω1 such that f (A) ∩ Uβ = ∅. Pick some α < ω1 such that α > β. Then α, dβ ∈ / St(A, U), since Uβ is the only element of U containing α, dβ . Therefore, X × Y is not selectively star-ccc. The proof is complete. 2 We finish the paper by the following questions which still are open. Question 3.12. ([2]) Does ccc property of a topological space implies selectively 2-star-ccc property? Question 3.13. ([16]) Is there a normal example in ZFC of a selectively 2-star-ccc space which is not strongly star Lindelöf? Acknowledgements We would like to thank the referee for his (or her) valuable remarks and suggestions which improved the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
L.F. Aurichi, Selectively c.c.c. spaces, Topol. Appl. 160 (2013) 2243–2250. P. Bal, Lj.D.R. Kočinac, On selectively star-ccc spaces, Topology Appl., in press. R. Engelking, General Topology, revised and completed edition, Heldermann Verlag, Berlin, 1989. P.M. Gartside, Cardinal invariants of monotonically normal spaces, Topol. Appl. 77 (3) (1997) 303–314. Lj.D.R. Kočinac, Star-Menger and related spaces, Publ. Math. (Debr.) 55 (3–4) (1999) 421–431. Lj.D.R. Kočinac, Star-Menger and related spaces II, Filomat 13 (1999) 129–140. Lj.D.R. Kočinac, Star selection principles: a survey, Khayyam J. Math. 1 (2015) 82–106. Lj.D.R. Kočinac, Selected results on selection principles, in: Proceedings of the Third Seminar on Geometry and Topology, Tabriz, Iran, July 15–17, 2004, pp. 71–104. Lj.D.R. Kočinac, M. Scheepers, Combinatorics of open covers (VII): groupability, Fundam. Math. 179 (2003) 131–155. M.V. Matveev, A survey on star covering properties, Topology Atlas, preprint, April 15, 1998. M.V. Matveev, On absolutely countably compact spaces, Topol. Appl. 58 (1994) 81–91. K. Menger, Einige Überdeckungssätze der Punltmengenlehre, Sitzungsber. Abt. 2a, Math. Astron. Phys. Meteorol. Mech. (Wiener Akademie, Wien) 133 (1924) 421–444. S. Mrowka, On completely regular spaces, Fundam. Math. 41 (1) (1955) 105–106. F. Rothberger, Eine Vershärfung der Eigenschaft C, Fundam. Math. 30 (1938) 50–55. M. Scheepers, The combinatorics of open covers (I): Ramsey theory, Topol. Appl. 69 (1996) 31–62. Yan-Kui Song, Wei-Feng Xuan, A note on selectively star-ccc spaces, Topol. Appl. 263 (2019) 343–349. B.E. Šapirovskij, On separability and metrizability of spaces with Souslin condition, Sov. Math. Dokl. 13 (6) (1972) 1633–1638. J. van Mill, V.V. Tkachuk, R.G. Wilson, Classes defined by stars and neighbourhood assignments, Topol. Appl. 154 (2007) 2127–2134. J.E. Vaughan, On the product of a compact space with an absolutely countably compact space, Ann. N.Y. Acad. Sci. 788 (1996) 203–208.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
More on selectively star-ccc spaces ✩ Yan-Kui Song a , Wei-Feng Xuan b,∗ a
Institute of Mathematics, School of Mathematical Science, Nanjing Normal University, Nanjing, 210046, China b School of statistics and Mathematics, Nanjing Audit University, Nanjing, 211815, China
a r t i c l e
i n f o
Article history: Received 26 June 2019 Received in revised form 28 September 2019 Accepted 29 September 2019 Available online 1 October 2019 MSC: primary 54D20 secondary 54E35 Keywords: Selectively star-ccc ccc Selectively 2-star-ccc Pseudocompact Strongly star-Lindelöf Regular closed Gδ -subspace
a b s t r a c t Bal and Kočinac in [2] introduced and studied the class of selectively star-ccc spaces. A space X is called selectively star-ccc if for every open cover U of X and for every sequence (An : n ∈ ω) of maximal pairwise disjoint open families in X there exists a sequence (An : n ∈ ω) such that An ∈ An for every n ∈ ω and St( n∈ω An , U ) = X. In this paper, we first provide some sufficient conditions for ccc spaces to be selectively 2-star-ccc, which partially answer Problem 4.4 of Bal and Kočinac [2]. We give a Tychonoff example of a pseudocompact selectively 2-star-ccc which is not strongly star-Lindelöf, which gives a positive answer to Problem 4.8 from [2] and Question 3.11 from [16]. We also show that a regular closed Gδ -subspace of a Tychonoff pseudocompact selectively star-ccc space may not be selectively star-ccc. We finally prove that the product of a selectively star-ccc space and a Lindelöf space may not be selectively star-ccc. © 2019 Elsevier B.V. All rights reserved.
