Displaying similar documents to “Some new versions of an old game”

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 ∈ 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 ∈ ω * . We prove that if { X μ : μ < ω 1 } is a set of spaces whose product X = ∏ μ < ω 1 X μ is a 𝒢 -space, then there is A ∈ [ ω 1 ] ≤ ω such that X μ is countably compact for every μ ∈ 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 , ⋯ , X M , g 1 , ⋯ , 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 ∈ X i , for any i = 1 , ⋯ , 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,...

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...

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 ∈ X and a set U n ∈ 𝒮 such that x n ∈ U n . The game stops after the moves { x n , U n : n ∈ ø } have been made and the player P wins if ⋃ n ∈ 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...

On remote points, non-normality and π -weight ω 1

Sergei Logunov (2001)

Commentationes Mathematicae Universitatis Carolinae

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We show, in particular, that every remote point of X is a nonnormality point of β X if X is a locally compact Lindelöf separable space without isolated points and π w ( X ) ≤ ω 1 .