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A study of selectively star-ccc spaces
Topology and its Applications 273 (2020) 107103
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Topology and its Applications www.elsevier.com/locate/topol
A study of selectively star-ccc spaces ✩ Wei-Feng Xuan a,∗ , Yan-Kui Song b a
School of statistics and Mathematics, Nanjing Audit University, Nanjing, 211815, China Institute of Mathematics, School of Mathematical Science, Nanjing Normal University, Nanjing, 210046, China b
a r t i c l e
i n f o
Article history: Received 20 October 2019 Received in revised form 28 January 2020 Accepted 4 February 2020 Available online 6 February 2020 MSC: primary 54D20 secondary 54E35 Keywords: Selectively star-ccc CCC DCCC Normal Regular closed Gδ -subspace Monotonically normal Extent First countable Gδ -diagonal Rank 2-diagonal Cardinal
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 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 prove that: (1) A selectively star-ccc space is DCCC and a selectively star-ccc perfect space is CCC. (2) There exists a pseudocompact (hence, DCCC) space that is not selectively starccc. (3) Every selectively star-ccc subspace of the product of finitely many scattered monotonically normal spaces has countable extent. (4) Every selectively star-ccc subspace of ω1ω has countable extent. (5) Under 2ℵ0 = 2ℵ1 , there exists a selectively star-ccc normal space having a regular closed Gδ -subset which is not selectively star-ccc. (6) Every first countable selectively star-ccc space with a Gδ -diagonal has cardinality at most c. (7) Every selectively star-ccc space with a rank 2-diagonal has cardinality at most c. © 2020 Elsevier B.V. All rights reserved.
1. Introduction In 1996, Scheepers [16] started a systematic study of classical selection principles, which attracted quite a few mathematicians to this field. Every selection principle give rise of some particular type of topological spaces, such as Rothberger spaces [15], Menger spaces [14], and star-Menger spaces [8–10]. In this paper, we use the following selection principle of the Rotheberger type. Let A and B be families of sets. Then the selection principle S1 (A, B) says that for any sequence (An : n ∈ ω) of elements of A one can select ✩
The first author is supported by NSFC project 11801271. The second author is supported by NSFC project 11771029.
* Corresponding author. E-mail addresses: [email protected] (W.-F. Xuan), [email protected] (Y.-K. Song). https://doi.org/10.1016/j.topol.2020.107103 0166-8641/© 2020 Elsevier B.V. All rights reserved.
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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 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 to introduce several selection principles (see [8–12]); nowadays, they are called “star-selection principles”. 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, Bal and Kočinac introduced in [2] selectively star ccc spaces which constitute a more general class than selectively ccc spaces. Definition 1.1. ([2]) A space X is called selectively star-ccc if for every open cover U of X and 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. Answering a question of Bal and Kočinac, Song and Xuan in [17] and [18] proved that there exists a Tychonoff selectively 2-star-ccc space which is neither strongly star Lindelöf nor selectively star-ccc and gave some sufficient conditions for CCC spaces to be selectively 2-star-ccc. In this paper, we prove that: (1) A selectively star-ccc space is DCCC (Theorem 3.6) and a selectively star-ccc perfect space is CCC (Theorem 3.10). (2) There exists a pseudocompact (hence, DCCC) space that is not selectively star-ccc (Example 3.7). (3) Every selectively star-ccc subspace of the product of finitely many scattered monotonically normal spaces has countable extent (Theorem 4.4). (4) Every selectively star-ccc subspace of ω1ω has countable extent (Theorem 4.6). (5) Under 2ℵ0 = 2ℵ1 , there exists a selectively star-ccc normal space having a regular closed Gδ -subset which is not selectively star-ccc (Example 5.1). (6) Every first countable selectively star-ccc space with a Gδ -diagonal has cardinality at most c (Theorem 6.4). (7) Every selectively star-ccc space with a rank 2-diagonal has cardinality at most c (Theorem 6.5). 2. Notation and terminology All topological spaces in this paper are assumed to be Hausdorff unless otherwise is stated. Given a space X, the family τ (X) is its topology and τ (x, X) = {U ∈ τ (X) : x ∈ U } for any x ∈ X. 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. We say that a space X satisfies the discrete countable chain condition (or X is DCCC for short) if every discrete family of non-empty open subsets of X is countable.
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Definition 2.2. A space X is weakly Lindelöf if every open cover of X has a countable subfamily whose union is dense in X. Definition 2.3. A space X satisfies the countable chain condition (or X is CCC for short) if any disjoint family of non-empty open sets in X is countable, that is, the Souslin number (or cellularity) of X is at most ω. Definition 2.4. A space X has a Gδ -diagonal [22] if there exists a sequence {Un : n ∈ ω} of open covers of X such that for each x ∈ X, {x} = {St(x, Un ) : n ∈ ω}. Definition 2.5. A space X has a rank 2-diagonal if there exists a sequence {Un : n ∈ ω} of open covers of X such that for each x ∈ X, {x} = {St2 (x, Un ) : n ∈ ω}. Definition 2.6. A space X is called perfect if every closed subset of X is a Gδ -set. Definition 2.7. We say that a space X has a development if there is a sequence of open covers {Un : n ∈ ω} of X such that for every x ∈ X, the sequence {St(x, Un ) : n ∈ ω} is a base at x. A Moore space is a regular space which has a development. Definition 2.8. The extent of a space X, denoted by e(X), is the supremum of the cardinalities of closed discrete subsets of X. Definition 2.9. ([13]) 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. Definition 2.10. A space X is weakly ω1 -collectionwise Hausdorff if for any closed discrete set D ⊂ X with |D| = ω1 there exists a set E ⊂ D such that |E| = ω1 and E has a disjoint open expansion. All notations and terminology not explained in the paper are given in [4]. 3. Selectively star-ccc and chain conditions In this section, we mainly study the relations between star-ccc selectivity and chain conditions. Theorem 3.1. If X is a Baire space such that for any open cover U of X and any dense subset Y 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. Clearly, An is dense in X for each n ∈ ω. Since X is Baire, Y = { An : n ∈ ω} is dense in X. By our hypothesis, there is a countable set D = {dn : n ∈ ω} of Y such that St(D, U) = X. Now for each n ∈ ω, let An ∈ An such that dn ∈ An . Since D ⊂ {An : n ∈ ω}, St( n∈ω An , U) = X. This shows that X is selectively star-ccc and completes the proof. 2 The following Proposition 3.2 will be often used in the sequel. If X is a space and A ⊂ X, we say that the family U is an open expansion of A if U = {Ua : a ∈ A} and Ua ∈ τ (a, X) for any a ∈ A. Proposition 3.2. If D is an uncountable closed discrete set of a space X which has a disjoint open expansion, then X is not selectively star-ccc.
