SongAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 9, No. 2, 2008 pp. 293-299 On σ-starcompact spaces Yan-Kui Song ∗ Abstract. A space X is σ-starcompact if for every open cover U of X, there exists a σ-compact subset C of X such that St(C, U) = X. We investigate the relations between σ-starcompact spaces and other related spaces, and also study topological properties of σ-starcompact spaces. 2000 AMS Classification: 54D20, 54B10, 54D55. Keywords: Lindelöf, σ-starcompact, L-starcompact. 1. Introduction By a space, we mean a topological space. Let us recall that a space X is countably compact if every countable open cover of X has a finite subcover. Fleischman [3] defined a space X to be starcompact if for every open cover U of X, there exists a finite subset F of X such that St(F, U) = X, where St(F, U) = ⋃ {U ∈ U : U ∩ F 6= ∅}, and he proved that every countably compact space is starcompact. Conversely, van Douwen-Reed-Roscoe-Tree [1] proved that every Hausdorff starcompact space is countably compact, but this does not hold for T1-space (see [7]). As generalizations of starcompactness, the following classes of spaces were given: Definition 1.1 ([1, 6]). A space X is star-Lindelöf if for every open cover U of X, there exists a countable subset F of X such that St(F, U) = X. Definition 1.2. A space X is σ-starcompact if for every open cover U of X, there exists a σ-compact subset C of X such that St(C, U) = X. Definition 1.3 ([3, 6, 8]). A space X is L-starcompact if for every open cover U of X, there exists a Lindelöf subset L of X such that St(L, U) = X. ∗The author acknowledges support from the NSF of China(Grants 10571081) and the National Science Foundation of Jiangsu Higher Education Institutions of China (Grant No 07KJB110055) 294 Y.-K. Song In [1], a star-Lindelöf space is called strong star-Lindelöf, in [3], L-starcompactness is called sLc property. From the above definitions, we have the following diagram: star-Lindelöf ⇒ σ-starcompact ⇒ L-starcompact. In the following section, we give examples showing that the converses in the above Diagram do not hold. Thorough this paper, the symbol β(X) means the Čech-Stone compactifica- tion of a Tychonoff space X. The cardinality of a set A is denoted by |A|. Let ω be the first infinite cardinal, ω1 the first uncountable cardinal and c the car- dinality of the set of all real numbers. As usual, a cardinal is the initial ordinal ordinals. For each 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}. Every cardinal is often viewed as a space with the usual order topology. Other terms and symbols follow [2]. 2. σ-starcompact spaces and related spaces In this section, we give two examples which show the converses in the above diagram in the section 1 do not hold. Example 2.1. There exists a Tychonoff σ-starcompact space which is not star-Lindelöf. Proof. Let D be a discrete space of the cardinality c. Define X = (β(D) × (ω + 1)) \ ((β(D) \ D) × {ω}). Then, X is σ-starcompact, since β(D) × ω is a σ-compact dense subset of X. Next, we show that X is not star-Lindelöf. Since |D| = c, then we can enumerate D as {dα : α < c}. For each α < c, let Uα = {dα} × [0, ω]. Then Uα ∩ Uα′ = ∅ for α 6= α ′. Let us consider the open cover U = {Uα : α < c} ∪ {β(D) × ω}. of X. Let F be a countable subset of X. Then, there exists a α0 < c such that F ∩ Uα0 = ∅. Since Uα0 is the only element of U containing the point 〈dα0 , ω〉 and Uα0 ∩ F = ∅, then 〈dα0 , ω〉 /∈ St(F, U), which shows that X is not star-Lindelöf. � Example 2.2. There exists a Tychonoff L-starcompact space which is not σ-star-compact. Proof. Let D = {dα : α < c} be a discrete space of the cardinality c and let Y = D ∪ {∞}, where ∞ /∈ D be the one-point Lindelöfication of