@ Appl. Gen. Topol. 23, no. 2 (2022), 315-323 doi:10.4995/agt.2022.17021 © AGT, UPV, 2022 On set star-Lindelöf spaces Sumit Singh Department of Mathematics, Dyal Singh College, University of Delhi, Lodhi Road, New Delhi- 110003, India. (sumitkumar405@gmail.com) Communicated by F. Mynard Abstract A space X is said to be set star-Lindelöf if for each nonempty subset A of X and each collection U of open sets in X such that A ⊆ ⋃ U, there is a countable subset V of U such that A ⊆ St( ⋃ V,U). The class of set star-Lindelöf spaces lie between the class of Lindelöf spaces and the class of star-Lindelöf spaces. In this paper, we investigate the relationship between set star-Lindelöf spaces and other related spaces by providing some suitable examples and study the topological properties of set star- Lindelöf spaces. 2020 MSC: 54D20; 54E35. Keywords: Menger; star-Lindelöf; strongly star-Lindelöf; set star-Lindelöf; covering; star-Covering; topological space. 1. Introduction and Preliminaries Arhangel’skii [1] defined a cardinal number sL(X) of X: the minimal infinite cardinality τ such that for every subset A ⊂ X and every open cover U of A, there is a subfamily V ⊂U such that |V|≤ τ and A ⊆ ⋃ V. If sL(X) = ω, then the space X is called sLindelöf space. Following this idea, Kočinac and Konca [7] introduced and studied the new types of selective covering properties called set-covering properties (for a similar studies, see [4, 14, 15, 16, 17]). A space X is said to have the set-Menger [7] property if for each nonempty subset A of X and each sequence (Un : n ∈ N) of collections of open sets in X such that for each n ∈ N, A ⊆ ⋃ Un, there is a sequence (Vn : n ∈ N) such that for each n ∈ N, Vn is a finite subset of Un and A ⊆ ⋃ n∈N ⋃ Vn. The author [13] noticed Received 16 January 2022 – Accepted 18 April 2022 http://dx.doi.org/10.4995/agt.2022.17021 https://orcid.org/0000-0001-9701-3091 S. Singh that the set-Menger property is nothing but another view of Menger covering property. Recently, the author [12] defined and studied set starcompact and set strongly starcompact spaces (also see [8]). In this paper, we consider the classes of set star-Lindelöf spaces and set strongly star-Lindelöf spaces already introduced in [9] and recently studied in [4]. Note that in fact in the class of T1 spaces, set strongly star-Lindelöfness is equivalent to the property having countable extent [[4], Proposition 3.1]. If A is a subset of a space X and U is a collection of subsets of X, then St(A,U)= ⋃ {U ∈U : U ∩A 6= ∅}. We usually write St(x,U) = St({x},U). Throughout the paper, by “a space” we mean “a topological space”, N, R and Q denotes the set of natural numbers, set of real numbers, and set of rational numbers, respectively, the cardinality of a set is denoted by |A|. Let ω denote the first infinite cardinal, ω1 the first uncountable cardinal, c the cardinality of the set of all real numbers. An open cover U of a subset A ⊂ X means elements of U open in X such that A ⊆ ⋃ U = ⋃ {U : U ∈U}. We first recall the classical notions of spaces that are used in this paper. Definition 1.1 ([5]). A space X is said to be (1) starcompact if for each open cover U of X, there is a finite subset V of U such that X = St( ⋃ V,U). (2) strongly starcompact if for each open cover U of X, there is a finite subset F of X such that X = St(F,U). Definition 1.2 ([12, 8]). A space X is said to be (1) set starcompact if for each nonempty subset A of X and each collection U of open sets in X such that A ⊆ ⋃ U, there is a finite subset V of U such that A ⊆ St( ⋃ V,U). (2) set strongly starcompact