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Proof. Assume the contrary. Let D = {dα : α < ω1 } ⊂ X be a closed discrete set of cardinality ω1 and let U = {Uα : α < ω1 } be a disjoint open expansion of D. Obviously, O = U ∪ (X \ D) is an open cover of X. Let A = U ∪ (X \ U). It is evident that A is a maximal pairwise disjoint family of open sets of X. Let An = A for every n ∈ ω. By star-ccc selectivity of X, there is a sequence (An : n ∈ ω) such that An ∈ An for every n ∈ ω and St( n∈ω An , O) = X. Since n∈ω An intersects only countably many elements of U, there exists some Uα ∈ U such that Uα ∩ n∈ω An = ∅. Since Uα is the unique element of O containing dα , it follows that dα ∈ / St( n∈ω An , O) = X which leads to a contradiction. This shows that X is not selectively star-ccc and completes the proof. 2 The following results are immediate corollaries of Proposition 3.2. Corollary 3.3. If X is a weakly ω1 -collectionwise Hausdorff selectively star-ccc space, then e(X) ≤ ω. Corollary 3.4. If X is a collectionwise normal selectively star-ccc space, then e(X) ≤ ω. Corollary 3.5. Every paracompact selectively star-ccc space X is Lindelöf. Proof. Since a paracompact space is collectionwise normal, e(X) ≤ ω by Corollary 3.4. Using the fact that every paracompact space of countable extent is Lindelöf, we can conclude that the space X is Lindelöf. 2 Theorem 3.6. If X is a selectively star-ccc space, then X is DCCC. Proof. Assume the contrary. We may find an uncountable discrete (and hence, disjoint) family U = {Uα : α < ω1 } of non-empty open sets of X. For each α < ω1 , pick dα ∈ Uα arbitrarily and let D = {dα : α < ω1 }; clearly, D is an uncountable closed discrete subset of X. By Proposition 3.2, we can deduce that X is not selectively star-ccc which is a contradiction. This proves that X is DCCC. 2 The following example shows that the converse of Theorem 3.6 is not true. Example 3.7. There is a Tychonoff DCCC space which is not selectively star-ccc. Proof. Let I(ω1 ) denote the set of isolated ordinals in ω1 . Let X = (ω1 + 1) × ω1 ∪ I(ω1 ) × {ω1 } . It is obvious that X is Tychonoff. Using the observation that (ω1 + 1) × ω1 is a dense countably compact subspace of X, one can easily prove that X is DCCC (in fact, X is pseudocompact). Now we show that X is not selectively star-ccc. For each n ∈ ω, let An = {x} : x ∈ I(ω1 ) × I(ω1 ) . Clearly, each An is a maximal pairwise disjoint family of open sets of X. We define an open cover U of X as follows: U = (ω1 + 1) × ω1 ∪ {γ} × (ω1 + 1) : γ ∈ I(ω1 ) . Let (An : n ∈ ω) be a sequence such that An ∈ An for every n ∈ ω; then there exists γ0 ∈ I(ω1 ) such that
An ∩ {γ0 } × (ω1 + 1) = ∅.
An is countable. Obviously,
W.-F. Xuan, Y.-K. Song / Topology and its Applications 273 (2020) 107103
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Since {γ0 } × (ω1 + 1) is the unique element of U containing γ0 , ω1 , it follows that γ0 , ω1 ∈ / St( An , U) showing that X is not selectively star-ccc. The proof is complete. 2 Lemma 3.8. ([18]) 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. Example 3.9. There is a selectively star-ccc space which is not weakly Lindelöf. Proof. Since the space ω1 is absolutely countably compact (see [20]) and has a dense subset of isolated points of ω1 , it follows that ω1 is selectively star-ccc (see Lemma 3.8). But the open cover U = {[0, γ] : γ ∈ ω1 } of ω1 witnesses that ω1 is not weakly Lindelöf, since U does not have any countable subfamily whose union is dense in ω1 . 2 But for perfect spaces, we have the following result. Theorem 3.10. If X is a perfect selectively star-ccc space, then X is CCC. Proof. Assume the contrary. We may find an uncountable pairwise disjoint family U = {Uα : α < ω1 } of non-empty open sets of X. For each α < ω1 , pick dα ∈ Uα arbitrarily and let D = {dα : α < ω1 }; clearly, D is an uncountable discrete subset of X. Since X is perfect, there is an uncountable set E ⊂ D which is closed discrete in X by Proposition 3.4 of [21]. We may assume, without loss of generality, that D is closed discrete in X. By Proposition 3.2, we can deduce that X is not selectively star-ccc which leads to a contradiction. This proves that X is CCC. 2 Corollary 3.11. Every selectively star-ccc space that is a union of countably many closed discrete subsets is CCC. Proof. We note that every space that is a union of countably many closed discrete subset is perfect. 2 Corollary 3.12. Every selectively star-ccc semi-stratifiable space is CCC. Proof. This follows from the fact that every semi-stratifiable space is perfect. 2 Corollary 3.13. Every selectively star-ccc Moore space is CCC. Proof. Simply use the fact that every Moore space is semi-stratifiable. 2 4. On the products of monotonically normal spaces In this section, we obtain several results on the products of monotonically normal spaces. Let us first recall the definition of monotone normality. Definition 4.1. ([7]) A space X is called monotonically normal if it admits an operator H (called the monotone normality operator) that assigns to any point x ∈ X and any U ∈ τ (x, X) a set H(x, U ) ∈ τ (x, X) such that H(x, U ) ⊂ U and for any points x, y ∈ X and sets U, V ∈ τ (X) such that x ∈ U and y ∈ V , it follows from H(x, U ) ∩ H(y, V ) = ∅ that x ∈ V or y ∈ U . Proposition 4.2. If X is a monotonically normal selectively star-ccc space, then e(X) ≤ ω.
