Topological games and product spaces

Salvador García-Ferreira; R. A. González-Silva; Artur Hideyuki Tomita

Displaying similar documents to “Topological games and product spaces”

Infinite $G_{δ}$ -games with imperfect information

D. Blackwell (1969)

Applicationes Mathematicae

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More on the Ehrenfeucht-Fraisse game of length ω₁

Tapani Hyttinen, Saharon Shelah, Jouko Vaananen (2002)

Fundamenta Mathematicae

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By results of [9] there are models and for which the Ehrenfeucht-Fraïssé game of length ω₁, $E F G_{ω ₁} (,)$ , is non-determined, but it is consistent relative to the consistency of a measurable cardinal that no such models have cardinality ≤ ℵ₂. We now improve the work of [9] in two ways. Firstly, we prove that the consistency strength of the statement “CH and $E F G_{ω ₁} (,)$ is determined for all models and of cardinality ℵ₂” is that of a weakly compact cardinal. On the other hand, we show that if $2^{ℵ ₀} < 2^{ℵ ₃}$ , T is a countable...

On the Variational Inequality and Tykhonov Well-Posedness in Game Theory

C. A. Pensavalle, G. Pieri (2010)

Bollettino dell'Unione Matematica Italiana

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Consider a M-player game in strategic form $G=(X_{1},\cdots,X_{M},g_{1},\cdots,g_{M})$ where the set $X_{i}$ is a closed interval of real numbers and the payoff function $g_{i}$ is concave and differentiable with respect to the variable $x_{i}\in X_{i}$ , for any $i=1,\cdots,M$ . The aim of this paper is to find appropriate conditions on the payoff functions under the well-posedness with respect to the related variational inequality is equivalent to the formulation of the Tykhonov well-posedness in a game context. The idea of the proof is to appeal to a third equivalence,...

Some applications of the point-open subbase game

D. Guerrero Sánchez, Vladimir Vladimirovich Tkachuk (2017)

Commentationes Mathematicae Universitatis Carolinae

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Given a subbase $𝒮$ of a space $X$ , the game $P O (𝒮, X)$ is defined for two players $P$ and $O$ who respectively pick, at the $n$ -th move, a point $x_{n} \in X$ and a set $U_{n} \in 𝒮$ such that $x_{n} \in U_{n}$ . The game stops after the moves ${x_{n}, U_{n} : n \in ø}$ have been made and the player $P$ wins if $⋃_{n \in ø} U_{n} = X$ ; otherwise $O$ is the winner. Since $P O (𝒮, X)$ is an evident modification of the well-known point-open game $P O (X)$ , the primary line of research is to describe the relationship between $P O (X)$ and $P O (𝒮, X)$ for a given subbase $𝒮$ . It turns out that, for any subbase $𝒮$ , the player $P$ has a winning...

Lindelöf indestructibility, topological games and selection principles

Marion Scheepers, Franklin D. Tall (2010)

Fundamenta Mathematicae

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Arhangel’skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most $2^{ℵ ₀}$ . Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are $G_{δ}$ has been more elusive. In this paper we continue the agenda started by the second author, [Topology Appl. 63 (1995)], of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinite games and selection principles, especially the Rothberger...

Infinite games and chain conditions

Santi Spadaro (2016)

Fundamenta Mathematicae

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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 the weak Lindelöf number of the $G_{δ}$ topology on a compact space. 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...

Chocolate games that satisfy the inequality $y \leq ⌊ \frac{z}{k} ⌋$ for $k = 1, 2$ and Grundy numbers

Shunsuke Nakamura, Ryo Hanafusa, Wataru Ogasa, Takeru Kitagawa, Ryohei Miyadera (2013)

Visual Mathematics

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Empirical approximation in Markov games under unbounded payoff: discounted and average criteria

Fernando Luque-Vásquez, J. Adolfo Minjárez-Sosa (2017)

Kybernetika

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This work deals with a class of discrete-time zero-sum Markov games whose state process $\{x_{t}\}$ evolves according to the equation $x_{t + 1} = F (x_{t}, a_{t}, b_{t}, ξ_{t}),$ where $a_{t}$ and $b_{t}$ represent the actions of player 1 and 2, respectively, and $\{ξ_{t}\}$ is a sequence of independent and identically distributed random variables with unknown distribution $θ$ . Assuming possibly unbounded payoff, and using the empirical distribution to estimate $θ$ , we introduce approximation schemes for the value of the game as well as for optimal strategies considering...