D. Then, every compact subset of Y is finite by the construction of the topology of Y . Hence, Y is not σ-compact. Define X = (Y × (ω + 1)) \ (〈∞, ω〉). Then, X is L-starcompact, since Y × ω is a Lindelöf dense subset of X. On σ-starcompact spaces 295 Now, we show that X is not σ-starcompact. For each α < c, let Uα = {dα} × [0, ω]. Then Uα ∩ Uα′ = ∅ for α 6= α ′. Let us consider the open cover U = {Uα : α < c} ∪ {Y × {n} : n ∈ ω}. of X. Let C be σ-compact subset of X. Then, C ∩ (D × {ω}) is countable, since D × {ω} is discrete closed in X. On the other hand, for each n ∈ ω, C ∩ (Y × {n}) is countable in Y × {n}, since Y × {n} is open and close in X. Thus, C is a countable subset of X. Since C is countable, then {α : C∩Uα 6= ∅} is countable, Hence, there exists a αω ∈ c such that C ∩ Uα = ∅ for each α > αω. If we pick α′ > αω. Then, 〈dα′ , ω〉 /∈ St(C, U), since Uα′ is the only element of U containing 〈dα′ , ω〉 and Uα′ ∩ C = ∅, which shows that X is not σ- starcompact. � Remark 2.3. The author does not know if there exists a normal L-starcompact which is not σ-starcompact and a normal σ-starcompact space which is not star-Lindelöf. 3. Properties of σ-starcompact spaces In Example 2.1, the closed subset D ×{ω} of X is not σ-starcompact, which shows that a closed subset of a σ-starcompact space need not be σ-starcompact. In the following, we construct an example which shows that a regular-closed Gδ-subspace of a σ-starcompact space need not be σ-starcompact. Example 3.1. There exists a star-Lindelöf (hence, σ-starcompact) Tychonoff space having a regular-closed Gδ-subspace which is not σ-starcompact. Proof. Let S1 = (Y × (ω + 1)) \ (〈∞, ω〉). be the same space as the space X in the proof of Example 2.2. As we prove above, S1 is not σ-starcompact. Let S2 = ω ∪ R be the Isbell-Mrówka space [7], where R is a maximal almost disjoint family of infinite subsets of ω with |R| = c. Then, S2 is star-Lindelöf, since ω is a countable dense subset of S2. Hence, it is σ-starcompact. We assume S1 ∩ S2 = ∅. Let π : D × {ω} → R be a bijection and let X be the quotient image of the disjoint sum S1 ⊕ S2 by identifying 〈dα, ω〉 of S1 with π(〈dα, ω〉) of S2 for each 〈dα, ω〉 of D × {ω}. Let ϕ : S1 ⊕ S2 → X be the quotient map. Then, ϕ(S1) is a regular-closed Gδ-subspace of X which is not σ-starcompact. We shall show that X is star-Lindelöf. To this end, let U be an open cover of X. Since ϕ(ω) is a countable dense subset of π(S2), then ϕ(S2) ⊆ St(ϕ(ω), U). On the other hand, since ϕ(Y × ω) is Lindelöf there exists a countable subset F1 of ϕ(Y × ω) such that ϕ(Y × ω) ⊆ St(F1, U). Let F = ϕ(ω) ∪ F1. Then, X = St(F, U). Hence, X is star-Lindelöf, which completes the proof. � 296 Y.-K. Song We give a positive result: Theorem 3.2. An open Fδ-subset of a σ-starcompact space is σ-starcompact. Proof. Let X be an σ-starcompact space and let Y = ∪{Hn : n ∈ ω} be an open Fδ-subset of X, where the set Hn is closed in X for each n ∈ ω. To show that Y is σ-starcompact, let U be an open cover of Y . we have to find a σ-compact subset C of Y such that St(C, U) = Y . For each n ∈ ω, consider the open cover Un = U ∪ {X \ Hn} of X. Since X is σ-starcompact, there exists a σ-compact subset Cn of X such that St(Cn, Un) = X. Let Dn = Cn ∩ Y . Since Y is a Fδ-set, Dn is σ-compact, and clearly Hn ⊆ St(Dn, U). Thus, if we put C = ∪{Dn : n ∈ ω}, then C is a σ-compact subset of Y and St(C, U) = Y . Hence, Y is σ-starcompact. � A cozero-set in a space X is a set of the form f −1(R \ {0}) for some real- valued continuous function f on X. Since a cozero-set is an open Fσ-set, we have the following corollary: Corollary 3.3. A cozero-set of a σ-starcompact space is σ-starcompact. Since a continuous image of a σ-compact space is σ-compact, then