CUBO, A Mathematical Journal Vol. 23, no. 02, pp. 287–298, August 2021 DOI: 10.4067/S0719-06462021000200287 Weakly strongly star-Menger spaces Gaurav Kumar 1 Brij K. Tyagi 2 1 Department of Mathematics, University of Delhi, New Delhi-110007, India. gaurav.maths.du@gmail.com 2 Atma Ram Sanatan Dharma College, University of Delhi, New Delhi-110021, India. brijkishore.tyagi@gmail.com ABSTRACT A space X is called weakly strongly star-Menger space if for each sequence (Un : n ∈ ω) of open covers of X, there is a sequence (Fn : n ∈ ω) of finite subsets of X such that ⋃ n∈ω St(Fn, Un) is X. In this paper, we investigate the re- lationship of weakly strongly star-Menger spaces with other related spaces. It is shown that a Hausdorff paracompact weakly star Menger P-space is star-Menger. We also study the images and preimages of weakly strongly star-Menger spaces under various type of maps. RESUMEN Un espacio X se llama débilmente fuertemente estrella- Menger si para cada sucesión (Un : n ∈ ω) de cubrimientos abiertos de X, existe una sucesión (Fn : n ∈ ω) de subcon- juntos finitos de X tales que ⋃ n∈ω St(Fn, Un) es X. En este art́ıculo, investigamos la relación entre espacios débilmente fuertemente estrella-Menger con otros espacios relaciona- dos. Se muestra que un P-espacio Hausdorff paracompacto débilmente estrella Menger es estrella-Menger. También es- tudiamos las imágenes y preimágenes de espacios débilmente fuertemente estrella-Menger bajo diversos tipos de aplica- ciones. Keywords and Phrases: Stronlgy star-Menger, star-Menger, almost star-Menger, Weakly star-Menger, covering topological spaces. 2020 AMS Mathematics Subject Classification: 54C10, 54D20, 54G10. Accepted: 03 June, 2021 Received: 28 January, 2021 c©2021 G. Kumar et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462021000200287 https://orcid.org/0000-0002-4010-7879 https://orcid.org/0000-0003-2660-2432 288 Gaurav Kumar & Brij K. Tyagi CUBO 23, 2 (2021) 1 Introduction The study of selection principles in topology and their relations to game theory and Ramsey theory was started by Scheepers [24] (see also [12]). In the last two decades, these have gained enough importance to become one of the most active areas of set theoretic topology. Several covering properties are defined based on these selection principles ([17, 18]). A number of results in the literature show that many topological properties can be described and characterized in terms of star covering properties ([7, 21, 22]). The method of stars has been used to study the problem of metrization of topological spaces, and for definitions of several important classical topological notions. Let us recall that a space X is countably compact (CC) if every countable open cover of X has a finite subcover. Fleischman [10] 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= φ}. He proved that every countably compact space is starcompact. Van Douwen in [7] showed that every T2 starcompact space is countably compact, but this does not hold for T1-spaces (see [26, Example 2.5]). Matveev [20] defined a space X to be absolutely countably compact (ACC) if for each open cover U of X and each dense subset D of X, there exists a finite subset F of D such that St(F,U) = X. It is clear that every T2-absolutely countably compact space is countably compact. Kočinac et al. ([1, 2, 15, 16]), defined a space X to be strongly star-Menger (SSM) if for each sequence (Un : n ∈ ω) of open covers of X, there exists