if for each nonempty subset A of X and each collection U of open sets in X such that A ⊆ ⋃ U, there is a finite subset F of A such that A ⊆ St(F,U). Definition 1.3. A space X is said to be (1) star-Lindelöf [5] if for each open cover U of X, there is a countable subset V of U such that X = St( ⋃ V,U). (2) strongly star-Lindelöf [5] if for each open cover U of X, there is a countable subset F of X such that X = St(F,U). Note that the star-Lindelöf spaces have a different name such as 1-star- Lindelöf and 1 1 2 -star-Lindelöf in different papers (see [5, 10]) and the strongly star-Lindelöf space is also called star countable in [10, 21]. It is clear that, every strongly star-Lindelöf space is star-Lindelöf. Recall that a collection A ⊆ P(ω) is said to be almost disjoint if each set A ∈A is infinite and the sets A ⋂ B are finite for all distinct elements A,B ∈A. For an almost disjoint family A, put ψ(A) = A ⋃ ω and topologize ψ(A) as follows: for each element A ∈ A and each finite set F ⊂ ω, {A} ⋃ (A \ F) is a basic open neighborhood of A and the natural numbers are isolated. The © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 316 On set star-Lindelöf spaces spaces of this type are called Isbell-Mrówka ψ-spaces [2, 11] or ψ(A) space. For other terms and symbols, we follow [6]. The following result was proved in [8]. Theorem 1.4 ([8]). Every countably compact space is set strongly starcompact. Note that in the class of Hausdorff spaces strongly starcompactness, set strongly starcompactness and countable compactness are equivalent [4, Propo- sition 2.2]. 2. set star-Lindelöf and related spaces In this section, we give some examples showing the relationship among set star-Lindelöf spaces, set strongly star-Lindelöf spaces, and other related spaces. First we define our main definition. Definition 2.1. A space X is said to be (1) set star-Lindelöf if for each nonempty subset A of X and each collection U of open sets in X such that A ⊆ ⋃ U, there is a countable subset V of U such that A ⊆ St( ⋃ V,U). (2) set strongly star-Lindelöf if for each nonempty subset A of X and each collection U of open sets in X such that A ⊆ ⋃ U, there is a countable subset F of A such that A ⊆ St(F,U). Note that in the class of T1 spaces the set strongly star-Lindelöfness is equiv- alent to the property to have a countable extent [4, Proposition 3.1]. Note that there is a misprint in the statement of the definition of relatively∗ set star strongly-compact in [4]: the authors write that set F is a finite subset of A but the original definition asks that F is contained in A and Bonanzinga and Maesano use exactly this last fact during all the paper. We have the following diagram from the definitions and [4, Proposition 3.1]. However, the following examples show that the converse of these implications are not true. set strongly starcompact → set starcompact ↓ ↓ Lindelof → countable extent ↔T1 set strongly star −Lindelof → set star −Lindelof ↓ ↓ strongly star −Lindelof → star −Lindelof Example 2.2. (i) The discrete space ω has countable extent but it is not set starcompact space. (ii) The space [0,ω1) has countable extent but it is not Lindelöf. (iii) Let Y be a discrete space with cardinality c. Let X = Y ∪{y∗}, where y∗ /∈ Y topologized as follows: each y ∈ Y is an isolated point and a set U © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 317 S. Singh containing y∗ is open if and only if X \U is countable. Then X has countable extent but it is not countably compact. Bonanzinga [3] proved that every Isbell-Mrówka space is a Tychonoff strongly star-Lindelöf