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Proof. Since a monotonically normal space is (hereditarily) collectionwise normal, Corollary 3.4 does the rest. 2 Example 3.9 shows that we cannot replace the extent with the Lindelöf number in Proposition 4.2, since ω1 is monotonically normal. But we have Theorem 4.3. If X is a stratifiable selectively star-ccc space, then X is Lindelöf. Proof. Note that a space is stratifiable if and only if it is monotonically normal and semi-stratifiable ([6, Theorem 2.5]). Thus, the space X has countable extent. It was shown in [3] that every semi-stratifiable space of countable extent is Lindelöf, so we can deduce that X is Lindelöf. 2 Theorem 4.4. If a subspace X ⊂ Y = {Yi : i ≤ n} is selectively star-ccc, where Yi is a scattered monotonically normal space for any i = 0, 1, ..., n, then e(X) ≤ ω. Proof. Suppose not. For each i ≤ n let fi : Y → Yi be the projection onto the i-th factor. There exists an uncountable closed discrete subset D of X of cardinality ω1 . There exists some k ≤ n such that |fk (D)| = ω1 ; denote by I the set of isolated points of fk (D). Since the space Yk is scattered, I is dense in fk (D). It is routine to see that I is uncountable. Since Yk is hereditary collectionwise normal, there exists an open disjoint expansion {U (x) : x ∈ I} of the set I. Choose a point dx ∈ D such that fk (dx ) = x for every x ∈ I. It follows that U = {fk−1 (U (x)) : x ∈ I} is an open disjoint expansion of uncountable subset E = {dx : x ∈ I} of the set D. Being as a subset of D the set E is closed discrete in X. By Proposition 3.2, we can deduce that X is not selectively star-ccc which leads to a contradiction. This shows that X has countable extent and completes the proof. 2 Corollary 4.5. If X is a selectively star-ccc subspace of λn for some ordinal λ and n ∈ ω, then e(X) ≤ ω. Theorem 4.6. If a subspace X ⊂ ω1ω is selectively star-ccc, then e(X) ≤ ω. Proof. Suppose not. For each n ∈ ω let fn : ω1ω → ω1 be the projection onto the n-th factor. There exists an uncountable closed discrete subset D of X of cardinality ω1 . If fn (D) is countable for all n ∈ ω, then D ⊂ n∈ω fn (D) is second countable which is not possible, since every second countable space has countable extent. Thus, there exists n ∈ ω such that |fn (D)| = ω1 . If I is the set of isolated points of fn (D), then I has to be uncountable. Since ω1 is hereditary collectionwise normal, there exists an open disjoint expansion {U (x) : x ∈ I} of the set I. Choose a point dx ∈ D such that fn (dx ) = x for every x ∈ I. It follows that U = {fn−1 (U (x)) : x ∈ I} is an open disjoint expansion of uncountable subset E = {dx : x ∈ I} of the set D. Obviously, the set E is closed discrete in X. By Proposition 3.2, we can deduce that X is not selectively star-ccc which leads to a contradiction. This shows that X has countable extent and completes the proof. 2 5. On regular closed Gδ -subspaces Song and Xuan in [18] showed that there exists a selectively star-ccc pseudocompact space having a regular closed Gδ -subset which is not selectively star-ccc. But note that this example is only Tychonoff. In this section, we give a normal example. Example 5.1. Under 2ω0 = 2ω1 , there exists a selectively star-ccc normal space having a regular closed Gδ -subset which is not selectively star-ccc.