a sequence (Fn : n ∈ ω) of finite subsets of X such that {St(Fn,Un) : n ∈ ω} is an open cover of X. The SSM property is stronger than the star-Menger (SM) property. Pansera [23], defined a space X to be weakly star-Menger (WSM) if for each sequence (Un : n ∈ ω) of open covers of X there is a sequence (Vn : n ∈ ω) with Vn a finite subset of Un for each n ∈ ω, and ⋃ n∈ω St(∪Vn,Un) = X. WSM is weaker than the SSM property. In this paper we introduce a star property which lies between SSM and WSM called weakly strongly star-Menger (WSSM). The paper is organized as follows. Section 2 contains some preliminaries used in the paper. In Section 3 we investigate the relationship of WSSM spaces with other related spaces. Section 4 contains the information on subspaces and product spaces of WSSM and in the last Section 5 we study the image and preimage of WSSM spaces under continuous maps. CUBO 23, 2 (2021) Weakly strongly star-Menger spaces 289 2 Preliminaries Throughout this paper a space means topological space. The cardinality of a set A is denoted by |A| . Let ω be the first infinite cardinal and ω1 the first uncountable cardinal, c the cardinality of the set of all real numbers. 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. Other terms and symbols that we define follow [9]. We make use of two of the cardinals defined in [8]. Define ωω as the set of all functions from ω to itself. For all f,g ∈ ωω, we say f ≤∗ g if and only if f(n) ≤ g(n) for all but finitely many n. The unbounding number, denoted by b, is the smallest cardinality of an unbounded subset of (ωω,≤∗). The dominating number, denoted by d, is the smallest cardinality of a cofinal subset of (ωω,≤∗). It is not difficult to show that ω1 ≤ b ≤ d ≤ c and it is known that ω1 < b = c, ω1 < d = c and ω1 ≤ b < d = c are all consistent with the axioms of ZFC (see [8] for details). A space X is said to be absolutely strongly star-Menger (ASSM) [6], if for each sequence (Un : n ∈ ω) of open covers of X and each dense subset D of X, there exists a sequence (Fn : n ∈ ω) of finite subsets of D such that {St(Fn,Un) : n ∈ ω} is an open cover of X. A space X is called star-Menger (SM) [15], if for each sequence (Un : n ∈ ω) of open covers of X there is a sequence (Vn : n ∈ ω) with Vn a finite subset of Un for each n ∈ ω, and {St(∪Vn,Un) : n ∈ ω} is a cover of X. A space X is called almost star-Menger (ASM) [14], if for each sequence (Un : n ∈ ω) of open covers of X there is a sequence (Vn : n ∈ ω) with Vn a finite subset of Un for each n ∈ ω, and {St(∪Vn,Un) : n ∈ ω} is a cover of X. Definition 2.1. A space X is called weakly strongly star-Menger (WSSM) if for each sequence (Un : n ∈ ω) of open covers of X, there is a sequence (Fn : n ∈ ω) of finite subsets of X such that ⋃ n∈ω St(Fn,Un) = X. From the above definitions we have the following diagram of implications: ACC ASSM ASM CC Starcompact SSM WSSM WSM SM T2−space T2−space P −space T2+P aracompact+P −space The purpose of this paper is to investigate the relationships of weakly strongly star-Menger spaces with other spaces. In Example 3.3, we have shown that the WSSM property need not be SSM 290 Gaurav Kumar & Brij K. Tyagi CUBO 23, 2 (2021) property. Presently, we do not know that the WSM property implies WSSM and ASM property. On the other hand, there are several examples in the literature on star-selection principles showing that other reverse implications need not be true. 3 Weakly strongly star-Menger