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,...

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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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,...

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...

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 tight quantitative version of Arrow’s impossibility theorem

Nathan Keller (2012)

Journal of the European Mathematical Society

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The well-known Impossibility Theorem of Arrow asserts that any generalized social welfare function (GSWF) with at least three alternatives, which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a dictatorship, is necessarily non-transitive. In 2002, Kalai asked whether one can obtain the following quantitative version of the theorem: For any $ϵ > 0$ , there exists $δ = δ (ϵ)$ such that if a GSWF on three alternatives satisfies the IIA condition and its probability of...

Characterizations of $z$ -Lindelöf spaces

Ahmad Al-Omari, Takashi Noiri (2017)

Archivum Mathematicum

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A topological space $(X, τ)$ is said to be $z$ -Lindelöf [1] if every cover of $X$ by cozero sets of $(X, τ)$ admits a countable subcover. In this paper, we obtain new characterizations and preservation theorems of $z$ -Lindelöf spaces.

MAD families and $P$ -points

Salvador García-Ferreira, Paul J. Szeptycki (2007)

Commentationes Mathematicae Universitatis Carolinae

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The Katětov ordering of two maximal almost disjoint (MAD) families $𝒜$ and $ℬ$ is defined as follows: We say that $𝒜 \leq_{K} ℬ$ if there is a function $f : ω \to ω$ such that $f^{- 1} (A) \in ℐ (ℬ)$ for every $A \in ℐ (𝒜)$ . In [Garcia-Ferreira S., Hrušák M., Ordering MAD families a la Katětov, J. Symbolic Logic 68 (2003), 1337–1353] a MAD family is called $K$ -uniform if for every $X \in ℐ {(𝒜)}^{+}$ , we have that ${𝒜 |}_{X} \leq_{K} 𝒜$ . We prove that CH implies that for every $K$ -uniform MAD family $𝒜$ there is a $P$ -point $p$ of $ω^{*}$ such that the set of all Rudin-Keisler predecessors of $p$ is dense...

Finiteness problems on Nash manifolds and Nash sets

José F. Fernando, José Manuel Gamboa, Jesús M. Ruiz (2014)

Journal of the European Mathematical Society

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We study here several finiteness problems concerning affine Nash manifolds $M$ and Nash subsets $X$ . Three main results are: (i) A Nash function on a semialgebraic subset $Z$ of $M$ has a Nash extension to an open semialgebraic neighborhood of $Z$ in $M$ , (ii) A Nash set $X$ that has only normal crossings in $M$ can be covered by finitely many open semialgebraic sets $U$ equipped with Nash diffeomorphisms $(u_{1}, \dots, u_{m}) : U \to ℝ^{m}$ such that $U \cap X = {u_{1} \dots u_{r} = 0}$ , (iii) Every affine Nash manifold with corners $N$ is a closed subset of an affine Nash...

More reflections on compactness

Lúcia R. Junqueira, Franklin D. Tall (2003)

Fundamenta Mathematicae

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We consider the question of when $X_{M} = X$ , where $X_{M}$ is the elementary submodel topology on X ∩ M, especially in the case when $X_{M}$ is compact.