space with uncountable extent (hence, it is not set strongly star- Lindelöf). Note that in [3] strongly star-Lindelöf is called star-Lindelöf. The following lemma was proved by Song [18]. Lemma 2.3 ([18, Lemma 2.2]). A space X having a dense Lindelöf subspace is star-Lindelöf. The following example shows that the Lemma 2.3 does not hold if we replace star-Lindelöf space by a set star-Lindelöf space. Example 2.4. There exists a Tychonoff space X having a dense Lindelöf sub- space such that X is not set star-Lindelöf. Proof. Let D(c) = {dα : α < c} be a discrete space of cardinality c and let Y = D(c) ∪{d∗} be one-point compactification of D(c). Let X = (Y × [0,ω)) ∪ (D(c) ×{ω}) be the subspace of the product space Y × [0,ω]. Then Y × [0,ω) is a dense Lindelöf subspace of X and by Lemma 2.3, X is star-Lindelöf. In [4, Proposition 3.4] shows that if X is a space such that there exists a closed and discrete subspace D of X having uncountable cardinality and a disjoint family U = {Oa : a ∈ D} of open neighborhoods of points a ∈ D, then X is not set star-Lindelöf. So, we conclude that X is not set star-Lindelöf. � Bonanzinga and Maesano [4, Example 3.5] constructed an example of a Tychonoff separable (hence set star-Lindelöf) non set strongly star-Lindelöf space. Remark 2.5. (1) In [12], Singh gave an example of a Tychonoff set starcompact space X that is not set strongly starcompact. (2) It is known that there are star-Lindelöf spaces that are not strongly star-Lindelöf (see [5, Example 3.2.3.2] and [5, Example 3.3.1]). Now we give some conditions under which star-Lindelöfness coincides with set star-Lindelöfness and strongly star-Lindelöfness coincide with set strongly star-Lindelöfness. Recall that a space X is paraLindelöf if every open cover U of X has a locally countable open refinement. Song and Xuan [19] proved the following result. Theorem 2.6 ([19, Theorem 2.24]). Every regular paraLindelöf star-Lindelöf spaces are Lindelöf. We have the following theorem from Theorem 2.6 and the diagram. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 318 On set star-Lindelöf spaces Theorem 2.7. If X is a regular paraLindelöf space, then the following state- ments are equivalent: (1) X is Lindelöf; (2) X is set strongly star-Lindelöf; (3) e(X) = ω; (4) X is set star-Lindelöf; (5) X is strongly star-Lindelöf; (6) X is star-Lindelöf. A space is said to be metaLindelöf if every open cover of it has a point- countable open refinement. Bonanzinga [3] proved the following result. Theorem 2.8 ([3]). Every strongly star-Lindelöf metaLindelöf spaces are Lin- delöf. We have the following theorem from Theorem 2.8 and the diagram. Theorem 2.9. If X is a metaLindelöf space, then the following statements are equivalent: (1) X is Lindelöf; (2) X is set strongly star-Lindelöf; (3) e(X) = ω; (4) X is strongly star-Lindelöf. 3. Properties of set star-Lindelöf spaces In this section, we study the topological properties of set star-Lindelöf spaces. Theorem 3.1. If X is a set star-Lindelöf space, then every open and closed subset of X is set star-Lindelöf. Proof. Let X be a set star-Lindelöf space and A ⊆ X be an open and closed set. Let B be any subset of A and U be a collection of open sets in (A,τA) such that ClA(B) ⊆ ⋃ U. Since A is open, then U is a collection of open sets in X. Since A is closed, ClA(B) = ClX(B). Applying the set star-Lindelöfness property of X, there exists a countable subset V of U such that B ⊆ St( ⋃ V,U). Hence A is a set star-Lindelöf. � Consider the Alexandorff duplicate A(X) = X ×{0, 1} of a space X. The 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 