spaces and related spaces In this section, we give some results and examples showing relationships of weakly strongly star- Menger with other properties. A subspace (subset) Y of a space X is WSSM if Y is WSSM as a subspace. Theorem 3.1. If X has a dense WSSM subspace, then X is WSSM. Proof. If D = X then we are done. Let D be a non-trivial dense WSSM subspace of X and (Un : n ∈ ω) be a sequence of open covers of X. Then (U ′ n : n ∈ ω) is a sequence of open covers of D, where U ′ n = {U ∩ D : U ∈ Un}. Therefore there exists a sequence (F ′ n : n ∈ ω) of finite subsets of D with ⋃ n∈ω St(F ′ n,U ′ n) = D. Hence ⋃ n∈ω St(F ′ n,Un) = X as D is dense in X. Corollary 3.2. Every separable topological space is WSSM. Given an almost disjoint family A of infinite subsets of ω (that is, the intersection of every two distinct elements of A is finite) the ψ-space or the Isbell-Mrówka space associated to A (denoted by ψ(A) has ω ∪ A as the underlying set, the points of ω being isolated, while the basic open neighborhoods of A ∈ A are of the form {A} ∪ (A\ F), where F ranges over all finite subsets of ω. For more details (see [3, 11]). Example 3.3. There exists a Tychonoff WSSM space X which is not SSM. Proof. Let X = ω ∪ A be the Isbell-Mrówka space, where A is the maximal almost disjoint family of infinite subsets of ω with |A| = c. Then X is not strongly star-Menger ([25, Example 2.3]). But X is WSSM, ω being a countable dense subset of X. Recall that a topological space X is a P-space [13] if every intersection of countably many open subsets of X is open. Proposition 3.4. A WSSM P-space X is almost star-Menger. Proof. Let (Un : n ∈ ω) be sequence of open covers of X. Then there exists a sequence (Fn : n ∈ ω) of finite subsets of X with ⋃ n∈ω St(Fn,Un) = X as X is WSSM. Since X is P-space, ⋃ n∈ω St(Fn,Un) is a closed subset of X which contains ⋃ n∈ω St(Fn,Un). Hence ⋃ n∈ω St(Fn,Un) = ⋃ n∈ω St(Fn,Un). Then we can find a sequence Vn of finite subsets of Un containing Fn such that ⋃ n∈ω St(∪Vn,Un) = X. CUBO 23, 2 (2021) Weakly strongly star-Menger spaces 291 Theorem 3.5. A Hausdorff paracompact weakly star-Menger P-space X is star-Menger. Proof. Let (Un : n ∈ ω) be a sequence of open covers of X. Since a Hausdorff paracompact space is regular, for each x ∈ X there exists an open neighborhood say Vn,x of x with Vn,x ⊆ U for some U ∈ Un. Let Vn be a locally finite open refinement of the open cover {Vn,x : x ∈ X}. Since X is WSM there exists a sequence (V ′ n : n ∈ ω) such that V ′ n is a finite subset of Vn for each n ∈ ω with ⋃ n∈ω St(∪V ′ n,Vn) = X. Now for each V ∈ Vn there is a UV ∈ Un with V ⊆ UV . Then for each fixed n ∈ ω, St(∪V ′ n,Vn) ⊆ St(∪U ′ n,Un), where U ′ n is finite subset of Un such that for every V ∈ V ′ n there is U ∈ U ′ n contaning V . Therefore, X = ⋃ n∈ω St(∪V ′ n,Vn) = ⋃ n∈ω St(∪V ′ n,Vn) = ⋃ n∈ω St(∪U ′ n,Un), because X is a P-space. In [15], Kočinac has shown that the property strongly star Menger is equivalent to the property star-Menger in Hausdorff paracompact space X. Then we have the following corollary: Corollary 3.6. For a Hausdorff paracompact P-space X, the following statements are equivalent: (1) X is strongly star-Menger; (2) X is weak strongly star-Menger; (3) X is almost star-Menger; (4) X is weakly star-Menger; (5) X is star-Menger. In [13], Kocev defined d-paracompact space. A space X is said to be d-paracompact if every dense family of subsets of X has a locally finite refinement. Theorem 