1. Introduction In 1996, Scheepers [15] started a systematic study of classical selection principles, which attracted a huge number of mathematicians to this field. Every selection principle give rise of some particular type of topological spaces, such as Rothberger spaces [14], Menger spaces [12], and star-Menger spaces [5–7]. In this paper, we use the following selection principle of the Rotheberger type. Let A and B be families of sets. Then S1 (A, B) is the selection principle: for every sequence (An : n ∈ ω) of elements of A one can select bn ∈ An , n ∈ ω, so that {bn : n ∈ ω} ∈ B. If O is the family of all open covers of a space X, then S1 (O, O) is the Rothberger covering property. If A is a subset of a space X and U is a family of subsets of X, then the star of a subset A ⊂ X with respect to U is the set ✩
The first author is supported by NSFC project 11771029. The second author is supported by NSFC project 11801271.
* Corresponding author. E-mail addresses: [email protected] (Y.-K. Song), [email protected] (W.-F. Xuan). https://doi.org/10.1016/j.topol.2019.106905 0166-8641/© 2019 Elsevier B.V. All rights reserved.
2
Y.-K. Song, W.-F. Xuan / Topology and its Applications 268 (2019) 106905
St(A, U) =
{U ∈ U : U ∩ A = ∅}.
We also put St2 (A, U) = St(St(A, U), U). If A = {x} for some x ∈ X, then we write St(x, U) instead of St({x}, U) for simplicity. Kočinac applied this star operator in the field of selection principles and introduced and studied many selection principles using this operator (see [5–9]), which are termed as “star-selection principles theory” nowadays. Aurichi in [1] introduced and studied the class of selectively ccc spaces. Recall that a space X is a selectively ccc space if for every sequence (An : n ∈ ω) of maximal pairwise disjoint open families in X there is a sequence (An : n ∈ ω) such that An ∈ An for every n ∈ ω and n∈ω An is dense in X. Recently, based on the notion of Aurichi’s selectively ccc spaces, Bal and Kočinac [2] introduced a star version of selectively ccc spaces. Definition 1.1. ([2]) A space X is called selectively star-ccc if for every open cover U of Xand for every sequence (An : n ∈ ω) of maximal pairwise disjoint open families in X, there is a sequence (An : n ∈ ω) such that An ∈ An for every n ∈ ω and St( n∈ω An , U) = X. In [2], Bal and Kočinac discussed the relations between selectively star-ccc spaces and some other existing selective properties. Several examples were also given which makes selectively star-ccc spaces different from other spaces. In [16], Song and Xuan showed that there exists a Tychonoff selectively 2-star-ccc space which is neither strongly star Lindelöf nor selectively star-ccc, which gives a positive answer to a question of Bal and Kočinac [2]. Under 2ℵ0 = 2ℵ1 , Song and Xuan [16] even provide a normal example of a selectively 2-star-ccc space which is neither strongly star Lindelöf nor selectively star-ccc. It was also shown in the same paper that every open Fσ -subset of a selectively star-ccc space is selectively star-ccc. In this paper, we first provide some sufficient conditions for ccc spaces to be selectively 2-star-ccc, which partially answer Problem 4.4 of Bal and Kočinac [2]. We also give a Tychonoff example of a pseudocompact selectively 2-star-ccc which is not strongly star-Lindelöf, which gives a positive answer to Problem 4.8 from [2] and Question 3.11 from [16]. Moreover, we show that a regular closed Gδ -subspace of a Tychonoff pseudocompact selectively star-ccc space may not be selectively star-ccc. Finally, we prove that the product of a selectively star-ccc space and a Lindelöf space may not be selectively star-ccc. 2. Notation and terminology All topological spaces in this paper are assumed to be Hausdorff unless otherwise is stated. Throughout the paper, the cardinality of a set A is denoted by |A|. Let c denote the cardinality of the continuum, ω1 the first uncountable cardinal and ω the first infinite cardinal. For a pair of ordinals α, β with α < β, we write (α, β) = {γ : α < γ < β}, (α, β] = {γ : α < γ ≤ β} and [α, β] = {γ : α ≤ γ ≤ β}. As usual, a cardinal is an initial ordinal and an ordinal is the set of smaller ordinals. Every cardinal is often viewed as a space with the usual order topology. Definition 2.1. A space X satisfies the countable chain condition (abbreviated as ccc) if any disjoint family of open sets in X is countable, that is, the Souslin number (or cellularity) of X is at most ω. We note that the following property here called strongly star-Lindelöf, has different names in articles such as [10] and [18]. Definition 2.2. We say that a space X is strongly star-Lindelöf if for every open cover U of X, there exists a countable subset A of X such that St(A, U) = X.
