SongAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 10, No. 1, 2009 pp. 13-20 Embedding into discretely absolutely star-Lindelöf spaces II Yan-Kui Song ∗ Abstract. A space X is discretely absolutely star-Lindelöf if for every open cover U of X and every dense subset D of X, there exists a countable subset F of D such that F is discrete closed in X and St(F, U) = X, where St(F, U) = ⋃ {U ∈ U : U ∩F 6= ∅}. We show that every Hausdorff star-Lindelöf space can be represented in a Hausdorff discretely absolutely star-Lindelöf space as a closed Gδ-subspace. Keywords: star-Lindelöf, absolutely star-Lindelöf, centered-Lindelöf 2000 AMS Classification: 54D20, 54G20 1. Introduction By a space, we mean a topological space. A space X is absolutely star- Lindelöf (see [1]) (discretely absolutely star-Lindelöf)(see [12, 13]) if for every open cover U of X and every dense subset D of X, there exists a countable subset F of D such that St(F, U) = X (F is discrete and closed in X and St(F, U) = X, respectively), where St(F, U) = ⋃ {U ∈ U : U ∩ F 6= ∅}. A space X is star-Lindelöf (see [4, 7] under different names) (discretely star- Lindelöf)(see [9, 16]) if for every open cover U of X, there exists a count- able subset (a countable discrete closed subset, respectively) F of X such that St(F, U) = X. It is clear that every separable space and every discretely star- Lindelöf space are star-Lindelöf as well as every space of countable extent(in particular, every countably compact space or every Lindelöf space). A family of subsets is centered (linked) provided every finite subfamily (every two elements, respectively) has nonempty intersection and a family is called ∗The author acknowledges support from the NSF of China Grant 10571081 and Project supported by the National Science Foundation of Jiangsu Higher Education Institutions of China (Grant No 07KJB110055) 14 Y.-K. Song σ-centered (σ-linked) if it is the union of countably many centered subfami- lies(linked subfamilies, respectively). A space X is centered-Lindelöf (linked- Lindelöf) (see [2, 3]) if for every open cover U of X has σ-centered (σ-linked) subcover. From the above definitions, it is not difficult to see that every discretely absolutely star-Lindelöf space is absolutely star-Lindelöf, every discretely ab- solutely star-Lindelöf space is discretely star-Lindelöf, every absolutely star- Lindelöf space is star-Lindelöf, every star-Lindelöf space is centered-Lindelöf, every centered-Lindelöf space is linked-Lindelöf. Bonanzinga and Matveev [2] proved that every Hausdorff (regular, Ty- chonoff) linked-Lindelöf space can be represented as a closed subspace in a Hausdorff (regular, Tychonoff, respectively)star-Lindelöf space. They asked if every Hausdorff (regular, Tychonoff) linked-Lindelöf space can be represented as a closed Gδ-subspace in a Hausdorff (regular, Tychonoff, respectively) star- Lindelöf space. The author [10] gave a positive answer to their question. The author [10] showed that every Hausdorff (regular, Tychonoff) linked-Lindelöf space can be represented as a closed Gδ-subspace in a Hausdorff (regular, Ty- chonoff, respectively) absolutely star-Lindelöf space. The author [13] showed that every separable Hausdorff (regular, Tychonoff, normal) star-Lindelöf