3.7. A WSM and d-paracompact space X is almost star-Menger. Proof. Let (Un : n ∈ ω) be a sequence of open covers of X. As X is WSM, there exists a se- quence (Vn : n ∈ ω), where Vn is a finite subset of Un with ⋃ {St(∪Vn,Un) : n ∈ ω} dense in X. By the assumption {St(∪Vn,Un) : n ∈ ω} has a locally finite refinement say, W. Then ∪W = ⋃ n∈ω St(∪Vn,Un) and therefore ∪W = ⋃ n∈ω St(∪Vn,Un). As W is a locally finite fam- ily, we have ∪W = ⋃ W∈W W. Since each W ∈ W is contained in St(∪Vn,Un) for some n ∈ ω, ⋃ n∈ω St(∪Vn,Un) = X. Corollary 3.8. For a Hausdorff paracompact d-paracompact space X, the following statements are equivalent: (1) X is strongly star-Menger; (2) X is weak strongly star-Menger; 292 Gaurav Kumar & Brij K. Tyagi CUBO 23, 2 (2021) (3) X is weakly star-Menger; (4) X is almost star-Menger; (5) X is star-Menger. At the end of this section, we study the relation of WSSM property to Lindelöf covering properties. Recall, a space X is called Lindelöf if for each open cover U of X there is countable subset V of U such that X = ⋃ V. Let X be a space of the Example 4.1, then X is WSSM space but it is not Lindelöf, because X has a uncountable discrete closed subset. That means WSSM property does not imply Lindelöf property. Theorem 3.9. Every T2-paracompact WSSM space is Lindelöf. Proof. Let U be an open cover of X. For each x ∈ X there exists an open neighborhood say Vx of x such that Vx ⊆ U for some U ∈ U, because a T2-paracompact space is regular. Let V be a locally finite open refinement of the cover {Vx : x ∈ X}. Then (Vn : n ∈ ω) be a sequence of open covers of X, where Vn = V for each n ∈ ω. Since X is WSSM, there exists a sequence (Fn : n ∈ ω) of finite subsets of X such that ⋃ n∈ω St(Fn,Vn) = X. Since Vn is locally finite family, there exist finite subset V ′ n of Vn such that St(Fn,Vn) ⊂ ∪V ′ n, so X = ⋃ n∈ω St(Fn,Vn) ⊂ ⋃ n∈ω ∪{V ′ : V ′ ∈ V′n} = ⋃ n∈ω ∪{V ′ : V ′ ∈ V′n} = ⋃ n∈ω ∪{V ′ : V ′ ∈ V′n}. For each V ∈ Vn there is a UV ∈ U with V ⊆ UV . Hence we can constuct a countable subset U′ of U such that ⋃ n∈ω ∪{V ′ : V ′ ∈ V′n} ⊂ ⋃ U′. Definition 3.10. [7] A space X is called strongly star-Lindelöf (in short, SSL) if for each open cover U of X there is a countable subset F of X such that St(F,U) = X. Clearly, SSM property implies SSL property. But next we will show that WSSM property need not be SSL property. A space X is almost star countable [28], if for each open cover U of X there exists a countable subset F of X such that ⋃ x∈F St(x,U) = X. Evidently, strongly star-Lindelöf ⇒ almost star-countable. Example 3.11. A WSSM space need not be SSL. Proof. Let D be a discrete space of cardinality ω1, X = (βD × (ω + 1)) \ ((βD \ D) × {ω}) is a subspace of the product space βD × (ω + 1). Then X is WSSM by Lemma 4.4., because βD × ω is a dense σ-compact (hence, σ-countably compact) subset of X. But X is not SSL, because X is not almost star countable (see [28, Example 2.5]). CUBO 23, 2 (2021) Weakly strongly star-Menger spaces 293 Theorem 3.12. Every T2-paracompact WSSM space is SSL. Proof. The proof follows the same constructions of Theorem 3.9, thus omitted. 