Gδ -sets in topological spaces and games

Winfried Just; Marion Scheepers; Juris Steprans; Paul Szeptycki

Fundamenta Mathematicae (1997)

  • Volume: 153, Issue: 1, page 41-58
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Just, Winfried, et al. "Gδ -sets in topological spaces and games." Fundamenta Mathematicae 153.1 (1997): 41-58. <http://eudml.org/doc/212214>.

@article{Just1997,
abstract = {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.},
author = {Just, Winfried, Scheepers, Marion, Steprans, Juris, Szeptycki, Paul},
journal = {Fundamenta Mathematicae},
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; $\{b\}$; $\{d\}$; Lusin set; Sierpiński set},
language = {eng},
number = {1},
pages = {41-58},
title = {Gδ -sets in topological spaces and games},
url = {http://eudml.org/doc/212214},
volume = {153},
year = {1997},
}

TY - JOUR
AU - Just, Winfried
AU - Scheepers, Marion
AU - Steprans, Juris
AU - Szeptycki, Paul
TI - Gδ -sets in topological spaces and games
JO - Fundamenta Mathematicae
PY - 1997
VL - 153
IS - 1
SP - 41
EP - 58
AB - 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.
LA - eng
KW - 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; ${b}$; ${d}$; Lusin set; Sierpiński set
UR - http://eudml.org/doc/212214
ER -

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