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Definition 2.3. ([11]) A space X is absolutely countably compact if for every open cover U of X and every dense subset D of X, there exists a finite subset F of D such that St(F, U) = X. Clearly, a compact space is absolutely countably compact, and an absolutely countably compact Hausdorff space is countably compact. Definition 2.4. ([2]) A space X is called selectively 2-star-ccc if for every open cover U of Xand for every sequence (An : n ∈ ω) of maximal pairwise disjoint open families in X, there is a sequence (An : n ∈ ω) such that An ∈ An for every n ∈ ω and St2 ( n∈ω An , U) = X. All notations and terminology not explained in the paper are given in [3]. 3. General results and examples In this section, we begin with some easy observations which look interesting. Theorem 3.1. Let X be a space which has a dense subset Y of isolated points. If for any open cover U of X there is a countable set D ⊂ Y such that St(D, U) = X, then X is selectively star-ccc. Proof. Let U be an open cover of X and let (An : n ∈ ω) be a sequence of maximal pairwise disjoint open families in X. It follows from our hypothesis that there is a countable set D = {dn : n ∈ ω} ⊂ Y such that St(D, U) = X. For each n ∈ ω, since the disjoint family An is maximal and dn is an isolated point of X, there exists An ∈ An such that dn ∈ An , and thus St(dn , U) ⊂ St(An , U). One can easily verify that X = St(D, U) = St(
An , U).
n∈ω
This proves that X is selectively star-ccc.
2
The following result is an immediate consequence of Theorem 3.1. Corollary 3.2. If X is a space with a countable dense subset of isolated points, then X is selectively star-ccc. The next group of results give partial answers to Problem 4.4 of [2]. Theorem 3.3. If X is a Baire ccc space, then X is selectively 2-star-ccc. Proof. Let U be an open cover of X and let (An : n ∈ ω) be a sequence of maximal pairwise disjoint open families in X. Clearly, An is dense in X for each n ∈ ω. Since X is Baire, it follows that D = { An : n ∈ ω} is dense in X. Since X is ccc, D is also ccc. Thus, for the open cover U of Y , there is a countable subfamily W = {Un : n ∈ ω} ⊂ U such that D ⊂ W. Clearly, Un ∩ D = ∅ for each n ∈ ω. For each Un ∈ W, we may choose some dn ∈ D and An ∈ An such that dn ∈ Un and dn ∈ An . It follows that Un ⊂ St(An , U), and so D ⊂ W ⊂ St( n∈ω An , U). Since D is dense in X, the set St( n∈ω An , U) is also dense in X, which implies that St2 ( n∈ω An , U) = X. The proof is complete. 2 Theorem 3.4. If X is a ccc space which has a dense paracompact subspace, then X is selectively 2-star-ccc. Proof. Let Y be a dense paracompact subspace of X. Since X is ccc, the subspace Y has to be ccc. We claim that Y being ccc paracompact is Lindelöf. It is well known that every paracompact space with countable extent is Lindelöf, so if Y is not Lindelöf then Y must have an uncountable closed discrete subset D. Using
