Uncountable γ-sets under axiom $CPA_{cube}^{game}$

Krzysztof Ciesielski; Andrés Millán; Janusz Pawlikowski

Displaying similar documents to “Uncountable γ-sets under axiom $C P A_{c u b e}^{g a m e}$ ”

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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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,...

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...

Possibly there is no uniformly completely Ramsey null set of size $2^{ω}$

Andrzej Nowik (2002)

Colloquium Mathematicae

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We show that under the axiom $C P A_{c u b e}$ there is no uniformly completely Ramsey null set of size $2^{ω}$ . In particular, this holds in the iterated perfect set model. This answers a question of U. Darji.

Covering Property Axiom $C P A_{c u b e}$ and its consequences

Krzysztof Ciesielski, Janusz Pawlikowski (2003)

Fundamenta Mathematicae

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We formulate a Covering Property Axiom $C P A_{c u b e}$ , which holds in the iterated perfect set model, and show that it implies easily the following facts. (a) For every S ⊂ ℝ of cardinality continuum there exists a uniformly continuous function g: ℝ → ℝ with g[S] = [0,1]. (b) If S ⊂ ℝ is either perfectly meager or universally null then S has cardinality less than . (c) cof() = ω₁ < , i.e., the cofinality of the measure ideal is ω₁. (d) For every uniformly bounded sequence $⟨ f ₙ \in ℝ^{ℝ} ⟩_{n < ω}$ of Borel functions...

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...

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...

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 $X$ .

On the Set-Theoretic Strength of Countable Compactness of the Tychonoff Product $2^{ℝ}$

Eleftherios Tachtsis (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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We work in ZF set theory (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) and show the following: 1. The Axiom of Choice for well-ordered families of non-empty sets ( $A C^{W O}$ ) does not imply “the Tychonoff product $2^{ℝ}$ , where 2 is the discrete space 0,1, is countably compact” in ZF. This answers in the negative the following question from Keremedis, Felouzis, and Tachtsis [Bull. Polish Acad. Sci. Math. 55 (2007)]: Does the Countable Axiom of Choice for families of non-empty sets...

A new Lindelöf space with points $G_{δ}$

Alan S. Dow (2015)

Commentationes Mathematicae Universitatis Carolinae

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We prove that $♦^{*}$ implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality $2^{ℵ_{1}}$ which has points $G_{δ}$ . In addition, this space has the property that it need not be Lindelöf after countably closed forcing.

On certain non-constructive properties of infinite-dimensional vector spaces

Eleftherios Tachtsis (2018)

Commentationes Mathematicae Universitatis Carolinae

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In set theory without the axiom of choice ( $AC$ ), we study certain non-constructive properties of infinite-dimensional vector spaces. Among several results, we establish the following: (i) None of the principles AC $^{LO}$ (AC for linearly ordered families of nonempty sets)—and hence AC $^{WO}$ (AC for well-ordered families of nonempty sets)— $DC (< κ)$ (where $κ$ is an uncountable regular cardinal), and “for every infinite set $X$ , there is a bijection $f : X \to {0, 1} \times X$ ”, implies the statement “there exists a field $F$ such that...

On the solvability of systems of linear equations over the ring $ℤ$ of integers

Horst Herrlich, Eleftherios Tachtsis (2017)

Commentationes Mathematicae Universitatis Carolinae

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We investigate the question whether a system ${(E_{i})}_{i \in I}$ of homogeneous linear equations over $ℤ$ is non-trivially solvable in $ℤ$ provided that each subsystem ${(E_{j})}_{j \in J}$ with $| J | \leq c$ is non-trivially solvable in $ℤ$ where $c$ is a fixed cardinal number such that $c < | I |$ . Among other results, we establish the following. (a) The answer is ‘No’ in the finite case (i.e., $I$ being finite). (b) The answer is ‘No’ in the denumerable case (i.e., $| I | = ℵ_{0}$ and $c$ a natural number). (c) The answer in case that $I$ is uncountable and $c \leq ℵ_{0}$ is ‘No...

Displaying similar documents to “Uncountable γ-sets under axiom C P A c u b e g a m e ”

Displaying similar documents to “Uncountable γ-sets under axiom $C P A_{c u b e}^{g a m e}$ ”