it is not difficult to show the following result. Theorem 3.4. A continuous image of a σ-starcompact space is σ-starcompact. Next, we turn to consider preimages. To show that the preimage of a σ-starcompact space under a closed 2-to-1 continuous map need not be σ- starcompact we use the Alexandorff duplicate A(X) of a space X. The underly- ing set of A(X) is X ×{0, 1}; each point of X ×{1} is isolated and a basic neigh- borhood 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. Example 3.5. There exists a closed 2-to-1 continuous map f : X → Y such that Y is a σ-starcompact space, but X is not σ-starcompact. Proof. Let Y be the space X in the proof of Example 2.1. Then Y is σ- starcompact and has the infinite discrete closed subset F = D × {ω}. Let X be the Alexandroff duplicate A(Y ) of Y . Then, X is not σ-starcompact, since F × {1} is an infinite discrete, open and closed set in X. Let f : X → Y be the natural map. Then, f is a closed 2-to-1 continuous map, which completes the proof. � Now, we give a positive result: Theorem 3.6. Let f be an open perfect map from a space X to a σ-starcompact space Y . Then, X is σ-starcompact Proof. Since f (X) is open and closed in Y , we may assume that f (X) = Y . Let U be an open cover of X and let y ∈ Y . Since f −1(y) is compact, there exists a finite subcollection Uy of U such that f −1(y) ⊆ ∪Uy and U ∩ f −1(y) 6= ∅ for On σ-starcompact spaces 297 each U ∈ Uy. Pick an open neighbourhood Vy of y in Y such that f −1(Vy ) ⊆ ∪{U : U ∈ Uy}, and we can assume that (1) Vy ⊆ ∩{f (U ) : U ∈ Uy} because f is open. Taking such open set Vy for each y ∈ Y , we have an open cover V = {Vy : y ∈ Y } of Y . Hence, there exists a σ-compact subset C of Y such that St(C, V) = Y , since Y is σ-compact. Since f is perfect, the set f −1(C) is a σ-compact subset of X. To show that St(f −1(C), V) = X, let x ∈ X. Then, there exists y ∈ Y such that f (x) ∈ Vy and Vy ∩ C 6= ∅. Since x ∈ f −1(Vy ) ⊆ ∪{U : U ∈ Uy}, we can choose U ∈ Uy with x ∈ U . Then Vy ⊆ f (U ) by (1), and hence U ∩ f −1(C) 6= ∅. Therefore, x ∈ St(f −1(C), U). Consequently , we have that St(f −1(C), U) = X. � By Theorem 3.6, we have the following Corollary 3.7. Corollary 3.7. Let X be a σ-starcompact space and Y a compact space. Then, X × Y is C-starcompact. The following theorem is a generalization of Corollary 3.7. Theorem 3.8. Let X be a σ-starcompact space and Y a locally compact, Lin- delöf space. Then, X × Y is σ-starcompact. Proof. Let U be an open cover of X × Y . For each y ∈ Y , there exists an open neighbourhood Vy of y in Y such that clY Vy is compact. By the Corollary 3.7, the subspace X × clY Vy is σ-starcompact. Thus, there exists a σ-compact subset Cy ⊆ X × clY Vy such that X × clY Vy ⊆ St(Cy, U). Since Y is Lindelöf, there exists a countable cover {Vyi : i ∈ ω} of Y . Let C = ∪{Cyi : i ∈ ω}. Then, C is a σ-compact subset of X × Y such that St(C, U) = X × Y . Hence, X × Y is σ-starcompact. � In the following, we give an example showing that the condition of the locally compact space in Theorem 3.8 is necessary. Example 3.9. There exist a countably compact space X and a Lindelöf space Y such that X × Y is not σ-starcompact. Proof. Let X = ω1 with the usual order topology. Y = ω1+1 with the following topology. Each point α with α < ω1 is isolated and a set U containing ω1 is open if and only if Y \ U is countable. Then, X is countably compact and Y is Lindelöf. Now, we show that X × Y is not σ-starcompact. For each α < ω1, let Uα = [0, α] × [α, ω1], and Vα = [α, ω1) × {α}. Consider the open cover U = {Uα : α < ω1} ∪ {Vα : α < ω1} 298 Y.