and each point 〈x, 1〉 ∈ X ×{1} is an isolated point. Theorem 3.2. If X is a T1-space and A(X) is a set star-Lindelöf space. Then e(X) < ω1. Proof. Suppose that e(X) ≥ ω1. Then there exists a discrete closed subset B of X such that |B| ≥ ω1. Hence B ×{1} is an open and closed subset of © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 319 S. Singh A(X) and every point of B ×{1} is an isolated point. Thus A(X) is not set star-Lindelöf by Theorem 3.1. � Theorem 3.3. Let X be a space such that the Alexandorff duplicate A(X) of X is set star-Lindelöf. Then X is a set star-Lindelöf space. Proof. Let B be any nonempty subset of X and U be an open cover of B. Let C = B ×{0} and A(U) = {U ×{0, 1} : U ∈U}. Then A(U) is an open cover of C. Since A(X) is set star-Lindelöf, there is a countable subset A(V) of A(U) such that C ⊆ St( ⋃ A(V),A(U)). Let V = {U ∈U : U ×{0, 1}∈ A(V)}. Then V is a countable subset of U. Now we have to show that B ⊆ St( ⋃ V,U). Let x ∈ B. Then 〈x, 0〉 ∈ St( ⋃ A(V),A(U)). Choose U × {0, 1} ∈ A(U) such that 〈x, 0〉 ∈ U ×{0, 1} and U ×{0, 1}∩ ( ⋃ A(V)) 6= ∅, which implies U ∩ ( ⋃ V) 6= ∅ and x ∈ U. Therefore x ∈ St( ⋃ V,U), which shows that X is set star-Lindelöf space. � On the images of set star-Lindelöf spaces, we have the following result. Theorem 3.4. A continuous image of set star-Lindelöf space is set star- Lindelöf. Proof. Let X be a set star-Lindelöf space and f : X → Y is a continuous mapping from X onto Y . Let B be any subset of Y and V be an open cover of B. Let A = f−1(B). Since f is continuous, U = {f−1(V ) : V ∈ V} is the collection of open sets in X with A = f−1(B) ⊆ f−1(B) ⊆ f−1( ⋃ V) = ⋃ U. As X is set star-Lindelöf, there exists a countable subset U′ of U such that A ⊆ St( ⋃ U′,U). Let V′ = {V : f−1(V ) ∈ U′}. Then V′ is a countable subset of V and B = f(A) ⊆ f(St( ⋃ U′,U)) ⊆ St( ⋃ f({f−1(V ) : V ∈ V′}),V) = St( ⋃ V′,V). Thus Y is set star-Lindelöf space. � Next, we turn to consider preimages of set strongly star-Lindelöf and set star-Lindelöf spaces. We need a new concept called nearly set star-Lindelöf spaces. A space X is said to be nearly set star-Lindelöf in X if for each subset Y of X and each open cover U of X, there is a countable subset V of U such that Y ⊆ St( ⋃ V,U). For the strong version of this property (see [4]). Theorem 3.5. If f : X → Y is an open and perfect continuous mapping and Y is a set star-Lindelöf space, then X is nearly set star-Lindelöf. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 320 On set star-Lindelöf spaces Proof. Let A ⊆ X be any nonempty set and U be an open cover of X. Then B = f(A) is a subset of Y . Let y ∈ B. Then f−1{y} is a compact subset of X, thus there is a finite subset Uy of U such that f−1{y}⊆ ⋃ Uy. Let Uy = ⋃ Uy. Then Vy = Y \ f(X \ Uy) is a neighborhood of y, since f is closed. Then V = {Vy : y ∈ B} is an open cover of B. Since Y is set star-Lindelöf, there exists a countable subset V′ of V such that B ⊆ St( ⋃ V′,V). Without loss of generality, we may assume that V′ = {Vyi : i ∈ N′ ⊆ N}. Let W = ⋃ i∈N′ Uyi . Since f −1(Vyi ) ⊆ ⋃ {U : U ∈Uyi} for each i ∈ N′. Then W is a countable subset of U and f−1( ⋃ V′) = ⋃ W. Next, we show that A ⊆ St( ⋃ W,U). Let x ∈ A. Then there exists a y ∈ B such that f(x) ∈ Vy and Vy ⋂ ( ⋃ V′) 6= ∅. Since x ∈ f−1(Vy) ⊆ ⋃ {U : U ∈Uy}, we can choose U ∈Uy with x ∈ U. Then Vy ⊆ f(U). Thus U ⋂ f−1( ⋃ V′) 6= ∅. Hence x ∈ St(f−1( ⋃ V′),U). Therefore x ∈ St( ⋃ W,U), which shows that A ⊆ St( ⋃ W,U). Thus X is nearly set star-Lindelöf. � It is known that the product of star-Lindelöf space and compact space is a