space can be represented in a Hausdorff (regular, Tychonoff, normal, respectively) discretely absolutely star-Lindelöf space as a closed Gδ-subspace. The author [14] showed that every Hausdorff linked-Lindelöf space can be represented in a Hausdorff discretely absolutely star-Lindelöf space as a closed subspace and asked the following question: Question 1.1. Is it true that every Hausdorff (regular, Tychonoff) linked- Lindelöf-space can be represented a closed Gδ-subspace in a Hausdorff (regular, Tychonoff, respectively) discretely absolutely star-Lindelöf space? The purpose of this note is to give a construction showing every Hausdorff linked-Lindelöf space can be represented in a Hausdorff discretely absolutely star-Lindelöf space as a closed Gδ-subspace, which give a positive answer to the above question in the class of Hausdorff spaces. Throughout this paper, the cardinality of a set A is denoted by |A|. Let ω denote the first infinite cardinal. For a cardinal κ, let κ+ be the smallest cardinal greater than κ. As usual, a cardinal is the initial ordinal and an ordinal is the set of smaller ordinals. When viewed as a space, every cardinal has the usual order topology. For each pair of ordinals α, β with α < β, we write [α, β] = {γ : α ≤ γ ≤ β} and (α, β) = {γ : α < γ < β}. Other terms and symbols that we do not define will be used as in [5]. 2. Embedding into discretely absolutely star-Lindelöf spaces as a closed Gδ-subspaces First, we show that every Hausdorff star-Lindelöf space can be represented in a Hausdorff discretely absolutely star-Lindelöf space as a closed Gδ-subspace. Embedding into discretely absolutely star-Lindelöf spaces II 15 Recall the Alexandorff duplicate A(X) of a space X. 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 from (U ×{0})∪((U ×{1})\{〈x, 1〉}), where U is a neighborhood of x in X. It is well-known that A(X) is Hausdorff(regular, Tychonoff, normal) iff X is, A(X) is compact iff X is and A(X) is Lindelöf iff X is. Recall from [6] that a space X is absolutely countably compact (=acc) 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. It is not difficult to show that every Hausdorff acc space is countably compact (see [6]). In our construction, we use the following lemma. Lemma 2.1 ([8, 15]). If X is countably compact, then A(X) is acc. Moreover, for any open cover U of A(X), there exists a finite subset F of X × {1} such that A(X) \ St(F, U) ⊆ X × {0} is a finite subset consisting of isolated points of X × {0}. Theorem 2.2. Every Hausdorff star-Lindelöf space can be represented in a Hausdorff discretely absolutely star-Lindelöf space as a closed Gδ-subspace. Proof. If |X| ≤ ω, then X is separable. The author [13] showed that every separable Hausdorff (regular, Tychonoff, normal) space can be represented in Hausdorff (regular, Tychonoff, normal, respectively) discretely absolutely star- Lindelöf space as a closed Gδ-subspace. Let X be a star-Lindelöf space with |X| > ω and let T be X with the discrete topology and let Y = T ∪ {∞}, where ∞ /∈ T be the one-point Lindelöfication of T . Pick a cardinal κ with κ ≥ |X|. Define S(X, κ) = X ∪ (Y × κ+). We topologize S(X, κ) as follows: Y × κ+ has the usual product topology and is an open subspace of