Infinite games and chain conditions

Santi Spadaro

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Displaying similar documents to “Infinite games and chain conditions”

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...

Topological games and product spaces

Salvador García-Ferreira, R. A. González-Silva, Artur Hideyuki Tomita (2002)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we deal with the product of spaces which are either $𝒢$ -spaces or $𝒢_{p}$ -spaces, for some $p \in ω^{*}$ . These spaces are defined in terms of a two-person infinite game over a topological space. All countably compact spaces are $𝒢$ -spaces, and every $𝒢_{p}$ -space is a $𝒢$ -space, for every $p \in ω^{*}$ . We prove that if ${X_{μ} : μ < ω_{1}}$ is a set of spaces whose product $X = \prod_{μ < ω_{1}} X_{μ}$ is a $𝒢$ -space, then there is $A \in {[ω_{1}]}^{\leq ω}$ such that $X_{μ}$ is countably compact for every $μ \in ω_{1} ∖ A$ . As a consequence, $X^{ω_{1}}$ is a $𝒢$ -space iff $X^{ω_{1}}$ is countably compact, and if $X^{2^{𝔠}}$ is a $𝒢$ -space,...

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...

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

D. Blackwell (1969)

Applicationes Mathematicae

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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,...

Applications of limited information strategies in Menger's game

Steven Clontz (2017)

Commentationes Mathematicae Universitatis Carolinae

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As shown by Telgársky and Scheepers, winning strategies in the Menger game characterize $σ$ -compactness amongst metrizable spaces. This is improved by showing that winning Markov strategies in the Menger game characterize $σ$ -compactness amongst regular spaces, and that winning strategies may be improved to winning Markov strategies in second-countable spaces. An investigation of 2-Markov strategies introduces a new topological property between $σ$ -compact and Menger spaces.

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,...

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 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...

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...

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...

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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Combinatorics of open covers (VII): Groupability

Ljubiša D. R. Kočinac, Marion Scheepers (2003)

Fundamenta Mathematicae

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We use Ramseyan partition relations to characterize: ∙ the classical covering property of Hurewicz; ∙ the covering property of Gerlits and Nagy; ∙ the combinatorial cardinal numbers and add(ℳ ). Let X be a $T_{31 / 2}$ -space. In [9] we showed that $C_{p} (X)$ has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the Gerlits-Nagy covering property. Now we show that the following are equivalent: 1. $C_{p} (X)$ has countable fan tightness and the Reznichenko...

A solution to Comfort's question on the countable compactness of powers of a topological group

Artur Hideyuki Tomita (2005)

Fundamenta Mathematicae

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In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $α \leq 2^{}$ , a topological group G such that $G^{γ}$ is countably compact for all cardinals γ < α, but $G^{α}$ is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $M A_{c o u n t a b l e}$ . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $M A_{c o u n t a b l e}$ . However, the question has...