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Proof. Let Z = L ∪ ω be a separable normal, uncountable T1 -space, where L is a closed and discrete subset Z of cardinality equal to ω1 and each element of ω is isolated (see [19, Example E]). Let Y = L ∪ (ω1 × ω) and topologize Y as follows: ω1 × ω has the usual product topology and is an open subspace of Y . On the other hand, a basic neighborhood of l ∈ L in Y is a set of the form GU,α (l) = (U ∩ L) ∪ ((α, ω1 ) × (U ∩ ω)) for a neighborhood U of l in Z and α < ω1 . Then Y is normal, and it was shown in [17, Example 3.4] that Y is not selectively star-ccc. Consider the discrete sum Y ⊕ Z and denote by LY and LZ the copies of L in the subspaces of Y ⊕ Z that correspond to Y and Z respectively. Let f : LY → LZ be the respective copy of the identity map and denote by X the quotient space of Y ⊕ Z obtained by identifying l and f (l) for any l ∈ LY . Now we show that the space X is as required. It is evident that the space X is normal by the construction of the topology of X. Let ϕ : Y ⊕ Z → X be the quotient map. Then ϕ(Y ) is a regular closed subspace of X. For each n ∈ ω, let Un = ϕ(Y ∪ {m : m ≥ n}). Then Un is open in X and ϕ(Y ) = n∈ω Un . Thus ϕ(Y ) is a regular closed Gδ -subset of X. However ϕ(Y ) is not selectively star-ccc, since it is homeomorphic to Y . Finally, we only need to show that X is selectively star-ccc. Let U be an open cover of X. Let I be the set of all the isolated points of ω1 and let E = ϕ((I × ω) ∪ ω). Then E is a dense subset of isolated points of X. Thus it remains to show that there exists a countable subset C of E such that St(C, U) = X by Lemma 3.8. Since ϕ(Z) is homeomorphic to Z, we have the inclusion ϕ(Z) ⊂ St(ϕ(ω), U). On the other hand, the set ϕ(ω1 × {n}) is absolutely countably compact being homeomorphic to ω1 so we can find a finite set Fn ⊂ ϕ(I × {n}) such that ϕ(ω1 × {n}) ⊂ St(Fn , U). Let C = ϕ(ω) ∪ (
Fn ).
n∈ω
Therefore, we obtain a countable subset C of E such that St(C, U) = X. This shows that X is selectively star-ccc and completes the proof. 2 6. Cardinal inequalities Finally, we establish some cardinal inequalities for selectively star-ccc spaces. Proposition 6.1. If X is a first countable, selectively star-ccc perfect space, then |X| ≤ c.
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Proof. It follows from Theorem 3.10 that X is CCC. Since the cardinality of every first countable CCC space is at most c, we have the conclusion. 2 Corollary 6.2. If X is a selectively star-ccc Moore space, then |X| ≤ c. Proof. We only note that a Moore space is first countable and perfect. 2 We will use the following countable version of a set-theoretic theorem due to Erdös and Radó. Lemma 6.3. ([5, p.8]) Let X be a set with |X| > 2ω and suppose that [X]2 = exist n < ω and a subset S of X with |S| > ω such that [S]2 ⊂ Pn .
{Pn : n ∈ ω}. Then there
Theorem 6.4. If X is a first countable selectively star-ccc space with a Gδ -diagonal, then X has cardinality at most c. Proof. Assume that |X| > c. For each x ∈ X, let B(x) = {Bn (x) : n ∈ ω} be a local decreasing base at the point x. Let {Un : n ∈ ω} be a sequence of open covers of X such that {x} = {St(x, Un ) : n ∈ ω} for every x ∈ X. For every n ∈ ω, we define a subset Pn of [X]2 by Pn = {x, y} ∈ [X]2 : Bn (x) ∩ Bn (y) = ∅ and y ∈ / St(x, Un ) . Clearly the sets Pn are well-defined. It is evident that [X]2 = {Pn : n ∈ ω}. Therefore, Lemma 6.3 implies that there exist a subset S of X with |S| > ω and k ∈ ω such that [S]2 ⊂ Pk . Let x ∈ X be a limit point of S. Since X is T1 , each neighborhood U ∈ Uk of x meets infinitely many members of S. Therefore, there exist distinct y, z ∈ S ∩ U . Thus y ∈ U ⊂ St(z, Uk ) which contradicts the fact that {y, z} ∈ Pk . We conclude that S is closed discrete in X. It is evident that U = {Bk (x) : x ∈ S} is a disjoint open expansion of S. By Proposition 3.2, we can deduce that X is not selectively star-ccc which leads to a contradiction. This shows that |X| ≤ c and completes the proof. 2 Theorem 6.5. If X is selectively star-ccc space with a rank 2-diagonal, then X has cardinality at most c. Proof. Assume that |X| > c. Let {Un : n ∈ ω} be a sequence of open covers of X such that {x} = 2 {St (x, Un ) : n ∈ ω} for every x ∈ X. For every n ∈ ω, we define a subset Pn of [X]2 by Pn = {x, y} ∈ [X]2 : y ∈ / St2 (x, Un ) . Clearly the sets Pn are well-defined. It is evident that [X]2 = {Pn : n ∈ ω}. Therefore, Lemma 6.3 implies that there exist a subset S of X with |S| > ω and k ∈ ω such that [S]2 ⊂ Pk . In a manner similar to that used in the proof of Theorem 6.4, we conclude that S is closed discrete in X. Note that U = {St(x, Uk ) : x ∈ S} is a disjoint open expansion of S. We can deduce that X is not selectively star-ccc By Proposition 3.2 which leads to a contradiction. This shows that |X| ≤ c and completes the proof. 2 7. Open problems We finish the paper by the following questions which we could not answer while working on this paper. Recall that a space X is weakly perfect if every closed subset of X contains a dense set which is a Gδ -set in X.