4 Subspaces and product spaces In this section we study subspaces of a WSSM space and also show that product of two WSSM spaces need not be WSSM. For some relative version of star selection principles see ([4, 5, 19]). Example 4.1. A closed subset of WSSM space need not be WSSM. Proof. Let R be the set of real numbers, I the set of irrational numbers and Q the set of rational numbers. For each irrational x we choose a sequence {xi : i ∈ ω} of rational numbers converging to x in the Euclidean topology. The rational sequence topology τ (see [29, Example 65]) is then defined by declaring each rational open and selecting the sets Un(x) = {xn,i : i ∈ ω} ∪ {x} as a basis for the irrational point x. Then the set of irrational points I is a closed subset of (R,τ) and I as a subspace of the space (R,τ) is not WSSM, because it is uncountable discrete subspace. On the other hand, (R,τ) is WSSM, because Q is dense in (R,τ). Proposition 4.2. Every clopen subset of a WSSM space is WSSM. Proof. Let Y be a clopen subset of a WSSM space X and let (Un : n ∈ ω) be a sequence of open covers of Y. Then (Vn : n ∈ ω), where Vn = Un ∪ {X \ Y } is a sequence of open covers of X. Since X is WSSM, there exists a sequence of finite subsets Fn of X with ⋃ n∈ω St(Fn,Vn) = X. Put F ′ n = Y ∩ Fn. Then clearly, ⋃ n∈ω St(F ′ n,Un) = Y. Song [25] gave an example showing that the product of two countably compact spaces is not strongly star compact. This example also shows that the product of two WSSM spaces need not be WSSM. We sketch it below. Example 4.3. There exist two countably compact (and hence WSSM) spaces X and Y such that X × Y is not WSSM. Proof. Let D be the discrete space of the cardinality c. We 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 three conditions: (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). 294 Gaurav Kumar & Brij K. Tyagi CUBO 23, 2 (2021) These sets Eα and Fα are well-defined since every infinite closed set in β(D) has the cardinality 2c, for detail see [30]. Then, X ×Y is not WSSM since the diagonal {< d,d >: d ∈ D} is a discrete open and closed subset of X × Y with the cardinality c and the property WSSM is preserved by open and closed subsets. Hence product of WSSM spaces need not be WSSM. Now we give some positive results. Recall that a subset A of a space X is said to be σ-countably compact if it is union of countably many countably compact subset of X. Lemma 4.4. If a space X has a σ-countably compact dense subset, then X is WSSM. Proof. Let D = ⋃ n∈ω Dn be a dense subset of X, where each Dn is countably compact subset of X. Let (Un : n ∈ ω) be a sequence of open covers of X. Then for each n ∈ ω there exists a finite subset say Fn of Dn such that Dn ⊂ St(Fn,Un). So (Fn : n ∈ ω) is a sequence of finite subsets of D such that D = ⋃ n∈ω St(Fn,Un). Theorem 4.5. Let {Xn : n ∈ ω} be a countable collection of countably compact subsets of space X such that X = ⋃ n∈ω Xn. Then X is a WSSM space. Corollary 4.6. If {Xn : n ∈ ω} is a countable collection of mutually disjoint countably compact spaces, then the topological sum ⊕ n∈ω Xn is WSSM if and only if each Xn is WSSM. Corollary 4.7. If X is countably compact space, then X × ω, where ω has discrete topology is WSSM. Proof. Since X × ω is homeomorphic to ⊕ n∈ω X × {n} and X × {n} is homeomorphic to X for each n ∈ ω. Then, by Corollary 4.6, ⊕ n∈ω X × {n} is WSSM. 5 Images and Preimages In this section we study the images and preimages of WSSM spaces under continuous maps. Theorem 5.1. A continuous image of a WSSM space is WSSM. Proof. Let f : X → Y be a continuous surjection and X be a WSSM space. Let (Un : n ∈ ω) be sequence of open covers of Y . Then (U ′ n : n ∈ ω), where U ′ n = {f −1(U) : U ∈ Un} is a sequence of open covers of X. Thus there exists a sequence (F ′ n : n ∈ ω) of