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 correct a flawed argument in a previous paper which attempted to show that a Pawlikowski theorem is true for $k$-covers.
Let X be a Tychonoff space and MC(X) be the space of convex minimal usco maps with values in R, the space of real numbers. Such set-valued maps are important in the study of subdifferentials of convex functions. Using the strong Choquet game we prove complete metrizability of MC(X) with the upper Vietoris topology. If X is normal, elements of MC(X) can be approximated in the Vietoris topology by continuous functions. We also study first countability, second countability and other properties of the upper Vietoris topology on MC(X).
We consider a version of the open-open game, indicating its connections with universally Kuratowski-Ulam spaces. We show that: Every I-favorable space is universally Kuratowski-Ulam, (Theorem 8); If a compact space Y is I-favorable, then the hyperspace exp(Y) with the Vietoris topology is I-favorable, and hence universally Kuratowski-Ulam, (Theorems 6 and 9). Notions of uK-U and uK-U* spaces are compared.
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 game…
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.
In this note we relate some selection principles to metrisability and separability of a manifold. In particular we show that $\sf{S}_{fin}(\mathcal K,\mathcal O)$, $\sf{S}_{fin}(Ω,Ω)$ and $\sf{S}_{fin}(Λ,Λ)$ are each equivalent to metrisability for a manifold, while $\sf{S}_1(\mathcal D,\mathcal D)$ is equivalent to separability for a manifold.
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…
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 $(C(X), τ_k, τ_p)$.
We present a selection principle $S_1(\mathcal{O},\mathcal{H})$ that characterizes the $G_δ$-diagonal property. We also present a topological game induced by this selection principle and we study the relations between this game and the $G_δ$-property. Finally, we give some applications and examples.
We investigate game-theoretic properties of selection principles related to weaker forms of the Menger and Rothberger properties. For appropriate spaces some of these selection principles are characterized in terms of a corresponding game. We use generic extensions by Cohen reals to illustrate the necessity of some of the hypotheses in our theorems.
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 $G_1(Ω_x,Ω_x)$.
Arhangel'skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most $2^{\aleph_0}$. Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are ${\sf G}_δ$ has been more elusive. In this paper we continue the agenda started in F.D. 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. Infinite games and selection principles, especially the Rothberger property, a…
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.
For a topological space $X$ and a point $x \in X$, consider the following game -- related to the property of $X$ being countably tight at $x$. In each inning $n\inω$, the first player chooses a set $A_n$ that clusters at $x$, and then the second player picks a point $a_n\in A_n$; the second player is the winner if and only if $x\in\overline{\{a_n:n\inω\}}$. In this work, we study variations of this game in which the second player is allowed to choose finitely many points per inning rather than one, but in which the number of points they are allowed to choose in each inning has been fixed in ad…
