In this paper we connect selection principles on a topological space to corresponding selection principles on one of its hyperspaces. We unify techniques and generalize theorems from the known results about selection principles for common hyperspace constructions. This includes results of Lj.D.R. Kočinac, Z. Li, and others. We use selection games to generalize selection principles and we work with strategies of various strengths for these games. The selection games we work with are primarily abstract versions of the selection principles of Rothberger, Menger, and Hurewicz type, as well as games of countable fan tightness and selective separability. The hyperspace constructions that we work with are the Vietoris and Fell topologies, both upper and full, generated by ideals of closed sets. Using a new technique we are able to extend straightforward connections between topological constructs to connections between selection games related to those constructs. This extension process works regardless of the length of the game, the kind of selection being performed, or the strength of the strategy being considered.
In this paper we define some combinatorial principles to characterize spaces $X$ whose hyperspace satisfies some variation of some classical star selection principle. Specifically, the variations characterized are the selective and absolute versions of the star selection principles for the Menger and Rothberger cases; also, the hyperspaces considered in these characterizations are $CL(X)$, $\mathbb{K}(X)$, $\mathbb{F}(X)$ and $\mathbb{CS}(X)$ in both cases, endowed with either the Fell topology or the Vietoris topology.
We explore the connections between selection games on Hausdorff spaces and their corresponding Vietoris space of compact subsets. These considerations offer a similar relationship as the well-known relationship between $ω$-covers of $X$ and regular open covers of the finite powers of $X$. The primary utility of this method is to establish similar relationships with $k$-covers and the Vietoris space of compact subsets. Particularly, we show that some commonly studied selection principles are equivalent to a related hyperspace being Menger or Rothberger. We then apply these equivalences to corre…
In 2017, Tkachuk isolated the closed discrete selection property while working on problems related to function spaces [15]. In this paper we will study the closed discrete selection property and the related games and strategies on $C_k(X)$. Clontz and Holshouser showed previously that the closed discrete selection game on $C_p(X)$ is equivalent to a modification of the point-open game on $X$. In this paper we show that the closed discrete selection game on $C_k(X)$ is equivalent to a modification of the compact-open game on $X$. We also connect discrete selection properties on $C_k(X)$ to a va…
Lindelof indestructibility, topological games and selection principles
Arhangel'skii证明了如果第一个可数T2空间是Lindelöf，那么它的基数至多是$2^{\aleph_0}$。在点为${\sf G}_δ$的更大类空间中，Lindelöf空间的这种干净上界更难找到。在本文中，我们继续在F.D.开始的议程。Tall，On the cardinality of Lindelöf spaces with points $G_δ$，Topology and its Applications 63（1995），21 - 38，of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property.无限对策和选择原理，特别是Rothberger性质，是我们研究的重要工具
We relate the property of discrete selectivity and its corresponding game, both recently introduced by V.V. Tkachuck, to a variety of selection principles and point picking games. In particular we show that player II can win the discrete selection game on \(C_p(X)\) if and only if player II can win a variant of the point open game on \(X\). We also show that the existence of limited information strategies in the discrete selection game on \(C_p(X)\) for either player are equivalent to other well-known topological properties.
Arhangelskii's properties $α_2$ and $α_4$ defined for convergent sequences may be characterized in terms of Scheeper's selection principles. We generalize these results to hold for more general collections and consider these results in terms of selection games.
Often, a given selection game studied in the literature has a known dual game. In dual games, a winning strategy for a player in either game may be used to create a winning strategy for the opponent in the dual. For example, the Rothberger selection game involving open covers is dual to the point-open game. This extends to a general theorem: if $\{\operatorname{ran}{f}:f\in\mathbf C(\mathcal R)\}$ is coinitial in $\mathcal A$ with respect to $\subseteq$, where $\mathbf C(\mathcal R)=\{f\in(\bigcup\mathcal R)^{\mathcal R}:R\in\mathcal R\Rightarrow f(R)\in R\}$ collects the choice functions on t…
We present a new variation of the classical selection principles $\mathsf{S}_\mathrm{k}(\mathcal A, \mathcal B)$ ($k\in\mathbb N$) and $\mathsf{S}_\mathrm{fin}(\mathcal A, \mathcal B)$ that formally lies between these two properties. As in the case of the classical selection principles, we also obtain a new variation of topological games and discuss how new topological properties may emerge in the specific cases of covering and tightness.
In this paper we study the selection principle of closed discrete selection, first researched by Tkachuk in [13] and strengthened by Clontz, Holshouser in [3], in set-open topologies on the space of continuous real-valued functions. Adapting the techniques involving point-picking games on \(X\) and \(C_p(X)\), the current authors showed similar equivalences in [1] involving the compact subsets of \(X\) and \(C_k(X)\). By pursuing a bitopological setting, we have touched upon a unifying framework which involves three basic techniques: general game duality via reflections (Clontz), general game …
Two selection games from the literature, $G_c(\mathcal O,\mathcal O)$ and $G_1(\mathcal O_{zd},\mathcal O)$, are known to characterize countable dimension among certain spaces. This paper studies their perfect- and limited-information strategies, and investigates issues related to non-equivalent characterizations of zero-dimensionality for spaces that are not both separable and metrizable. To relate results on zero-dimensional and finite-dimensional spaces, a generalization of Telgársky's proof that the point-open and finite-open games are equivalent is demonstrated.
For a Tychonoff space $X$, we denote by $(C(X), τ_k, τ_p)$ the bitopological space of all real-valued continuous functions on $X$ where $τ_k$ is the compact-open topology and $τ_p$ is the topology of pointwise convergence. In papers [5, 6, 13] variations of selective separability and tightness in $(C(X), τ_k, τ_p)$ were investigated. In this paper we continued to study the selective properties and the corresponding topological games in the space $(C(X), τ_k, τ_p)$.
This work presents the selection principle $S_1^*(τ_x,CD)$ that characterizes $q$-points. We also discuss the induced topological game $G_1^*(τ_x,CD)$ and its relations with $W$-points and $\widetilde{W}$-points, as well as with the game $G_1(Ω_x,Ω_x)$.