finite subsets of X such that ⋃ n∈ω St(F ′ n,U ′ n) = X. Let Fn = f(F ′ n). Then (Fn : n ∈ ω) is a sequence of finite subsets of Y. Hence result follows from the fact that for an arbitrary y ∈ Y and each neighbourhood U of y, U ⋂ ⋃ n∈ω St(Fn,Un) 6= φ. CUBO 23, 2 (2021) Weakly strongly star-Menger spaces 295 Next we turn to consider preimages. We show that the preimage of a WSSM space under a closed 2-to-1 continuous map need not be WSSM. First we discuss examples. Recall the Alexandroff duplicate A(X) of a space X. The underlying set A(X) is X × {0,1}; each point of X × {1} is isolated and a basic neighborhood of < x,0 > ∈ X ×{0} is a set of the form (U × {0}) ∪ ((U × {1}) \ {< x,1 >}), where U is a neighborhood of x in X. Example 5.2. Assuming d = c, there exists a WSSM space X such that A(X) is not WSSM. Proof. Assume that d = c. Let X = ω ∪ A be the Isbell–Mrówka space with |A| = ω1. Then X is absolutely strongly star-Menger ([27, Example 3.5]), and hence WSSM. However A(X) is not WSSM. Since the set A × {1} is an open and closed subset of A(X) with |A × {1}| = ω1, and for each a ∈ A, the point < a,1 > is isolated in A(X) . Hence A(X) is not WSSM, since every open and closed subset of a WSSM space is WSSM, and A × {1} is not WSSM . Example 5.3. Assuming d = c, there exists a closed 2-to-1 continuous map f : X → Y such that Y is a WSSM space, but X is not a WSSM. Proof. Let Y be the space X of Example 5.2. Then Y is WSSM. Let X be the space A(Y ). Then X is not WSSM. Let f : X → Y be the projection. Then f is a closed 2-to-1 continuous map, which completes the proof. Theorem 5.4. Let f be an open and closed, finite-to-one continuous map from a space X onto a WSSM space Y. Then X is WSSM. Proof. Let (Un : n ∈ ω) be a sequence of open covers of X and let y ∈ Y. Since f −1(y) is finite, for each n ∈ ω there exists a finite sub-collection Uny of Un such that f −1(y) ⊂ ∪Uny and U ∩ f−1(y) 6= φ for each U ∈ Uny . Since f is closed, there exists an open neighbourhood Vny of y in Y such that f −1(Vny ) ⊆ ∪{U : U ∈ Uny }. Since f is open, we can assume that Vny ⊆ ∩{f(U) : U ∈ Uny }. For each n ∈ ω, take such open set Vny for each y ∈ Y, and put Vn = {Vny : y ∈ Y } of Y . Thus (Vn : n ∈ ω) is a sequence of open covers of Y. Since Y is WSSM, there exists a sequence (Fn : n ∈ ω) of finite subsets of Y such that ⋃ n∈ω St(Fn,Vn) = Y . Since f is finite to one, the sequence (f−1(Fn) : n ∈ ω) is a sequence of finite subsets of X. We show that ⋃ n∈ω St(f−1(Fn),Un) = X. Let x ∈ X and V be an arbitrary neighbourhood of x in X, then f(V ) is a neighbourhood of y = f(x) as f is an open map. Then there exist n ∈ ω and y′ ∈ Y such that y ∈ f(V ) ∩Vn y′ with Vn y′ ∩Fn 6= φ. Choose U ∈ Un y′ . Then Vn y′ ⊆ f(U). Hence U ∩f−1(Fn) 6= φ as Vn y′ ∩ Fn 6= φ. Therefore, x ∈ ⋃ n∈ω St(f−1(Fn),Un). This shows that X is WSSM. 296 Gaurav Kumar & Brij K. Tyagi CUBO 23, 2 (2021) Acknowledgements (1) The first author acknowledges the fellowship grant of University Grant Commission, India. (2) The authors would like to thank referees for their valuable suggestions which led to improve- ments of the paper in several places. CUBO 23, 2 (2021) Weakly strongly star-Menger spaces 297 References [1] M. Bonanzinga, F. Cammaroto and Lj. D. R. Kočinac, “Star-Hurewicz and related properties”, Appl. Gen. Topol., vol. 5, no. 1, pp. 79-89, 2004. [2] M. Bonanzinga, F. Cammaroto, Lj. D. R. Kočinac and M. V. 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