### Infinite ${G}_{\delta}$-games with imperfect information

D. Blackwell (1969)

Applicationes Mathematicae

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D. Blackwell (1969)

Applicationes Mathematicae

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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, $EF{G}_{\omega \u2081}(,)$, 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 $EF{G}_{\omega \u2081}(,)$ is determined for all models and of cardinality ℵ₂” is that of a weakly compact cardinal. On the other hand, we show that if ${2}^{\aleph \u2080}<{2}^{\aleph \u2083}$, T is a countable...

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 ${2}^{\aleph \u2080}$. Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are ${G}_{\delta}$ 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...

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 ${G}_{\delta}$ 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...

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

Visual Mathematics

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Krzysztof Ciesielski, Andrés Millán, Janusz Pawlikowski (2003)

Fundamenta Mathematicae

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We formulate a Covering Property Axiom $CP{A}_{cube}^{game}$, 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 $CP{A}_{cube}^{game}$ implies that every universally null has cardinality less than = ω₂. We also show that $CP{A}_{cube}^{game}$ implies the existence of a partition of ℝ into ω₁ null...

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 ${U}_{n}\ni {x}_{n}$. The game stops after $\omega $ moves and the first player wins if $\cup \{{U}_{n}:n\in \omega \}=X$. Otherwise the victory is ascribed to the second player. In this paper we introduce and study the games $\theta $ and $\Omega $. In $\theta $ the moves are made exactly as in the point-open game,...

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

Discussiones Mathematicae - General Algebra and Applications

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For a given square matrix $A\in {M}_{n}\left(\mathbb{R}\right)$ and the vector $e\in {\left(\mathbb{R}\right)}^{n}$ 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...

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 ${C}_{p}\left(X\right)$ 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. ${C}_{p}\left(X\right)$ has countable fan tightness and the Reznichenko...

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 $\u03f5>0$, there exists $\delta =\delta \left(\u03f5\right)$ such that if a GSWF on three alternatives satisfies the IIA condition and its probability of...

Ahmad Al-Omari, Takashi Noiri (2017)

Archivum Mathematicum

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

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 $\mathcal{A}$ and $\mathcal{B}$ is defined as follows: We say that $\mathcal{A}{\le}_{K}\mathcal{B}$ if there is a function $f:\omega \to \omega $ such that ${f}^{-1}\left(A\right)\in \mathcal{I}\left(\mathcal{B}\right)$ for every $A\in \mathcal{I}\left(\mathcal{A}\right)$. 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 $X\in \mathcal{I}{\left(\mathcal{A}\right)}^{+}$, we have that ${\mathcal{A}|}_{X}{\le}_{K}\mathcal{A}$. We prove that CH implies that for every $K$-uniform MAD family $\mathcal{A}$ there is a $P$-point $p$ of ${\omega}^{*}$ such that the set of all Rudin-Keisler predecessors of $p$ is dense...

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 $({u}_{1},\cdots ,{u}_{m}):U\to {\mathbb{R}}^{m}$ such that $U\cap X=\{{u}_{1}\cdots {u}_{r}=0\}$, (iii) Every affine Nash manifold with corners $N$ is a closed subset of an affine Nash...

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.

Wei-Feng Xuan, Wei-Xue Shi (2017)

Commentationes Mathematicae Universitatis Carolinae

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We prove that if $X$ is a first countable space with property $\left(DC\left({\omega}_{1}\right)\right)$ and with a ${G}_{\delta}$-diagonal then the cardinality of $X$ is at most $\U0001d520$. We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $\U0001d520$.