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the collectionwise normality of Y , D has an uncountable disjoint expansion, which contradicts the fact that Y is ccc. It was shown in [2] that every Lindelöf space is selectively star-ccc. Since Y is dense in X, we can deduce easily that X is selectively 2-star-ccc. The proof is complete. 2 Corollary 3.5. If X is a Čech-complete ccc space, then X is selectively 2-star-ccc. Proof. By a well known result of Šapirovskij [17], X contains a dense paracompact Čech-complete subspace. Theorem 3.4 does the rest. 2 Corollary 3.6. If X is a ccc space which has a dense metrizable subspace, then X is selectively 2-star-ccc. Proof. We only note that every metrizable space is paracompact. 2 Theorem 3.7. If X is a ccc space which has a dense monotonically normal subspace (hence, a dense GOspace), then X is selectively 2-star-ccc. Proof. The proof is similar to the one which was given in Theorem 3.4. We only note that every ccc monotonically normal space is (hereditarily) Lindelöf (see [4]). 2 The following example shows that a Tychonoff selectively 2-star-ccc pseudocompact space may not be strongly star Lindelöf, which answers the Problem 4.8 posed by Bal and Kočinac [2] and Question 3.11 by Song and Xuan [16]. We will use Alexandroff duplicate. Recall that A(X) is the Alexandroff duplicate of a space X if the underlying set of A(X) is X × {0, 1}; each point of X × {1} is isolated and a basic neighborhood of a point x, 0 ∈ X × {0} is of the form (U × {0}) ∪ ((U × {1}) \ {x, 1 }), where U is a neighborhood of x in X. It is evident that A(X) is countably compact if and only if X is countably compact. We also need the following lemma of Vaughan (see [19]). Lemma 3.8. If X is countably compact, then A(X) is absolutely countably compact, i.e., if for every open cover U of A(X) and every dense subset D of A(X), there exists a finite subset F of D such that St(F, U) = A(X). Example 3.9. There exists a Tychonoff pseudocompact selectively 2-star-ccc space which is not strongly star Lindelöf. Proof. Let D = {dα : α < c} be a discrete space of cardinality c and let βD be the Čech-stone compactification of D. Let Y = (βD × (c + 1)) \ ((βD \ D) × {c}) as a subspace of βD × (c + 1). Let X = A(Y ) \ ((D × {c}) × {1}). Clearly, the space X is a Tychonoff pseudocompact space, since A(βD × c) is a dense countably compact subspace of X. Next, we show that X is not strongly star Lindelöf. For each α < c, let Uα = A(({dα } × (c + 1)) \ {dα , c , 1 }.
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Then Uα is open in X and Uα ∩ Uα = ∅ whenever α = α . Let U = {Uα : α < c} ∪ {A(βD × c)}. Clearly, U is an open cover of X. Let B be a countable subset of X; then there exists β < c such that B ∩ Uβ = ∅, and thus dβ , c , 0 ∈ / St(B, U), since Uβ is the only element of U containing dβ , c , 0 and Uβ ∩ B = ∅, which shows that X is not strongly star Lindelöf. Finally, we show that X is selectively 2-star-ccc. We only need to show that A(βD × c) is selectively 1-star-ccc, since it is dense in X. To this end, let U be an open cover of X and (An : n ∈ ω) be a sequence of maximal pairwise disjoint open families in X. Let I be the set of all isolated points of c and let A = ((D × I) × {0}) ∪ {(βD × c) × {1}}. Then A be a dense subset of X and every point of A is isolated. Since βD×c is countably compact, A(βD×c) is absolutely countably compact by Lemma 3.8, there exists a finite subset F ⊂ A such that A(βD × c) ⊂ St(F, U). We can enumerate F as {xn : n ∈ ω}. For each n ∈ ω, since An is a maximal pairwise disjoint open family in X and xn is an isolated point of X, we may find some An ∈ An such that xn ∈ An , thus St(xn , U) ⊂ St(An , U). Therefore, we have A(βD × c) ⊂ St(F, U) ⊂ St(
An , U).
n∈ω
This shows that A(βD × c) is selectively star-ccc. The proof is complete.