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Question 7.1. Is a weakly perfect selectively star-ccc space CCC? Question 7.2. Is a weakly perfect selectively star-ccc space weakly Lindelöf? Question 7.3. Is there a countably compact monotonically normal space which is not selectively star-ccc? Question 7.4. Does every selectively star-ccc subspace of the product of finitely many monotonically normal spaces have countable extent? Question 7.5. Is there in ZFC a selectively star-ccc normal space having a regular closed Gδ -subset which is not selectively star-ccc? Recall that a space X has a regular Gδ -diagonal if there is a countable family {Un : n ∈ ω} of open neighborhoods of the diagonal ΔX in X 2 such that ΔX = {Un : n ∈ ω}, where ΔX = {(x, x) : x ∈ X}. Question 7.6. Is it true that a selectively star-ccc space with a regular Gδ -diagonal has cardinality at most c? Question 7.7. Is it true that a selectively star-ccc normal space with a regular Gδ -diagonal has cardinality at most c? Acknowledgement We would like to thank the referee for his (or her) valuable remarks and suggestions which greatly improved the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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Topology and its Applications www.elsevier.com/locate/topol
A study of selectively star-ccc spaces ✩ Wei-Feng Xuan a,∗ , Yan-Kui Song b a
School of statistics and Mathematics, Nanjing Audit University, Nanjing, 211815, China Institute of Mathematics, School of Mathematical Science, Nanjing Normal University, Nanjing, 210046, China b
a r t i c l e
i n f o
Article history: Received 20 October 2019 Received in revised form 28 January 2020 Accepted 4 February 2020 Available online 6 February 2020 MSC: primary 54D20 secondary 54E35 Keywords: Selectively star-ccc CCC DCCC Normal Regular closed Gδ -subspace Monotonically normal Extent First countable Gδ -diagonal Rank 2-diagonal Cardinal
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 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 prove that: (1) A selectively star-ccc space is DCCC and a selectively star-ccc perfect space is CCC. (2) There exists a pseudocompact (hence, DCCC) space that is not selectively starccc. (3) Every selectively star-ccc subspace of the product of finitely many scattered monotonically normal spaces has countable extent. (4) Every selectively star-ccc subspace of ω1ω has countable extent. (5) Under 2ℵ0 = 2ℵ1 , there exists a selectively star-ccc normal space having a regular closed Gδ -subset which is not selectively star-ccc. (6) Every first countable selectively star-ccc space with a Gδ -diagonal has cardinality at most c. (7) Every selectively star-ccc space with a rank 2-diagonal has cardinality at most c. © 2020 Elsevier B.V. All rights reserved.
1. Introduction In 1996, Scheepers [16] started a systematic study of classical selection principles, which attracted quite a few mathematicians to this field. Every selection principle give rise of some particular type of topological spaces, such as Rothberger spaces [15], Menger spaces [14], and star-Menger spaces [8–10]. In this paper, we use the following selection principle of the Rotheberger type. Let A and B be families of sets. Then the selection principle S1 (A, B) says that for any sequence (An : n ∈ ω) of elements of A one can select ✩
The first author is supported by NSFC project 11801271. The second author is supported by NSFC project 11771029.
* Corresponding author. E-mail addresses: [email protected] (W.-F. Xuan), [email protected] (Y.-K. Song). https://doi.org/10.1016/j.topol.2020.107103 0166-8641/© 2020 Elsevier B.V. All rights reserved.
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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 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 to introduce several selection principles (see [8–12]); nowadays, they are called “star-selection principles”. 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, Bal and Kočinac introduced in [2] selectively star ccc spaces which constitute a more general class than selectively ccc spaces. Definition 1.1. ([2]) A space X is called selectively star-ccc if for every open cover U of X and 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. Answering a question of Bal and Kočinac, Song and Xuan in [17] and [18] proved that there exists a Tychonoff selectively 2-star-ccc space which is neither strongly star Lindelöf nor selectively star-ccc and gave some sufficient conditions for CCC spaces to be selectively 2-star-ccc. In this paper, we prove that: (1) A selectively star-ccc space is DCCC (Theorem 3.6) and a selectively star-ccc perfect space is CCC (Theorem 3.10). (2) There exists a pseudocompact (hence, DCCC) space that is not selectively star-ccc (Example 3.7). (3) Every selectively star-ccc subspace of the product of finitely many scattered monotonically normal spaces has countable extent (Theorem 4.4). (4) Every selectively star-ccc subspace of ω1ω has countable extent (Theorem 4.6). (5) Under 2ℵ0 = 2ℵ1 , there exists a selectively star-ccc normal space having a regular closed Gδ -subset which is not selectively star-ccc (Example 5.1). (6) Every first countable selectively star-ccc space with a Gδ -diagonal has cardinality at most c (Theorem 6.4). (7) Every selectively star-ccc space with a rank 2-diagonal has cardinality at most c (Theorem 6.5). 2. Notation and terminology All topological spaces in this paper are assumed to be Hausdorff unless otherwise is stated. Given a space X, the family τ (X) is its topology and τ (x, X) = {U ∈ τ (X) : x ∈ U } for any x ∈ X. 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. We say that a space X satisfies the discrete countable chain condition (or X is DCCC for short) if every discrete family of non-empty open subsets of X is countable.
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Definition 2.2. A space X is weakly Lindelöf if every open cover of X has a countable subfamily whose union is dense in X. Definition 2.3. A space X satisfies the countable chain condition (or X is CCC for short) if any disjoint family of non-empty open sets in X is countable, that is, the Souslin number (or cellularity) of X is at most ω. Definition 2.4. A space X has a Gδ -diagonal [22] if there exists a sequence {Un : n ∈ ω} of open covers of X such that for each x ∈ X, {x} = {St(x, Un ) : n ∈ ω}. Definition 2.5. A space X has a rank 2-diagonal if there exists a sequence {Un : n ∈ ω} of open covers of X such that for each x ∈ X, {x} = {St2 (x, Un ) : n ∈ ω}. Definition 2.6. A space X is called perfect if every closed subset of X is a Gδ -set. Definition 2.7. We say that a space X has a development if there is a sequence of open covers {Un : n ∈ ω} of X such that for every x ∈ X, the sequence {St(x, Un ) : n ∈ ω} is a base at x. A Moore space is a regular space which has a development. Definition 2.8. The extent of a space X, denoted by e(X), is the supremum of the cardinalities of closed discrete subsets of X. Definition 2.9. ([13]) 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. Definition 2.10. A space X is weakly ω1 -collectionwise Hausdorff if for any closed discrete set D ⊂ X with |D| = ω1 there exists a set E ⊂ D such that |E| = ω1 and E has a disjoint open expansion. All notations and terminology not explained in the paper are given in [4]. 3. Selectively star-ccc and chain conditions In this section, we mainly study the relations between star-ccc selectivity and chain conditions. Theorem 3.1. If X is a Baire space such that for any open cover U of X and any dense subset Y 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. Clearly, An is dense in X for each n ∈ ω. Since X is Baire, Y = { An : n ∈ ω} is dense in X. By our hypothesis, there is a countable set D = {dn : n ∈ ω} of Y such that St(D, U) = X. Now for each n ∈ ω, let An ∈ An such that dn ∈ An . Since D ⊂ {An : n ∈ ω}, St( n∈ω An , U) = X. This shows that X is selectively star-ccc and completes the proof. 2 The following Proposition 3.2 will be often used in the sequel. If X is a space and A ⊂ X, we say that the family U is an open expansion of A if U = {Ua : a ∈ A} and Ua ∈ τ (a, X) for any a ∈ A. Proposition 3.2. If D is an uncountable closed discrete set of a space X which has a disjoint open expansion, then X is not selectively star-ccc.
