In this paper we study connections between topological games such as Rothberger, Menger and compact-open, and relate these games to properties involving covers by G_δ subsets. The results include: (1) If Two has a winning strategy in the Menger game on a regular space X, then X is an Alster space. (2) If Two has a winning strategy in the Rothberger game on a topological space X, then the G_δ-topology on X is Lindelof. (3) The Menger game and the compact-open game are (consistently) not dual.
No convenient internal characterization of spaces that are productively Lindelof is known. Perhaps the best general result known is Alster's internal characterization, under the Continuum Hypothesis, of productively Lindelof spaces which have a basis of cardinality at most $\aleph_1$. It turns out that topological spaces having Alster's property are also productively weakly Lindelof. The weakly Lindelof spaces form a much larger class of spaces than the Lindelof spaces. In many instances spaces having Alster's property satisfy a seemingly stronger version of Alster's property and consequently …
In this paper, we defined two new games - the mildly Menger game and the compact-clopen game. In a zero-dimensional space, the Menger game is equivalent to the mildly Menger game and the compact-open game is equivalent to the compact-clopen game. An example is given for a space on which the mildly Menger game is undetermined. Also we introduced a new game namely K-quasi-component-clopen game and proved that this game is equivalent to the compact-clopen game. Then we proved that if a topological space is a union of countably many quasi-components of compact sets, then TWO has a winning strategy…
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.
In this paper, we continue the study of function spaces equipped with topologies of (strong) uniform convergence on bornologies initiated by Beer and Levi \cite{beer-levi:09}. In particular, we investigate some topological properties these function spaces defined by topological games. In addition, we also give further characterizations of metrizability and completeness properties of these function spaces.
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.
We introduce a class of $β-v$-unfavorable spaces, which contains some known classes of $β$-unfavorable spaces for topological games of Choquet type. It is proved that every $β-v$-unfavorable space $X$ is a Namioka space, that is for any compact space $Y$ and any separately continuous function $f:X\times Y\to \mathbb R$ there exists a dense in $X$ $G_δ$-set $A\subseteq X$ such that $f$ is jointly continuous at each point of $A\times Y$.
The two main results of this work are the following: if a space $X$ is such that player II has a winning strategy in the game $\gone(Ω_x, Ω_x)$ for every $x \in X$, then $X$ is productively countably tight. On the other hand, if a space is productively countably tight, then $\sone(Ω_x, Ω_x)$ holds for every $x \in X$. With these results, several other results follow, using some characterizations made by Uspenskii and Scheepers.
Inspired by work of Scheepers and Tall, we use properties defined by topological games to provide bounds for the cardinality of topological spaces. We obtain a partial answer to an old question of Bell, Ginsburg and Woods regarding the cardinality of weakly Lindelöf first-countable regular spaces and answer a question recently asked by Babinkostova, Pansera and Scheepers. In the second part of the paper we study a game-theoretic version of cellularity, a special case of which has been introduced by Aurichi. We obtain a game-theoretic proof of Shapirovskii's bound for the number of regular open…
This article is a continuation of our investigations in the function space $C(X)$ with respect to the topology $τ^s_\mathfrak{B}$ of strong uniform convergence on $\mathfrak{B}$ in line of (Chandra et al. 2020 \cite{dcpdsd} and Das et al. 2022 \cite{pddcsd-3}) using the idea of strong uniform convergence (Beer and Levi, 2009 \cite{bl}) on a bornology. First we focus on the notion of tightness property of $(C(X),τ^s_\mathfrak{B})$ and some of its variations such as supertightness, Id-fan tightness and $T$-tightness. Certain situations are discussed when $C(X)$ is a {\rm k}-space with respect to…
An open question of Gruenhage asks if all strategically selectively separable spaces are Markov selectively separable, a game-theoretic statement known to hold for countable spaces. As a corollary of a result by Berner and Juh$\acute{a}$sz, we note that the strong version of this statement, where the second player is restricted to selecting single points in the rather than finite subsets, holds for all $T_3$ spaces without isolated points. Continuing this investigation, we also consider games related to selective sequential separability, and demonstrate results analogous to those for selective…
We give topological and game theoretic definitions and theorems nec- essary for defining a Banach-Mazur game, and apply these definitions to formalize the game. We then state and prove two theorems which give necessary conditions for existence of winning strategies for players in a Banach-Mazur game.
