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Fundamenta Mathematicae

1994 | 145 | 3 | 205-220
Tytuł artykułu

On the open-open game

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
205-220
Opis fizyczny
Daty
wydano
1994
otrzymano
1991-06-28
poprawiono
1993-08-24
poprawiono
1994-01-31
Twórcy
autor
• Department of Mathematics, Auburn University, Auburn, Alabama 36849-5310, U.S.A.
autor
autor
• Department of Mathematics, Incarnate Word College, 4301 Broadway, San Antonio, Texas 78209, U.S.A.
Bibliografia
• [B] A. Berner, Types of strategies in point-picking games, Topology Proc. 9 (1984), 227-242.
• [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.
• [CN] W. W. Comfort and S. Negrepontis, Chain Conditions in Topology, Cambridge University Press, 1982.
• [D] P. Daniels, Pixley-Roy spaces over subsets of the reals, Topology Appl. 29 (1988), 93-106.
• [DG] P. Daniels and G. Gruenhage, The point-open type of subsets of the reals, ibid. 37 (1990), 53-64.
• [J] T. Jech, Set Theory, Academic Press, New York, 1978.
• [Ju] I. Juhász, On point-picking games, Topology Proc. 10 (1985), 103-110.
• [K] K. Kunen, Set Theory, An Introduction to Independence Proofs, North-Holland, Amsterdam, 1980.
• [Ma] R. D. Mauldin, The Scottish Book, Birkhäuser, Boston, 1981.
• [M1] A. Miller, Special subsets of the real line, in: Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, 201-233.
• [M2] A. Miller, Some properties of measure and category, Trans. Amer. Math. Soc. 266 (1981), 93-114.
• [S] A. Szymański, Some applications of tiny sequences, Rend. Circ. Mat. Palermo (2) Suppl. 3 (1984), 321-338.
• [T] S. Todorčević, Trees and linear ordered sets, in: Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, 235-293.
Typ dokumentu
Bibliografia
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