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Proof. Assume the contrary. Let D = {dα : α < ω1 } ⊂ X be a closed discrete set of cardinality ω1 and let U = {Uα : α < ω1 } be a disjoint open expansion of D. Obviously, O = U ∪ (X \ D) is an open cover of X. Let A = U ∪ (X \ U). It is evident that A is a maximal pairwise disjoint family of open sets of X. Let An = A for every n ∈ ω. By star-ccc selectivity of X, there is a sequence (An : n ∈ ω) such that An ∈ An for every n ∈ ω and St( n∈ω An , O) = X. Since n∈ω An intersects only countably many elements of U, there exists some Uα ∈ U such that Uα ∩ n∈ω An = ∅. Since Uα is the unique element of O containing dα , it follows that dα ∈ / St( n∈ω An , O) = X which leads to a contradiction. This shows that X is not selectively star-ccc and completes the proof. 2 The following results are immediate corollaries of Proposition 3.2. Corollary 3.3. If X is a weakly ω1 -collectionwise Hausdorff selectively star-ccc space, then e(X) ≤ ω. Corollary 3.4. If X is a collectionwise normal selectively star-ccc space, then e(X) ≤ ω. Corollary 3.5. Every paracompact selectively star-ccc space X is Lindelöf. Proof. Since a paracompact space is collectionwise normal, e(X) ≤ ω by Corollary 3.4. Using the fact that every paracompact space of countable extent is Lindelöf, we can conclude that the space X is Lindelöf. 2 Theorem 3.6. If X is a selectively star-ccc space, then X is DCCC. Proof. Assume the contrary. We may find an uncountable discrete (and hence, disjoint) family U = {Uα : α < ω1 } of non-empty open sets of X. For each α < ω1 , pick dα ∈ Uα arbitrarily and let D = {dα : α < ω1 }; clearly, D is an uncountable closed discrete subset of X. By Proposition 3.2, we can deduce that X is not selectively star-ccc which is a contradiction. This proves that X is DCCC. 2 The following example shows that the converse of Theorem 3.6 is not true. Example 3.7. There is a Tychonoff DCCC space which is not selectively star-ccc. Proof. Let I(ω1 ) denote the set of isolated ordinals in ω1 . Let X = (ω1 + 1) × ω1 ∪ I(ω1 ) × {ω1 } . It is obvious that X is Tychonoff. Using the observation that (ω1 + 1) × ω1 is a dense countably compact subspace of X, one can easily prove that X is DCCC (in fact, X is pseudocompact). Now we show that X is not selectively star-ccc. For each n ∈ ω, let An = {x} : x ∈ I(ω1 ) × I(ω1 ) . Clearly, each An is a maximal pairwise disjoint family of open sets of X. We define an open cover U of X as follows: U = (ω1 + 1) × ω1 ∪ {γ} × (ω1 + 1) : γ ∈ I(ω1 ) . Let (An : n ∈ ω) be a sequence such that An ∈ An for every n ∈ ω; then there exists γ0 ∈ I(ω1 ) such that
An ∩ {γ0 } × (ω1 + 1) = ∅.
An is countable. Obviously,
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Since {γ0 } × (ω1 + 1) is the unique element of U containing γ0 , ω1 , it follows that γ0 , ω1 ∈ / St( An , U) showing that X is not selectively star-ccc. The proof is complete. 2 Lemma 3.8. ([18]) 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. Example 3.9. There is a selectively star-ccc space which is not weakly Lindelöf. Proof. Since the space ω1 is absolutely countably compact (see [20]) and has a dense subset of isolated points of ω1 , it follows that ω1 is selectively star-ccc (see Lemma 3.8). But the open cover U = {[0, γ] : γ ∈ ω1 } of ω1 witnesses that ω1 is not weakly Lindelöf, since U does not have any countable subfamily whose union is dense in ω1 . 2 But for perfect spaces, we have the following result. Theorem 3.10. If X is a perfect selectively star-ccc space, then X is CCC. Proof. Assume the contrary. We may find an uncountable pairwise disjoint family U = {Uα : α < ω1 } of non-empty open sets of X. For each α < ω1 , pick dα ∈ Uα arbitrarily and let D = {dα : α < ω1 }; clearly, D is an uncountable discrete subset of X. Since X is perfect, there is an uncountable set E ⊂ D which is closed discrete in X by Proposition 3.4 of [21]. We may assume, without loss of generality, that D is closed discrete in X. By Proposition 3.2, we can deduce that X is not selectively star-ccc which leads to a contradiction. This proves that X is CCC. 2 Corollary 3.11. Every selectively star-ccc space that is a union of countably many closed discrete subsets is CCC. Proof. We note that every space that is a union of countably many closed discrete subset is perfect. 2 Corollary 3.12. Every selectively star-ccc semi-stratifiable space is CCC. Proof. This follows from the fact that every semi-stratifiable space is perfect. 2 Corollary 3.13. Every selectively star-ccc Moore space is CCC. Proof. Simply use the fact that every Moore space is semi-stratifiable. 2 4. On the products of monotonically normal spaces In this section, we obtain several results on the products of monotonically normal spaces. Let us first recall the definition of monotone normality. Definition 4.1. ([7]) A space X is called monotonically normal if it admits an operator H (called the monotone normality operator) that assigns to any point x ∈ X and any U ∈ τ (x, X) a set H(x, U ) ∈ τ (x, X) such that H(x, U ) ⊂ U and for any points x, y ∈ X and sets U, V ∈ τ (X) such that x ∈ U and y ∈ V , it follows from H(x, U ) ∩ H(y, V ) = ∅ that x ∈ V or y ∈ U . Proposition 4.2. If X is a monotonically normal selectively star-ccc space, then e(X) ≤ ω.
