- Tytuł:
- $G_δ$-sets in topological spaces and games
- Autorzy:
-
Winfried, Just
Sheepers, Marion
Steprans, Juris
Szeptycki, Paul - Powiązania:
- https://bibliotekanauki.pl/articles/1205436.pdf
- Data publikacji:
- 1997
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Tematy:
-
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
${\ninegot b}$
${\ninegot d}$ - Opis:
-
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. - Źródło:
-
Fundamenta Mathematicae; 1997, 153, 1; 41-58
0016-2736 - Pojawia się w:
- Fundamenta Mathematicae
- Dostawca treści:
- Biblioteka Nauki
Artykuł