On the open-open game
Daniels, Peg ; Kunen, Kenneth ; Zhou, Haoxuan
Fundamenta Mathematicae, Tome 144 (1994), p. 205-220 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

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 ω1-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.

Publié le : 1994-01-01
EUDML-ID : urn:eudml:doc:212043 
@article{bwmeta1.element.bwnjournal-article-fmv145i3p205bwm,
     author = {Peg Daniels and Kenneth Kunen and Haoxuan Zhou},
     title = {On the open-open game},
     journal = {Fundamenta Mathematicae},
     volume = {144},
     year = {1994},
     pages = {205-220},
     zbl = {0811.54008},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv145i3p205bwm}
}

Daniels, Peg; Kunen, Kenneth; Zhou, Haoxuan. On the open-open game. Fundamenta Mathematicae, Tome 144 (1994) pp. 205-220. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv145i3p205bwm/

[00000] [B] A. Berner, Types of strategies in point-picking games, Topology Proc. 9 (1984), 227-242. | Zbl 0598.90110

[00001] [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. | Zbl 0573.54004

[00002] [CN] W. W. Comfort and S. Negrepontis, Chain Conditions in Topology, Cambridge University Press, 1982.

[00003] [D] P. Daniels, Pixley-Roy spaces over subsets of the reals, Topology Appl. 29 (1988), 93-106. | Zbl 0656.54007

[00004] [DG] P. Daniels and G. Gruenhage, The point-open type of subsets of the reals, ibid. 37 (1990), 53-64. | Zbl 0718.54015

[00005] [J] T. Jech, Set Theory, Academic Press, New York, 1978.

[00006] [Ju] I. Juhász, On point-picking games, Topology Proc. 10 (1985), 103-110. | Zbl 0604.54006

[00007] [K] K. Kunen, Set Theory, An Introduction to Independence Proofs, North-Holland, Amsterdam, 1980. | Zbl 0443.03021

[00008] [Ma] R. D. Mauldin, The Scottish Book, Birkhäuser, Boston, 1981. | Zbl 0485.01013

[00009] [M1] A. Miller, Special subsets of the real line, in: Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, 201-233.

[00010] [M2] A. Miller, Some properties of measure and category, Trans. Amer. Math. Soc. 266 (1981), 93-114.

[00011] [S] A. Szymański, Some applications of tiny sequences, Rend. Circ. Mat. Palermo (2) Suppl. 3 (1984), 321-338. | Zbl 0549.54003

[00012] [T] S. Todorčević, Trees and linear ordered sets, in: Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, 235-293.