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Proof. Since a monotonically normal space is (hereditarily) collectionwise normal, Corollary 3.4 does the rest. 2 Example 3.9 shows that we cannot replace the extent with the Lindelöf number in Proposition 4.2, since ω1 is monotonically normal. But we have Theorem 4.3. If X is a stratifiable selectively star-ccc space, then X is Lindelöf. Proof. Note that a space is stratifiable if and only if it is monotonically normal and semi-stratifiable ([6, Theorem 2.5]). Thus, the space X has countable extent. It was shown in [3] that every semi-stratifiable space of countable extent is Lindelöf, so we can deduce that X is Lindelöf. 2 Theorem 4.4. If a subspace X ⊂ Y = {Yi : i ≤ n} is selectively star-ccc, where Yi is a scattered monotonically normal space for any i = 0, 1, ..., n, then e(X) ≤ ω. Proof. Suppose not. For each i ≤ n let fi : Y → Yi be the projection onto the i-th factor. There exists an uncountable closed discrete subset D of X of cardinality ω1 . There exists some k ≤ n such that |fk (D)| = ω1 ; denote by I the set of isolated points of fk (D). Since the space Yk is scattered, I is dense in fk (D). It is routine to see that I is uncountable. Since Yk is hereditary collectionwise normal, there exists an open disjoint expansion {U (x) : x ∈ I} of the set I. Choose a point dx ∈ D such that fk (dx ) = x for every x ∈ I. It follows that U = {fk−1 (U (x)) : x ∈ I} is an open disjoint expansion of uncountable subset E = {dx : x ∈ I} of the set D. Being as a subset of D the set E is closed discrete in X. By Proposition 3.2, we can deduce that X is not selectively star-ccc which leads to a contradiction. This shows that X has countable extent and completes the proof. 2 Corollary 4.5. If X is a selectively star-ccc subspace of λn for some ordinal λ and n ∈ ω, then e(X) ≤ ω. Theorem 4.6. If a subspace X ⊂ ω1ω is selectively star-ccc, then e(X) ≤ ω. Proof. Suppose not. For each n ∈ ω let fn : ω1ω → ω1 be the projection onto the n-th factor. There exists an uncountable closed discrete subset D of X of cardinality ω1 . If fn (D) is countable for all n ∈ ω, then D ⊂ n∈ω fn (D) is second countable which is not possible, since every second countable space has countable extent. Thus, there exists n ∈ ω such that |fn (D)| = ω1 . If I is the set of isolated points of fn (D), then I has to be uncountable. Since ω1 is hereditary collectionwise normal, there exists an open disjoint expansion {U (x) : x ∈ I} of the set I. Choose a point dx ∈ D such that fn (dx ) = x for every x ∈ I. It follows that U = {fn−1 (U (x)) : x ∈ I} is an open disjoint expansion of uncountable subset E = {dx : x ∈ I} of the set D. Obviously, the set E is closed discrete in X. By Proposition 3.2, we can deduce that X is not selectively star-ccc which leads to a contradiction. This shows that X has countable extent and completes the proof. 2 5. On regular closed Gδ -subspaces Song and Xuan in [18] showed that there exists a selectively star-ccc pseudocompact space having a regular closed Gδ -subset which is not selectively star-ccc. But note that this example is only Tychonoff. In this section, we give a normal example. Example 5.1. Under 2ω0 = 2ω1 , there exists a selectively star-ccc normal space having a regular closed Gδ -subset which is not selectively star-ccc.
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Proof. Let Z = L ∪ ω be a separable normal, uncountable T1 -space, where L is a closed and discrete subset Z of cardinality equal to ω1 and each element of ω is isolated (see [19, Example E]). Let Y = L ∪ (ω1 × ω) and topologize Y as follows: ω1 × ω has the usual product topology and is an open subspace of Y . On the other hand, a basic neighborhood of l ∈ L in Y is a set of the form GU,α (l) = (U ∩ L) ∪ ((α, ω1 ) × (U ∩ ω)) for a neighborhood U of l in Z and α < ω1 . Then Y is normal, and it was shown in [17, Example 3.4] that Y is not selectively star-ccc. Consider the discrete sum Y ⊕ Z and denote by LY and LZ the copies of L in the subspaces of Y ⊕ Z that correspond to Y and Z respectively. Let f : LY → LZ be the respective copy of the identity map and denote by X the quotient space of Y ⊕ Z obtained by identifying l and f (l) for any l ∈ LY . Now we show that the space X is as required. It is evident that the space X is normal by the construction of the topology of X. Let ϕ : Y ⊕ Z → X be the quotient map. Then ϕ(Y ) is a regular closed subspace of X. For each n ∈ ω, let Un = ϕ(Y ∪ {m : m ≥ n}). Then Un is open in X and ϕ(Y ) = n∈ω Un . Thus ϕ(Y ) is a regular closed Gδ -subset of X. However ϕ(Y ) is not selectively star-ccc, since it is homeomorphic to Y . Finally, we only need to show that X is selectively star-ccc. Let U be an open cover of X. Let I be the set of all the isolated points of ω1 and let E = ϕ((I × ω) ∪ ω). Then E is a dense subset of isolated points of X. Thus it remains to show that there exists a countable subset C of E such that St(C, U) = X by Lemma 3.8. Since ϕ(Z) is homeomorphic to Z, we have the inclusion ϕ(Z) ⊂ St(ϕ(ω), U). On the other hand, the set ϕ(ω1 × {n}) is absolutely countably compact being homeomorphic to ω1 so we can find a finite set Fn ⊂ ϕ(I × {n}) such that ϕ(ω1 × {n}) ⊂ St(Fn , U). Let C = ϕ(ω) ∪ (
Fn ).