2
Bal and Kočinac in [2] pointed out that selectively star-ccc is not hereditary with respect to closed sets. We now give a Tychonoff pseudocompact example showing that selectively star-ccc even is not hereditary with respect to regular closed Gδ -subsets. Example 3.10. There exists a Tychonoff pseudocompact selectively star-ccc space having a regular closed Gδ -subset that is not selectively star-ccc. Proof. Let S1 be the space X in Example 3.9. Then S1 is Tychonoff pseudocompact. Let us first show that S1 is not selectively star-ccc. For each α < c, let Uα = A({dα } × (c + 1)) \ {dα , c , 1 }. Then Uα is open in S1 and Uα ∩ Uα = ∅ for α = α . Let U = {Uα : α < c} ∪ {A(βD × c)}. Clearly, U is an open cover of S1 . Let I be the set of all isolated points of c and let A = ((D × I) × {0}) ∪ {(βD × c) × {1}}. Then A is a dense subset of S1 and every point of A is isolated. For each n ∈ ω, let An = {a} : a ∈ A ; then (An : n ∈ ω) is a sequence of maximal pairwise disjoint open families in S1 . Now let (An : n ∈ ω) be a sequence such that for every n ∈ N, An ∈ An . Let B = n∈ω An ; clearly, B is countable. Hence, there
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exists β < c such that B ∩ Uβ = ∅, thus dβ , c , 0 ∈ / St(B, U), since Uβ is the only element of U containing dβ , c , 0 and Uβ ∩ B = ∅, which shows that S1 is not selectively star-ccc. Let S2 = ω ∪ R be the Isbell-Mrówka space (see [13]), where R is a maximal almost disjoint family of infinite subsets of ω with |R| = c. It is well-known that S2 is Tychonoff pseudocompact. Since ω is a countable dense subset of isolated points of S2 , then S2 is selectively star-ccc by Corollary 3.2. We assume that S1 ∩ S2 = ∅. Since |R| = c, we can enumerate R as {rα : α < c}. Let Z be the quotient space of the discrete sum S1 ⊕ S2 by identifying dα , c , 0 with rα for each α < c. Now we show that the space Z is as required. It is evident that the space Z is Tychonoff pseudocompact by the construction of the topology of Z. Let ϕ : S1 ⊕ S2 → Z be the quotient map. Then ϕ(S1 ) is a regular closed subspace of Z. For each n ∈ ω, let Un = ϕ(S1 ∪ {m : m ≥ n}). Then Un is open in Z and ϕ(S1 ) = n∈ω Un . Thus ϕ(S1 ) is a regular-closed Gδ -subset of Z. However ϕ(S1 ) is not selectively star-ccc, since it is homeomorphic to S1 . Finally, we prove that Z is selectively star-ccc. Let U be an open cover of Z. Let E = ϕ(A ∪ ω) (note that the set A has been defined before); then E is a dense subset of isolated points of Z. Thus we only need to show that there exists a countable subset C of E such that St(C, U) = Z by Theorem 3.1. Since ϕ(S2 ) is homeomorphic to S2 , then π(S2 ) ⊂ St(ϕ(ω), U). On the other hand, since ϕ(A(βD × c)) is homeomorphic to A(βD × c) and βD × c is countably compact, thus ϕ(A(βD × c)) is absolutely countably compact by Lemma 3.8, we may find a finite subset F ⊂ ϕ(A) such that ϕ(A(βD × c)) ⊂ St(F, U). Let C = ϕ(ω) ∪ F . Then C is a countable subset of E such that St(C, U) = Z. The proof is complete.
2
Note that the product of a selectively star-ccc space and a Lindelöf space may not be selectively star-ccc, as can be seen in the next example. Example 3.11. There exists a selectively star-ccc space X and a Lindelöf space Y such that the product X × Y is not selectively star-ccc. Proof. Let X = ω1 with the usual order topology. Being countably compact, the space X is absolutely countably compact. Thus X is selectively star-ccc by Theorem 3.1, since the set of all the isolated points of X is dense. Let D = {dα : α < ω1 } be a discrete space of cardinality ω1 and let Y = D ∪ {d∗ } be one-point Lindelöfication of D , where d∗ ∈ / D . Now we show that X × Y is not selectively star-ccc. For each α < ω1 , let Uα = (α, ω1 ) × {dα }. Then Uα is open in X × Y and Uα ∩ Uα = ∅ for α = α . Let V = (X × Y ) \ (
Uα ).
α<ω1
It is not difficult to see that V is open in X × Y . Let U = {Uα : α < ω1 } ∪ {V }. Then U is an open cover of X × Y . Let S be the set of all isolated points of ω1 and let D = S × D . Then D is a dense subset of X × Y and every point of D is isolated. For each n ∈ ω, let An = {d} : d ∈ D . Then, one can easily see that (An : n ∈ ω) is a sequence of maximal pairwise disjoint open families in X × Y .
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Let (An : n ∈ ω) be a sequence such that for every n ∈ ω, An ∈ An . Let A = n∈ω An ; then A is countable. Thus f (A) is a countable subset of X, where f : X × Y → X is the projection. Therefore, there exists β < ω1 such that f (A) ∩ Uβ = ∅. Pick some α < ω1 such that α > β. Then α, dβ ∈ / St(A, U), since Uβ is the only element of U containing α, dβ . Therefore, X × Y is not selectively star-ccc. The proof is complete. 2 We finish the paper by the following questions which still are open. Question 3.12. ([2]) Does ccc property of a topological space implies selectively 2-star-ccc property? Question 3.13. ([16]) Is there a normal example in ZFC of a selectively 2-star-ccc space which is not strongly star Lindelöf? Acknowledgements We would like to thank the referee for his (or her) valuable remarks and suggestions which improved the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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