On the open-open game

Peg Daniels; Kenneth Kunen; Haoxuan Zhou

Fundamenta Mathematicae (1994)

  • Volume: 145, Issue: 3, page 205-220
  • ISSN: 0016-2736

Abstract

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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 does or does not have a winning strategy. We investigate the games on trees and Boolean algebras in detail, completely characterizing the game for ω 1 -trees. An undetermined game is also defined. (In contrast, it is still open whether there is an undetermined game using the definition due to Berner and Juhász.) Finally, we show that various variations on the game yield equivalent games.

How to cite

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Daniels, Peg, Kunen, Kenneth, and Zhou, Haoxuan. "On the open-open game." Fundamenta Mathematicae 145.3 (1994): 205-220. <http://eudml.org/doc/212043>.

@article{Daniels1994,
abstract = {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 does or does not have a winning strategy. We investigate the games on trees and Boolean algebras in detail, completely characterizing the game for $ω_1$-trees. An undetermined game is also defined. (In contrast, it is still open whether there is an undetermined game using the definition due to Berner and Juhász.) Finally, we show that various variations on the game yield equivalent games.},
author = {Daniels, Peg, Kunen, Kenneth, Zhou, Haoxuan},
journal = {Fundamenta Mathematicae},
keywords = {open-open game of length ; ccc spaces; games on trees and Boolean algebras; undetermined game},
language = {eng},
number = {3},
pages = {205-220},
title = {On the open-open game},
url = {http://eudml.org/doc/212043},
volume = {145},
year = {1994},
}

TY - JOUR
AU - Daniels, Peg
AU - Kunen, Kenneth
AU - Zhou, Haoxuan
TI - On the open-open game
JO - Fundamenta Mathematicae
PY - 1994
VL - 145
IS - 3
SP - 205
EP - 220
AB - 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 does or does not have a winning strategy. We investigate the games on trees and Boolean algebras in detail, completely characterizing the game for $ω_1$-trees. An undetermined game is also defined. (In contrast, it is still open whether there is an undetermined game using the definition due to Berner and Juhász.) Finally, we show that various variations on the game yield equivalent games.
LA - eng
KW - open-open game of length ; ccc spaces; games on trees and Boolean algebras; undetermined game
UR - http://eudml.org/doc/212043
ER -

References

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  1. [B] A. Berner, Types of strategies in point-picking games, Topology Proc. 9 (1984), 227-242. Zbl0598.90110
  2. [BJ] A. Berner and I. Juhász, Point-picking games and HFD's, in: Models and Sets, Proc. Logic Colloq. 1983, Lecture Notes in Math. 1103, Springer, 1984, 53-66. Zbl0573.54004
  3. [CN] W. W. Comfort and S. Negrepontis, Chain Conditions in Topology, Cambridge University Press, 1982. 
  4. [D] P. Daniels, Pixley-Roy spaces over subsets of the reals, Topology Appl. 29 (1988), 93-106. Zbl0656.54007
  5. [DG] P. Daniels and G. Gruenhage, The point-open type of subsets of the reals, ibid. 37 (1990), 53-64. Zbl0718.54015
  6. [J] T. Jech, Set Theory, Academic Press, New York, 1978. 
  7. [Ju] I. Juhász, On point-picking games, Topology Proc. 10 (1985), 103-110. Zbl0604.54006
  8. [K] K. Kunen, Set Theory, An Introduction to Independence Proofs, North-Holland, Amsterdam, 1980. Zbl0443.03021
  9. [Ma] R. D. Mauldin, The Scottish Book, Birkhäuser, Boston, 1981. Zbl0485.01013
  10. [M1] A. Miller, Special subsets of the real line, in: Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, 201-233. 
  11. [M2] A. Miller, Some properties of measure and category, Trans. Amer. Math. Soc. 266 (1981), 93-114. 
  12. [S] A. Szymański, Some applications of tiny sequences, Rend. Circ. Mat. Palermo (2) Suppl. 3 (1984), 321-338. Zbl0549.54003
  13. [T] S. Todorčević, Trees and linear ordered sets, in: Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, 235-293. 

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