n∈ω
Therefore, we obtain a countable subset C of E such that St(C, U) = X. This shows that X is selectively star-ccc and completes the proof. 2 6. Cardinal inequalities Finally, we establish some cardinal inequalities for selectively star-ccc spaces. Proposition 6.1. If X is a first countable, selectively star-ccc perfect space, then |X| ≤ c.
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Proof. It follows from Theorem 3.10 that X is CCC. Since the cardinality of every first countable CCC space is at most c, we have the conclusion. 2 Corollary 6.2. If X is a selectively star-ccc Moore space, then |X| ≤ c. Proof. We only note that a Moore space is first countable and perfect. 2 We will use the following countable version of a set-theoretic theorem due to Erdös and Radó. Lemma 6.3. ([5, p.8]) Let X be a set with |X| > 2ω and suppose that [X]2 = exist n < ω and a subset S of X with |S| > ω such that [S]2 ⊂ Pn .
{Pn : n ∈ ω}. Then there
Theorem 6.4. If X is a first countable selectively star-ccc space with a Gδ -diagonal, then X has cardinality at most c. Proof. Assume that |X| > c. For each x ∈ X, let B(x) = {Bn (x) : n ∈ ω} be a local decreasing base at the point x. Let {Un : n ∈ ω} be a sequence of open covers of X such that {x} = {St(x, Un ) : n ∈ ω} for every x ∈ X. For every n ∈ ω, we define a subset Pn of [X]2 by Pn = {x, y} ∈ [X]2 : Bn (x) ∩ Bn (y) = ∅ and y ∈ / St(x, Un ) . Clearly the sets Pn are well-defined. It is evident that [X]2 = {Pn : n ∈ ω}. Therefore, Lemma 6.3 implies that there exist a subset S of X with |S| > ω and k ∈ ω such that [S]2 ⊂ Pk . Let x ∈ X be a limit point of S. Since X is T1 , each neighborhood U ∈ Uk of x meets infinitely many members of S. Therefore, there exist distinct y, z ∈ S ∩ U . Thus y ∈ U ⊂ St(z, Uk ) which contradicts the fact that {y, z} ∈ Pk . We conclude that S is closed discrete in X. It is evident that U = {Bk (x) : x ∈ S} is a disjoint open expansion of S. By Proposition 3.2, we can deduce that X is not selectively star-ccc which leads to a contradiction. This shows that |X| ≤ c and completes the proof. 2 Theorem 6.5. If X is selectively star-ccc space with a rank 2-diagonal, then X has cardinality at most c. Proof. Assume that |X| > c. Let {Un : n ∈ ω} be a sequence of open covers of X such that {x} = 2 {St (x, Un ) : n ∈ ω} for every x ∈ X. For every n ∈ ω, we define a subset Pn of [X]2 by Pn = {x, y} ∈ [X]2 : y ∈ / St2 (x, Un ) . Clearly the sets Pn are well-defined. It is evident that [X]2 = {Pn : n ∈ ω}. Therefore, Lemma 6.3 implies that there exist a subset S of X with |S| > ω and k ∈ ω such that [S]2 ⊂ Pk . In a manner similar to that used in the proof of Theorem 6.4, we conclude that S is closed discrete in X. Note that U = {St(x, Uk ) : x ∈ S} is a disjoint open expansion of S. We can deduce that X is not selectively star-ccc By Proposition 3.2 which leads to a contradiction. This shows that |X| ≤ c and completes the proof. 2 7. Open problems We finish the paper by the following questions which we could not answer while working on this paper. Recall that a space X is weakly perfect if every closed subset of X contains a dense set which is a Gδ -set in X.
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Question 7.1. Is a weakly perfect selectively star-ccc space CCC? Question 7.2. Is a weakly perfect selectively star-ccc space weakly Lindelöf? Question 7.3. Is there a countably compact monotonically normal space which is not selectively star-ccc? Question 7.4. Does every selectively star-ccc subspace of the product of finitely many monotonically normal spaces have countable extent? Question 7.5. Is there in ZFC a selectively star-ccc normal space having a regular closed Gδ -subset which is not selectively star-ccc? Recall that a space X has a regular Gδ -diagonal if there is a countable family {Un : n ∈ ω} of open neighborhoods of the diagonal ΔX in X 2 such that ΔX = {Un : n ∈ ω}, where ΔX = {(x, x) : x ∈ X}. Question 7.6. Is it true that a selectively star-ccc space with a regular Gδ -diagonal has cardinality at most c? Question 7.7. Is it true that a selectively star-ccc normal space with a regular Gδ -diagonal has cardinality at most c? Acknowledgement We would like to thank the referee for his (or her) valuable remarks and suggestions which greatly improved the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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