S(X, κ), and a basic neighborhood of a point x of X takes the form G(U, α) = U ∪ (U × (α, κ+)), where U is a neighborhood of x in X and α < κ+. Then, it is easy to see that X is a closed subset of S(X, κ) and S(X, κ) is Hausdorff if X is Hausdorff. Let R(X) = A(S(X, κ)) \ (X × {1}). Then, R(X) is Hausdorff if X is Hausdorff. Let P(R(X)) = ((X × {0}) × {ω}) ∪ (R(X) × ω) be the subspace of the product of R(X) × (ω + 1). For each n ∈ ω, let Xω = (X × {0}) × {ω} and Xn = R(X) × {n} for each n ∈ ω. Then, P(R(X)) = Xω ∪ ∪n∈ω Xn. 16 Y.-K. Song From the construction of the topology of P(R(X)), it is not difficult to see that X can be represented in P(R(X)) as a closed Gδ-subspace, since X is homeomorphic to Xω, and P(R(X)) is Hausdorff if X is Hausdorff. We show that P(R(X)) is discretely absolutely star-Lindelöf. To this end, let U be an open cover of P(R(X)). Without loss of generality, we assume that U consists of basic open sets of P(R(X)). Let S be the set of all isolated points of κ+ and let Dn1 = (((T × S) × {0}) × {n}) ∪ (((T × κ +) × {1}) × {n}), Dn2 = (({∞} × κ +) × {1}) × {n} and Dn = Dn1 ∪ Dn2 for each n ∈ ω. If we put D = ∪n∈ωDn. Then, every element of D is isolated in P(R(X)), and every dense subset of P(R(X)) contains D. Thus, it is sufficient to show that there exists a countable subset F of D such that F is discrete closed in P(R(X)) and St(F, U) = P(R(X)). For each x ∈ X, there exists a Ux ∈ U such that 〈〈x, 0〉, ω〉 ∈ Ux, Hence there exist αx < κ +, nx ∈ ω and an open neighborhood Vx of x in X such that ((Vx × {0}) × [nx, ω]) ∪ (A(Vx × (αx, κ +)) × [nx, ω)) ⊆ Ux. If we put V = {Vx : x ∈ X}, then V is an open cover of X. For each n ∈ ω, let X′ n = ∪{x : nx = n}, then X = ∪n∈ωX ′ n . For each x′ ∈ X \ X′ n , there exists a Ux′ ∈ U such that 〈〈x′, 0〉, n〉 ∈ Ux′ . Hence, there exist αx′ < κ + and an open neighborhood Vx′ of x ′ in X such that ((Vx′ × {0}) × {n}) ∪ (A(Vx′ × (αx′ , κ +)) × {n}) ⊆ Ux′ . If we put Vn = {Vx : x ∈ X ′ n } ∪ {Vx′ : x ′ ∈ X \ X′ n }. Then, Vn is an open cover of X. Hence, there exists a countable subset F ′ n of X such that X = St(F ′ n , U), since X is star-Lindelöf. If we pick αn0 > max{sup{αx : x ∈ X ′ n }, sup{αx′ : x ′ ∈ X \ X′ n }}. Then, αn0 < κ +, since |X| ≤ κ. Let Xn1 = ((X × {0}) × {n}) ∪ (A(T × [αn0, κ +)) × {n}); Xn2 = A(T × [0, αn0]) × {n} and Xn3 = A({∞} × κ +) × {n}). Then, Xn = Xn1 ∪ Xn2 ∪ Xn3. Let Fn1 = ((F ′ n × {αn0}) × {1}) × {n}. Then, Fn1 is a countable subset of Dn1 and ((X′ n × {0}) × {ω}) ∪ Xn1 ⊆ St(Fn1, U), since Ux ∩ Fn1 6= ∅ for each x ∈ X ′ n and Ux′ ∩ Fn1 6= ∅ for each x ′ ∈ X \ X′ n . Since Fn1 ⊆ Dn1 and Fn1 is countable. Then, Fn1 is closed in Xn by the Embedding into discretely absolutely star-Lindelöf spaces II 17 construction of the topology of Xn. Hence, Fn1 is closed in P(R(X)), since Xn is open and closed in P(R(X)). On the other hand, since Y is Lindelöf and [0, αn0] is compact, then Y × [0, αn0] is Lindelöf, hence Xn2 = A(Y × [0, αn0]) × {n} is Lindelöf. For each α ≤ αn0, there exists a Uα ∈ U such that 〈〈〈∞, α〉, 0〉, n〉 ∈ Uα. Hence, there exists an open neighborhood Vα of α in κ + and an open neigh- borhood V ′ α of ∞ in Y such that (A(V ′ α × Vα) × {n}) \ (〈〈〈∞, α〉, 1〉, n〉) ⊆ Uα. Let V′ n = {Vα : α ≤ αn0}. Then, V ′ n is an open