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Infinite games and chain conditions

Santi Spadaro (2016)

Fundamenta Mathematicae

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 Lindelöf...

Minimal predictors in hat problems

Christopher S. Hardin, Alan D. Taylor (2010)

Fundamenta Mathematicae

We consider a combinatorial problem related to guessing the values of a function at various points based on its values at certain other points, often presented by way of a hat-problem metaphor: there are a number of players who will have colored hats placed on their heads, and they wish to guess the colors of their own hats. A visibility relation specifies who can see which hats. This paper focuses on the existence of minimal predictors: strategies guaranteeing at least one player guesses correctly,...

Monotone extenders for bounded c-valued functions

Kaori Yamazaki (2010)

Studia Mathematica

Let c be the Banach space consisting of all convergent sequences of reals with the sup-norm, C ∞ ( A , c ) the set of all bounded continuous functions f: A → c, and C A ( X , c ) the set of all functions f: X → c which are continuous at each point of A ⊂ X. We show that a Tikhonov subspace A of a topological space X is strong Choquet in X if there exists a monotone extender u : C ∞ ( A , c ) → C A ( X , c ) . This shows that the monotone extension property for bounded c-valued functions can fail in GO-spaces, which provides a negative answer to a question...

On generalized games in H -spaces

Paolo Cubiotti, Giorgio Nordo (1999)

Commentationes Mathematicae Universitatis Carolinae

We show that a recent existence result for the Nash equilibria of generalized games with strategy sets in H -spaces is false. We prove that it becomes true if we assume, in addition, that the feasible set of the game (the set of all feasible multistrategies) is closed.

On some problem of A. Rosłanowski

Szymon Plewik (1996)

Colloquium Mathematicae

We present a negative answer to problem 3.7(b) posed on page 193 of [2], where, in fact, A. Rosłanowski asked: Does every set of Lebesgue measure zero belong to some Mycielski ideal?

On the open-open game

Peg Daniels, Kenneth Kunen, Haoxuan Zhou (1994)

Fundamenta Mathematicae

We modify a game due to Berner and Juhász to get what we call “the open-open game (of length ω)”: a round consists of player I choosing a nonempty open subset of a space X and II choosing a nonempty open subset of I’s choice; I wins if the union of II’s open sets is dense in X, otherwise II wins. This game is of interest for ccc spaces. It can be translated into a game on partial orders (trees and Boolean algebras, for example). We present basic results and various conditions under which I or II...

On β-favorability of the strong Choquet game

László Zsilinszky (2011)

Colloquium Mathematicae

In the main result, partially answering a question of Telgársky, the following is proven: if X is a first countable R₀-space, then player β (i.e. the EMPTY player) has a winning strategy in the strong Choquet game on X if and only if X contains a nonempty W δ -subspace which is of the first category in itself.

Simple games in Łukasiewicz calculus and their cores

Petr Cintula, Tomáš Kroupa (2013)

Kybernetika

We propose a generalization of simple coalition games in the context of games with fuzzy coalitions. Mimicking the correspondence of simple games with non-constant monotone formulas of classical logic, we introduce simple Łukasiewicz games using monotone formulas of Łukasiewicz logic, one of the most prominent fuzzy logics. We study the core solution on the class of simple Łukasiewicz games and show that cores of such games are determined by finitely-many linear constraints only. The non-emptiness...

Some weak covering properties and infinite games

Masami Sakai (2014)

Open Mathematics

We show that (I) there is a Lindelöf space which is not weakly Menger, (II) there is a Menger space for which TWO does not have a winning strategy in the game Gfin(O,Do). These affirmatively answer questions posed in Babinkostova, Pansera and Scheepers [Babinkostova L., Pansera B.A., Scheepers M., Weak covering properties and infinite games, Topology Appl., 2012, 159(17), 3644–3657]. The result (I) automatically gives an affirmative answer of Wingers’ problem [Wingers L., Box products and Hurewicz...

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