We present a unified approach, based on dominating families in binary relations, for the study of topological properties defined in terms of selection principles and the games associated to them.
In this note we study the open-point topological games in order to analyze the least upper bound for density of dense subsets of a topological space. This way we may also analyze the behavior of such cardinal invariants in taking products of spaces. Various related cardinal equalities and inequalities are given. As an application we take a look at Banach spaces with the property (CSP) which can be formulated by stating that each weak-star dense linear subspace of the dual is weak-star separable.
We apply the theory of infinite two-person games to two well-known problems in topology: Suslin's Problem and Arhangel'skii's problem on $G_δ$ covers of compact spaces. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable and 2) in every compact space satisfying the game-theoretic version of the weak Lindelöf property, every cover by $G_δ$ sets has a continuum-sized subcollection whose union is $G_δ$-dense.
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…
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 provide conceptual proofs of the two most fundamental theorems concerning topological games and open covers: Hurewicz's Theorem concerning the Menger game, and Pawlikowski's Theorem concerning the Rothberger game.
This work presents proofs for Hurewicz's and Pawlikowski's Theorems about topological games in the context of lattices, in which one cannot argue with points.
The Game of Cycles, introduced by Su (2020), is played on a simple connected planar graph together with its bounded cells, and players take turns marking edges with arrows according to a sink-source rule that gives the game a topological flavor. The object of the game is to produce a cycle cell---a cell surrounded by arrows all cycling in one direction---or to make the last possible move. We analyze the two-player game for various classes of graphs and determine who has a winning strategy. We also establish a topological property of the game: that a board with every edge marked must have a cyc…
The aim of this note is to investigate the open-open game of uncountable length. We introduce a cardinal number $μ(X)$, which says how long the Player I has to play to ensure a victory. It is proved that $\su(X)\leqμ(X)\leq\su(X)^+$. We also introduce the class $\mathcal C_κ$ of topological spaces that can be represented as the inverse limit of $κ$-complete system $\{X_σ,π^σ_ρ,Σ\}$ with $\w(X_σ)\leqκ$ and skeletal bonding maps. It is shown that product of spaces which belong to $\mathcal C_κ$ also belongs to this class and $μ(X)\leqκ$ whenever $X\in\mathcal C_κ$ .
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).
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…
Let S be a topological property of sequences (such as, for example, "to contain a convergent subsequence" or "to have an accumulation point"). We introduce the following open-point game OP(X,S) on a topological space X. In the n'th move, Player A chooses a non-empty open subet U_n of X, and Player B responds by selecting a point x_n in U_n. Player B wins the game if the sequence (x_n) satisfies property S in X; otherwise, Player A wins. The (non-)existence of regular or stationary winning strategies in OP(X,S) for both players defines new compactness properties of the underlying space X. We th…
We show that: 1. Rothberger bounded subgroups of sigma-compact groups are characterized by Ramseyan partition relations. 2. For each uncountable cardinal $κ$ there is a ${\sf T}_0$ topological group of cardinality $κ$ such that ONE has a winning strategy in the point-open game on the group and the group is not a subspace of any sigma-compact space. 3. For each uncountable cardinal $κ$ there is a ${\sf T}_0$ topological group of cardinality $κ$ such that ONE has a winning strategy in the point-open game on the group and the group is σ-compact.
Sprouts is a two-player topological game, invented in 1967 by Michael Paterson and John Conway. The game starts with p spots drawn on a sheet of paper, and lasts at most 3p-1 moves: the player who makes the last move wins. Sprouts is a very intricate game and the best known manual analysis only achieved to find a winning strategy up to p=7 spots. Recent computer analysis reached up to p=32. The standard game is played on a plane, or equivalently on a sphere. In this article, we generalize and study the game on any compact surface. First, we describe the possible moves on a compact surface, and…
