Abstract Given a topological property P $\mathcal{P}$ , a space X is called star- P $\mathcal{P}$ if for any open cover U $\mathcal{U}$ of the space X , there exists a set Y ⊆ X satisfying the property P $\mathcal{P}$ such that S t ( Y , U ) = X $\mathrm{S}\mathrm{t}(Y,\mathcal{U})=X$ ; the set Y is called a star kernel of the cover U $\mathcal{U}$ . In (J. Casas-de la Rosa and Á. Tamariz-Mascarúa, “On spaces with star kernel Menger,” Acta Math. Hung. , vol. 170, pp. 379–404, 2023), the authors introduced and studied spaces with a star kernel Menger. Motivated by this idea, in this paper we introduced several properties obtained by requiring certain kernel with Menger property. Namely, the dually Menger spaces and the cellular Menger spaces as well as some of the almost and weak versions of them. Some examples are given to show the relationship among these properties. Additionally, we give some results on the Pixley–Roy hyperspace that involve some of these properties. Finally, we make some comments on a Song’s example given in (Y. K. Song, “A pseudocompact Tychonoff space that is not star Lindelöf,” Bull. Aust. Math. Soc. , vol. 84, pp. 452–454, 2011).

In this paper, we study two classes of topological spaces: cellular-almost Lindelöf spaces and cellular-weakly Lindelöf spaces. We prove that the classes of cellular-Lindelöf, cellular weakly Lindelöf and cellular-almost Lindelöf are distinct. In addition, we present a comparative study of these classes. We also establish some cardinality results. In particular, we prove that, under the assumption of 2

In this paper we introduce the almost strongly star-Menger property and we provide some results and relationships with another known properties in literature. Furthermore, we characterize the almost strongly star-Menger and the almost star-Menger properties in hyperspaces endowed with the hit-and-miss topology $\tau_\Delta$, by using two new selection principles $\mathbf{S}_{\textsf{fin}}^{\star}(\Pi_{\Delta}(\Lambda), \mathscr{B})$ and $\mathbf{SS}_{\textsf{fin}}^{\star}(\Pi_{\Delta}(\Lambda), \mathscr{B})$, for an appropriate family $\mathscr{B}$.

In this article we have a couple of mains. The first one is to make a generic exposure on the classes that are obtained by using neighborhood assignments or applying the star operator to a given topological property P. The second one is to perform an analysis on the classes defined in the first part of the work, for the numerability property. It is important to note that in Example 3.13 we present a Hausdorff weakly star countable space which is not feebly Lindelof, which gives a negative response to the Question 3.14 from [1]. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data

In this paper we provide some general results about topological spaces X which satisfy starn- P , weakly starn- P and almost starn- P, for P∈ { κ ᴄᴄ , W κ L , ᴅ κ ᴄᴄ } , where κ is an infinite cardinal number. The particular cases when κ = ω , P ∈ { ᴄᴄᴄ , weakly Lindelöf , ᴅ ᴄᴄᴄ } are obtained. Furthermore, for the same classes of spaces defined by such P, by applying Erdős-Radó's theorem and using the rank l-diagonal notion, we establish some cardinal inequalities.

In this paper, we introduce and analyze the selection principles weakly (almost) R-star-Lindelof and weakly (almost) M-star-Lindelof, which turn out to be weaker versions of the R-star-Lindelof and M-star-Lindelof properties, respectively, and we provide some relationships with other known properties in literature. Also, we introduce the selection principles SS Pi triangle(A),1 star(C-triangle(A), ) and SS Pi triangle(A)star,fin(C-triangle(A), ) to characterize the properties weakly (almost) R-star-Lindelof and weakly (almost) M-star-Lindelof in the hyperspace (A, tau(+)(triangle)), respectively.

We introduce the selection principles H-star-Lindelöf, weakly H-star-Lindelöf and almost H-star-Lindelöf and we provide some propositions and relationships with other known properties in literature. Furthermore, in this paper we characterize the above selection principles in hyperspaces endowed with the hit-and-miss topology τΔ, by using a new property SSΠΔ(Λ),fin⋆(CΔ(Λ),B), for the corresponding choose of the family B.

In this paper, we introduce the selection principles wSS* 1(??(?),C?(?)), wSS* fin(??(?),C?(?)), wS* 1(??(?),C?(?)) and wS* fin(??(?),C?(?)) to characterize the properties of weakly strong-star Rothberger (Menger) and weakly star-Rothberger (Menger) in the hyperspace (?, ?+ ?), respectively. Furthermore, we introduce the notions H(C?(?)) and Ifin(C?(?),C?(?)) to characterize, respectively, the H-separability and the principle Ufin(D,D), in the same hyperspace.

In this paper, we introduce the notions of almost ??(?)-network and weakly ??(?)-network to characterize the properties of almost Rothberger (Menger) and weakly Rothberger (Menger), respectively, in the hyperspaces CL(X),K(X), F(X) and CS(X), endowed with the hit-and-miss topology. Also, we introduce the concepts of groupable c?(?)-cover and weakly (?,?)-groupable cover of X to give equivalences of the selection principles S1(D,D1p), Sfin(D,D1p), S1(D,Dw1p) and Sfin(D,Dw1p) in the same hyperspaces.

In this paper we continue the study of the characterization of selection principles in the hyperspaces CL(X), K(X), F(X) and CS(X), endowed with the hit-and-miss topology, by using ??(?)-networks and c?(?)-covers of a topological space X. Specifically, we prove theorems which characterize the covering properties Hurewicz, strongly star Hurewicz, star Hurewicz and absolutely strongly star Hurewicz in these hyperspaces.

