Gδ -sets in topological spaces and games

Just Winfried¹ ; Marion Sheepers¹ ; Juris Steprans¹ ; Paul Szeptycki¹

¹

Fundamenta Mathematicae, Tome 153 (1997) no. 1, pp. 41-58

Résumé

Players ONE and TWO play the following game: In the nth inning ONE chooses a set $O_n$ from a prescribed family ℱ of subsets of a space X; TWO responds by choosing an open subset $T_n$ of X. The players must obey the rule that $O_n ⊆ O_{n+1} ⊆ T_{n+1} ⊆ T_n$ for each n. TWO wins if the intersection of TWO's sets is equal to the union of ONE's sets. If ONE has no winning strategy, then each element of ℱ is a $G_δ$-set. To what extent is the converse true? We show that:  (A) For ℱ the collection of countable subsets of X:   1. There are subsets of the real line for which neither player has a winning strategy in this game.   2. The statement "If X is a set of real numbers, then ONE does not have a winning strategy if, and only if, every countable subset of X is a $G_δ$-set" is independent of the axioms of classical mathematics.   3. There are spaces whose countable subsets are $G_δ$-sets, and yet ONE has a winning strategy in this game.   4. For a hereditarily Lindelöf space X, TWO has a winning strategy if, and only if, X is countable.  (B) For ℱ the collection of $G_σ$-subsets of a subset X of the real line the determinacy of this game is independent of ZFC.

Détail

DOI : 10.4064/fm-153-1-41-58

Keywords: game, strategy, Lusin set, Sierpiński set, Rothberger's property C", concentrated set, λ-set, σ-set, perfectly meager set, Q-set, $s_0$-set, $A_1$-set, $A_2$-set, $A_3$-set, ${\ninegot b}$, ${\ninegot d}$