Access the full text.
Sign up today, get DeepDyve free for 14 days.
A. Besicovitch (1933)
Concentrated and rarified sets of pointsActa Mathematica, 62
J. Pawlikowski (1994)
Undetermined sets of point-open gamesFundamenta Mathematicae
M. Sakai, M. Scheepers (1996)
The Combinatorics of Open CoversTopology and its Applications, 73
S. Baldwin (1991)
Possible point-open types of subsets of the realsTopology and its Applications, 38
P. Daniels, G. Gruenhage (1990)
THE POINT-OPEN TYPE OF SUBSETS OF THE REALSTopology and its Applications, 37
R. Telgarsky (1975)
Spaces defined by topological gamesFundamenta Mathematicae, 88
M. Scheepers (1997)
Combinatorics of Open Covers (III): Games, C p ( X )Fundamenta Mathematicae, 152
Fritz Rothberger (1938)
Eine Verschärfung der Eigenschaft CFundamenta Mathematicae, 30
F. Galvin, A. Miller (1984)
γ-sets and other singular sets of real numbersTopology and its Applications, 17
R. Laver (1976)
On the consistency of Borel's conjectureActa Mathematica, 137
A. Berner, I. Juhász (1984)
Point-picking games and HFD's
For X a separable metric space and $\alpha$ an infinite ordinal, consider the following three games of length $\alpha$ : In $G^{\alpha}_1$ ONE chooses in inning $\gamma$ an $\omega$ –cover $O_{\gamma}$ of X; TWO responds with a $T_{\gamma}\in O_{\gamma}$ . TWO wins if $\{T_{\gamma}:\gamma<\alpha\}$ is an $\omega$ –cover of X; ONE wins otherwise. In $G^{\alpha}_2$ ONE chooses in inning $\gamma$ a subset $O_{\gamma}$ of ${\sf C}_p(X)$ which has the zero function $\underline{0}$ in its closure, and TWO responds with a function $T_{\gamma}\in O_{\gamma}$ . TWO wins if $\underline{0}$ is in the closure of $\{T_{\gamma}:\gamma<\alpha\}$ ; otherwise, ONE wins. In $G^{\alpha}_3$ ONE chooses in inning $\gamma$ a dense subset $O_{\gamma}$ of ${\sf C}_p(X)$ , and TWO responds with a $T_{\gamma}\in O_{\gamma}$ . TWO wins if $\{T_{\gamma}:\gamma<\alpha\}$ is dense in ${\sf C}_p(X)$ ; otherwise, ONE wins. After a brief survey we prove: 1. If $\alpha$ is minimal such that TWO has a winning strategy in $G^{\alpha}_1$ , then $\alpha$ is additively indecomposable (Theorem 4) 2. For $\alpha$ countable and minimal such that TWO has a winning strategy in $G^{\alpha}_1$ on X, the following statements are equivalent (Theorem 9): a) TWO has a winning strategy in $G^{\alpha}_2$ on ${\sf C}_p(X)$ . b) TWO has a winning strategy in $G^{\alpha}_3$ on ${\sf C}_p(X)$ . 3. The Continuum Hypothesis implies that there is an uncountable set X of real numbers such that TWO has a winning strategy in $G^{\omega^2}_1$ on X (Theorem 10).
Archive for Mathematical Logic – Springer Journals
Published: Feb 1, 1999
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.