cover of [0, αn0]. Hence, there exists a finite subcover Vα1 , Vα2 , ...Vαm , since [0, αn0] is compact. Let Tn = ∪{T \ V ′ αi : i ≤ m}. Then, Tn is a countable subset of T . For each i ≤ m, we pick xi ∈ Dn ∩ Uαi . Let F ′ n2 = {xi : i ≤ m}. Then, F ′ n2 is a finite subset of Dn and ((({∞} × [0, αn0]) × {0}) × {n}) ∪ (A((T \ Tn) × [0, αn0]) × {n}) ⊆ St(F ′ n2, U). For each t ∈ Tn, since {t} × [0, αn0] is compact, then A({t} × [0, αn0]) × {n} is compact, hence there exists a finite subset Ft of Dn such that A({t} × [0, α0]) × {n} ⊆ St(Ft, U). Let F ′′ n2 = ∪{Ft : t ∈ Tn}. Then, F ′′ n2 is countable, since Tn is countable. Since F ′′ n2 ∩ (A(Y × {α}) × {n}) is countable for each α < κ + and F ′′ n2 ∩ (A({t} × κ+) × {n} is finite for each t ∈ T , then F ′′ n2 is closed in Xn by the construction of the topology of Xn, hence Fn2 is closed in P(R(X)), since Xn is open closed in P(R(X)). By the definition of F ′′ n2, we have A(Tn × [0, αn0]) × {n} ⊆ St(F ′′ n2, U). If we put Fn2 = F ′ n2 ∪ F ′′ n2. Then, Fn2 is a countable subset of Dn and F ′′ n2 is closed in P(R(X)), since F ′ n1 is finite and and F ′′ n2 is closed in P(R(X)). By the definition of Fn2, we have Xn2 ∪ ((({∞} × [0, αn0]) × {0}) × {n}) ⊆ St(Fn2, U). Finally, we show that there exists a finite subset Fn of Dn such that Xn3 ⊆ St(Fn3, U). Since {∞}×κ + is countably compact, then, By Lemma 2.1, A({∞}× κ+) × {n} is acc and there exists a finite subset F ′ n3 ⊆ Dn2 such that En = Xn3 \ St(F ′ n3, U) ⊆ (({∞} × κ +) × {0}) × {n} is a finite subset and each point of En is an isolated point of (({∞}× κ +)×{0})×{n}. For each point x ∈ En, there exists a Ux ∈ U such that x ∈ Ux. For each point x ∈ En, pick dx ∈ Dn ∩ Ux. Let F ′′ n3 = {dx : x ∈ E}, then F ′′ n3 is a finite subset of Dn and E ⊆ St(F ′′ n3, U). If we put Fn3 = F ′ n3 ∪ F ′′ n3, then Fn3 is a finite subset of Dn and Xn3 ⊆ St(Fn3, U). 18 Y.-K. Song If we put Fn = Fn1 ∪ Fn2 ∪ Fn3, then Fn is a countable subset of Dn such that ((X′ n × {0}) × {ω}) ∪ Xn ⊆ St(Fn, U). Since Fn1 and Fn2 are closed in P(R(X)), Fn3 is finite and each point of Fn is isolated, then Fn is discrete closed in P(R(X)). Let F = ∪n∈ωFn. Then, F is a countable subset of D and St(F, U) = ∪n∈ωSt(Fn, U) ⊇ ∪n∈ω(((X ′ n × {0}) × {ω}) ∪ Xn) = P(R(X)). Since each point of F is isolated, then F is discrete in P(R(X)). Since Fn is discrete closed in Xn and Xn is open closed in P(R(X)) for each n ∈ ω, then F has not accumulation points in R(X) × ω. On the other hand, since F is countable and κ ≥ |X| > ω, then every point of Xω is not accumulation point of F by the construction of the topology of P(R(X)). This shows that F is closed in P(R(X)), which completes the proof. � Since every discretely absolutely star-Lindelöf space is discretely star-Lindelöf, the next corollary follows from Theorem 2.2. Corollary 2.3. Every Hausdorff star-Lindelöf space can be represented in a Hausdorff discretely star-Lindelöf space as a closed Gδ-subspace. Since every discretely absolutely star-Lindelöf space is absolutely star-Lindelöf, the next corollary follows from Theorem 2.2. Corollary 2.4. Every Hausdorff star-Lindelöf space can be represented in a Hausdorff absolutely star-Lindelöf space as a closed Gδ-subspace. The author [10] proved that every