In this we characterize selection properties of the Rothberger and Lindelof type in the hyperspaces CL(X), K (X), F (X) and CS (X), endowed with the hit-and-miss topology. To do so, we introduce several generic selection properties.

In this paper, we introduce the generic notions of c(Delta) (Lambda)-cover of Y (subspace of X), which generalizes the kF-cover and cV-cover of Y, defined by Li in [13]. In some others, we use this notion to characterize the countable fan tightness and the countable strong fan tightness in the hyperspaces CL(X), K(X), F(X) and CS(X) endowed with hit-and-miss topology. Also, we introduce the notions of Delta(gamma)-cover and Delta(gamma)-set, which permits to characterize properties like Frechet-Urysohn, strongly Frechet-Urysohn and sequential denseness in the hyperspaces before mentioned. We also use the concept of c(Delta)(Lambda)-covers to characterize tightness, set-tightness and T-tightness of the hyperspace (Lambda, Tau(+)(Delta)). (C) 2021 Elsevier B.V. All rights reserved.

In this paper, we introduce the generic notions of cΔ(Λ)-cover of Y (subspace of X), which generalizes the kF-cover and cV-cover of Y, defined by Li in [13]. In some others, we use this notion to characterize the countable fan tightness and the countable strong fan tightness in the hyperspaces CL(X), K(X), F(X) and CS(X) endowed with hit-and-miss topology. Also, we introduce the notions of Δγ-cover and Δγ-set, which permits to characterize properties like Frećhet-Urysohn, strongly Frećhet-Urysohn and sequential denseness in the hyperspaces before mentioned. We also use the concept of cΔ(Λ)-covers to characterize tightness, set-tightness and T-tightness of the hyperspace (Λ,τΔ+).

We employ the notion of pF (?)-network to define the combina-torial principles FELLM(?(F)(?), ?(F)(?)) and FELL*(M)(?(F)(?)), ?(F)(?)), which will be applied to characterize the spaces X whose hyperspace, endowed with the Fell topology, satisfies the SSM condition and the SM condition. We use the selection principle SS*(?) (O, O) to characterize the SSM property for the spaces K(X ), F(X) and [X](1), endowed with the lower Vietoris topology. Finally, we use the notion of ?-moving-off family, which generalizes the one of moving-off family, and we use it to characterize the Menger property for certain subspaces of CL(X).

In this paper we characterize the Rothberger property and the selection principles star-Rothberger and strongly star-Rothberger in the hyperspaces CL(X), K(X), F(X) and CS(X), endowed with the Vietoris topology. To characterize the corresponding principles type star, we introduce a couple of technical selection principles, which we have denoted by SΠV(ΠV(Λ),ΠV(Λ)) and SΠV⁎(ΠV(Λ),ΠV(Λ)). Also, we give an equivalence of the selection principle S1(D,D) in the same hyperspaces, by using cV-covers of a space.

In this paper, we introduce some selection principles, which are motivated by star and strong star Rothberger and star and strong star Menger principles. Theorems to characterize the star and strong star Rothberger-type properties in spaces using the family CΔ(Λ) of cΔ(Λ)-covers of a space X and the family D of dense subsets in hit-and-miss hyperspaces are proved. In a similar way are presented the versions to characterize the star and strong star Menger-type properties in spaces using again the families CΔ(Λ) and D in the respective spaces. Finally, we provide some relationships between the selection principles defined along the paper.

In this paper we introduce a couple of general technical results which are tailored to prove some cardinal functions inequalities related to the Arhangel'skiis inequality, many of these obtained in recent years. Fur-thermore, we define the non --y number of a space X, denoted by n-y(X), to establish, using one of our generic theorems (Theorem 3.1), that if X is a T1 -space such that n-y(X) < 2(7,6)-wL theta(X)chi(X), then |X| < 2(7,6)-wL theta(X)chi(X). This result gives a partial positive answer to the question posed in [14], namely, is |X| < 2(7,6)-wL theta(X)chi(X) if X is Urysohn?

In this paper we characterize the Menger property and the selection principles star-Menger and strongly star-Menger in the hyperspaces CL(X), K(X), F(X) and CS(X), endowed with the hit-and-miss topology. To characterize the corresponding principles type star, we introduce a couple of technical selection principles, which we have denoted by SM(ΠΔ(Γ),ΠΔ(Γ)) and SM⁎(ΠΔ(Γ),ΠΔ(Γ)). Also, we give an equivalence of the selection principle Sfin(D,D) in the same hyperspaces, by using cV-covers of a space.

We introduce the notion of πF(Δ)–network and denote the combinatorial principles FELL(ΠF(Δ),ΠF(Δ)) and FELL⁎(ΠF(Δ),ΠF(Δ)), which will be applied to characterize the spaces X whose hyperspace, endowed with the Fell topology, satisfies the SSR condition and the SR condition. We use the selection principle SSΔ⁎(O,O) to characterize the SSR property for the spaces K(X), F(X) and [X]1, endowed with the lower Vietoris topology. In addition, we introduce the selection principle SΔf(O,O) to obtain characterizations of the selective versions of metacompactness, mesocompactness and sequential mesocompactness. Finally, we introduce the notion of Δ-moving-off family, which generalizes the one of moving-off family, and we use it to characterize the Rothberger property for certain subspaces of CL(X).

In this paper we use some cardinal functions introduced in [3], [5] and [1], and the technique of elementary submodels to generalize some cardinal inequalities. Also, we introduce the cardinal functions, denoted by (γ,θ)−wLθ and Uθ⁎, and we prove that: |X|≤2(γ,θ)−wLθ(X)χ(X), for any T1-space X with Uθ⁎(X)≤ω.