-K. Song of X × Y and let C be a σ-compact subset of X × Y . Then, πX (C) is a σ- compact subset of X, where πX : X × Y → X is the projection. Thus, there exists β < ω1 such that πX (C) ∩ (β, ω1) = ∅ by the definition of the topology of X. Pick α0 with α0 > β. Then, Vα0 ∩C = ∅. If we pick α′ > α0, then 〈α ′, α0〉 /∈ St(C, U) since Vα0 is the only element of U containing 〈α′, α0〉. Hence, X × Y is not σ-starcompact, which completes the proof. � The Theorem 3.9 also shows the product of two σ-starcompact spaces need not be σ-starcompact. Next, we give a well-known example showing that the product of two countably compact spaces need not be σ-starcompact. We give the proof roughly for the sake of completeness. Example 3.10. There exist two countably compact spaces X and Y such that X × Y is not σ-starcompact. Proof. Let D be a discrete space of the cardinality c. We can define X = ∪α<ω1 Eα, Y = ∪α<ω1 Fα, where Eα and Fα are the subsets of β(D) which are defined inductively so as to satisfy the following conditions (1), (2) and (3): (1) Eα ∩ Fβ = D if α 6= β; (2) |Eα| ≤ c and |Fα| ≤ c; (3) every infinite subset of Eα (resp. Fα) has an accumulation point in Eα+1 (resp. Fα+1). Those sets Eα and Fα are well-defined since every infinite closed set in β(D) has the cardinality 2c (see [5]). Then, X × Y is not σ-starcompact, because the diagonal {〈d, d〉 : d ∈ D} is a discrete open and closed subset of X × Y with the cardinality c and σ-starcompactness is preserved by open and closed subsets. � Example 3.11. There exist a separable space X and a Lindelöf space Y such that X × Y is not σ-starcompact. Proof. Let X = Y be the same space Y in the proof of Example 2.2. Then, Y is Lindelöf, however is not σ-starcompact. Let Y = ω ∪R be the Isbell-Mrówka space [7], where R is a maximal almost disjoint family of infinite subsets of ω with |R| = c. Then, Y is separable. Since |R| = c, then we can enumerate R as {rα : α < c}. To show that X × Y is not σ-starcompact. For each α < c, let Uα = {dα}×Y and Vα = (X \{dα})×({rα}∪rα). For n ∈ ω, let Wn = X ×{n}. We consider the open cover U = {Uα : α < c} ∪ {Vα : α < c} ∪ {Wn : n ∈ ω} of X ×Y . Let C be a σ-compact subset of X ×Y . Then, πX (C) is a σ-compact subset of X, where πX : X × Y → X is the projection. Thus, there exists α < c such that C ∩ Uα = ∅. Hence, 〈dα, rα〉 /∈ St(C, U) since Uα is the only element of U containing 〈dα, rα〉. Hence, X × Y is not σ-starcompact. which completes the proof. � On σ-starcompact spaces 299 Theorem 3.12. Every Tychonoff space can be embedded in a σ-starcompact Tychonoff space as a closed Gδ-subspace. Proof. Let X be a Tychonoff space. If we put Z = (β(X) × (ω + 1)) \ ((β(X) \ X) × {ω}), then X × {ω} is a closed subset of Z, which is homeomorphic to X. Since β(D) × ω is a σ-compact dense subset of Z, then Z is σ-starcompact, which completes the proof. � Acknowledgements. The author is most grateful to the referee for his kind help and valuable suggestions References [1] E. K. van Douwen, G. M. Reed, A. W. Roscoe and I. J. Tree, Star covering properties, Topology Appl. 39 (1991), 71–103. [2] R. Engelking, General Topology, Revised and completed edition, Heldermann Verlag., 1989. (1991), 255–271. [3] G. R. Hiremath, On star with Lindelöf center property, J. Indian Math. Soc. 59 (1993), 227–242. [4] W. M. Fleischman, A new extension of countable compactness, Fund. Math. 67 (1971), 1–9. [5] R. C. Walker, The Stone-Čech compactification, Berlin, 1974. [6] M. V. Matveev, A survey on star covering properties, Topology Atlas., preprint No. 330, 1998. [7] S. Mrówka, On completely regular spaces, Fund. Math. 41 (1954), 105–106. [8] Y.-K. Song, On L-starcompact spaces, Czech. Math. J. 56 (2006), 781–788. Received September 2007 Accepted November 2007 Yan-Kui Song (songyankui@njnu.edu.cn) Department of Mathematics, Nanjing Normal University, Nanjing 210097, P. R. China.