Hausdorff (regular, Tychonoff) linked- Lindelöf space can be represented a closed Gδ-subspace in Hausdorff (regular, Tychonoff, respectively) star-Lindelöf space. Thus, we have the next corollary. Corollary 2.5. Every Hausdorff linked-Lindelöf space can be represented in a Hausdorff discretely absolutely star-Lindelöf space as a closed Gδ-subspace. On the separation of Theorem 2.2, Song [14] showed that R(X) is Tychonoff if X is a locally-countable (ie., each point of X has a neighborhood U with |U| ≤ ω) Tychonoff space. Thus, we have the following proposition by the construction of the topology of P(R(X)). Proposition 2.6. If X is a locally countable Tychonoff space, then P(R(X)) is Tychonoff. By Theorem 2.2 and Proposition 2.6, we have the next corollary. Corollary 2.7. Every locally-countable, star-Lindelöf Tychonoff space can be represented in a discretely absolutely star-Lindelöf Tychonoff space as a closed Gδ-subspace. The author [10] proved that every Hausdorff (regular, Tychonoff) linked- Lindelöf space can be represented a closed Gδ-subspace in Hausdorff (regular, Tychonoff, respectively) star-Lindelöf space. Thus, we have the following corol- lary by Corollary 2.7. Embedding into discretely absolutely star-Lindelöf spaces II 19 Corollary 2.8. Every locally-countable, linked-Lindelöf Tychonoff space can be represented in a discretely absolutely star-Lindelöf Tychonoff space as a closed Gδ-subspace. Remark 2.9. In Theorem 2.2, even if X is locally-countable normal, R(X) need not be normal (hence, P(R(X)) need not be normal). Indeed, X×{0} and A({∞} × κ+) are disjoint closed subsets of R(X) that can not be separated by disjoint open subsets of R(X). Thus, the author does not know if every locally countable, normal star-Lindelöf space can be represented in a normal discretely absolutely star-Lindelöf space as a closed Gδ-subspace. Remark 2.10. The author does not know if every regular (Tychonoff, nor- mal) star-Lindelöf space can be represented in a regular (Tychonoff, normal, respectively) discretely absolutely star-Lindelöf space as a closed subspace or as a closed Gδ-subspace. References [1] M. Bonanzinga, Star-Lindelöf and absolutely star-Lindelöf spaces, Quest. Answers Gen. Topology 16 (1998), 79–104. [2] M. Bonanzinga and M. V. Matveev, Closed subspaces of star-Lindelöf and related spaces, East-West J. Math. 2 (2000), no. 2, 171–179. [3] M. Bonanzinga and M. V. Matveev, Products of star-Lindelöf and related spaces, Hous- ton J. Math. 27 (2001), 45–57. [4] E. K. van Douwn, G. M. Reed, A. W. 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Song, Some notes on star-Lindelöf spaces, Questions Answers Gen. Topology 24 (2006), 11–15. [14] Y.-K. Song, Embedding into discretely absolutely star-Lindelöf spaces, Comment. Math. Univ. Carolinae 12 (2007), no. 2, 303–309. [15] J. E. Vaughan, Absolutely countably compactness and property (a), Talk at 1996 Praha symposium on General Topology. [16] Y. Yasui and Z.-M. Gao, Spaces in countable web, Houston J. Math. 25 (1999), 327–335. 20 Y.-K. Song Received November 2007 Accepted March 2009 Yan-Kui Song (songyankuinjnu.edu.cn) Institute of Mathematics, School of Mathematics and Computer Sciences, Nan- jing Normal University, Nanjing, 210097, P. R. China