Uncountable γ-sets under axiom $C P A_{c u b e}^{g a m e}$

Krzysztof Ciesielski, Andrés Millán, Janusz Pawlikowski (2003)

Fundamenta Mathematicae

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We formulate a Covering Property Axiom $C P A_{c u b e}^{g a m e}$ , which holds in the iterated perfect set model, and show that it implies the existence of uncountable strong γ-sets in ℝ (which are strongly meager) as well as uncountable γ-sets in ℝ which are not strongly meager. These sets must be of cardinality ω₁ < , since every γ-set is universally null, while $C P A_{c u b e}^{g a m e}$ implies that every universally null has cardinality less than = ω₂. We also show that $C P A_{c u b e}^{g a m e}$ implies the existence of a partition of ℝ into ω₁ null...

Some new versions of an old game

Vladimir Vladimirovich Tkachuk (1995)

Commentationes Mathematicae Universitatis Carolinae

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The old game is the point-open one discovered independently by F. Galvin [7] and R. Telgársky [17]. Recall that it is played on a topological space $X$ as follows: at the $n$ -th move the first player picks a point $x_{n} \in X$ and the second responds with choosing an open $U_{n} ∋ x_{n}$ . The game stops after $ω$ moves and the first player wins if $\cup {U_{n} : n \in ω} = X$ . Otherwise the victory is ascribed to the second player. In this paper we introduce and study the games $θ$ and $Ω$ . In $θ$ the moves are made exactly as in the point-open game,...

Norm continuity of pointwise quasi-continuous mappings

Alireza Kamel Mirmostafaee (2018)

Mathematica Bohemica

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Let $X$ be a Baire space, $Y$ be a compact Hausdorff space and $ϕ : X \to C_{p} (Y)$ be a quasi-continuous mapping. For a proximal subset $H$ of $Y \times Y$ we will use topological games $𝒢_{1} (H)$ and $𝒢_{2} (H)$ on $Y \times Y$ between two players to prove that if the first player has a winning strategy in these games, then $ϕ$ is norm continuous on a dense $G_{δ}$ subset of $X$ . It follows that if $Y$ is Valdivia compact, each quasi-continuous mapping from a Baire space $X$ to $C_{p} (Y)$ is norm continuous on a dense $G_{δ}$ subset of $X$ .

Applications of saddle-point determinants

Jan Hauke, Charles R. Johnson, Tadeusz Ostrowski (2015)

Discussiones Mathematicae - General Algebra and Applications

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For a given square matrix $A \in M_{n} (ℝ)$ and the vector $e \in {(ℝ)}^{n}$ of ones denote by (A,e) the matrix ⎡ A e ⎤ ⎣ $e^{T}$ 0 ⎦ This is often called the saddle point matrix and it plays a significant role in several branches of mathematics. Here we show some applications of it in: game theory and analysis. An application of specific saddle point matrices that are hollow, symmetric, and nonnegative is likewise shown in geometry as a generalization of Heron’s formula to give the volume of a general simplex, as well as...

Edge-sum distinguishing labeling

Jan Bok, Nikola Jedličková (2021)

Commentationes Mathematicae Universitatis Carolinae

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We study edge-sum distinguishing labeling, a type of labeling recently introduced by Z. Tuza (2017) in context of labeling games. An ESD labeling of an $n$ -vertex graph $G$ is an injective mapping of integers $1$ to $l$ to its vertices such that for every edge, the sum of the integers on its endpoints is unique. If $l$ equals to $n$ , we speak about a canonical ESD labeling. We focus primarily on structural properties of this labeling and show for several classes of graphs if they have or do not...

A countably cellular topological group all of whose countable subsets are closed need not be $ℝ$ -factorizable

Mihail G. Tkachenko (2023)

Commentationes Mathematicae Universitatis Carolinae

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We construct a Hausdorff topological group $G$ such that $ℵ_{1}$ is a precalibre of $G$ (hence, $G$ has countable cellularity), all countable subsets of $G$ are closed and $C$ -embedded in $G$ , but $G$ is not $ℝ$ -factorizable. This solves Problem 8.6.3 from the book “Topological Groups and Related Structures" (2008) in the negative.