star-Lindelöf (see [5]). Problem 3.6. Does the product of set star-Lindelöf space and a compact space is set star-Lindelöf ? The following example shows that the product of two countably compact (hence, set star-Lindelöf) spaces need not be set star-Lindelöf. Example 3.7. There exist two countably compact spaces X and Y such that X ×Y is not set star-Lindelöf. Proof. Let D(c) be a discrete space of the cardinality c. We can define X =⋃ α<ω1 Eα and Y = ⋃ α<ω1 Fα, where Eα and Fα are the subsets of β(D(c)) which are defined inductively to satisfy the following three conditions: (1) Eα ⋂ Fβ = D(c) 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). © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 321 S. Singh Those sets Eα and Fα are well-defined since every infinite closed set in β(D(c)) has the cardinality 2c (see [20]). Then X×Y is not set star-Lindelöf, since the diagonal {〈d,d〉 : d ∈ D(c)} is a discrete open and closed subset of X ×Y with the cardinality c. � van Douwen-Reed-Roscoe-Tree [5, Example 3.3.3] gave an example of a countably compact X (hence, set star-Lindelöf) and a Lindelöf space Y such that X × Y is not strongly star-Lindelöf. Now we use this example to show that X ×Y is not set star-Lindelöf. Example 3.8. There exists a countably compact space X and a Lindelöf space Y such that X ×Y is not set star-Lindelöf. Proof. Let X = [0,ω1) with the usual order topology. Let Y = [0,ω1] with the following topology. Each point 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. It is enough to show that X ×Y is not star-Lindelöf. For each α < ω1, Uα = X × {α} is open in X × Y . For each β < ω1, Vβ = [0,β] × (0,ω1] is open in X ×Y . Let U = {Uα : α < ω1}∪{Vβ : β < ω1}. Then U is an open cover of X ×Y . Let V be any countable subset of U. Since V is countable, there exists α′ < ω1 such that Uα /∈ V for each α > α′. Also, there exists α′′ < ω1 such that Vβ /∈ V for each β > α′′. Let β = sup{α′,α′′}. Then Uβ ⋂ ( ⋃ V) = ∅ and Uβ is the only element containing 〈β,β〉. Thus 〈β,β〉 /∈ St( ⋃ V,U), which shows that X is not star-Lindelöf. � van Douwen-Reed-Roscoe-Tree [5, Example 3.3.6] gave an example of Haus- dorff regular Lindelöf spaces X and Y such that X ×Y is star-Lindelöf. Now we use this example and show that the product of two Lindelöf spaces is not set star-Lindelöf. Example 3.9. There exists a Hausdorff regular Lindelöf spaces X and Y such that X ×Y is not set star-Lindelöf. Proof. Let X = R\Q have the induced metric topology. Let Y = R with each point of R\Q is isolated and points of Q having metric neighborhoods. Hence both spaces X and Y are Hausdorff regular Lindelöf spaces and first countable too, so X×Y Hausdorff regular and first countable. Now we show that X×Y is not set star-Lindelöf. Let A = {(x,x) ∈ X × Y : x ∈ X}. Then A is an uncountable closed and discrete set (see [[5], Example 3.3.6]). For (x,x) ∈ A, Ux = X ×{x} is the open subset of X × Y . Then U = {Ux : (x,x) ∈ A} is an open cover of A. Let V be any countable subset of U. Then there exists (a,a) ∈ A such that (a,a) /∈ ⋃ V and thus ( ⋃ V) ⋂ Ua = ∅. But Ua is the only element of U containing (a,a). Thus (a,a) /∈ St( ⋃ V,U), which completes the proof. � © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 322 On set star-Lindelöf spaces Acknowledgements. The author would like to thank the referee for several suggestions that led to an improvement of both the content and exposition of the paper. References [1] A. V. Arhangel’skii, An external disconnected bicompactum of weight c is inhomoge- neous, Dokl. Akad. 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