# Gδ -sets in topological spaces and games

Fundamenta Mathematicae (1997)

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

top

## Abstract

top
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 ${O}_{n}\subseteq {O}_{n+1}\subseteq {T}_{n+1}\subseteq {T}_{n}$ 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 ${G}_{\delta }$-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 ${G}_{\delta }$-set” is independent of the axioms of classical mathematics.   3. There are spaces whose countable subsets are ${G}_{\delta }$-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 ${G}_{\sigma }$-subsets of a subset X of the real line the determinacy of this game is independent of ZFC.

## How to cite

top

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 -

## References

top
1. [1] Z. Balogh, There is a Q-set space in ZFC, Proc. Amer. Math. Soc. 113 (1991), 557-561. Zbl0748.54012
2. [2] T. Bartoszyński and M. Scheepers, A-sets, Real Anal. Exchange 19 (1993-94), 521-528.
3. [3] A. S. Besicovitch, Concentrated and rarified sets of points, Acta Math. 62 (1934), 289-300. Zbl0009.10504
4. [4] E. Čech, Sur la dimension des espaces parfaitement normaux, Bull. Internat. Acad. Bohême (Prague) 33 (1932), 38-55.
5. [5] H F. Hausdorff, Dimension und äusseres Mass, Math. Ann. 79 (1919), 157-179. Zbl46.0292.01
6. [6] W. Just, A. Miller, M. Scheepers and P. J. Szeptycki, The combinatorics of open covers II, Topology Appl. 73 (1996), 241-266. Zbl0870.03021
7. [7] K. Kunen, Set Theory: An Introduction to Independence Proofs, North-Holland, 1984. Zbl0443.03021
8. [8] K. Kuratowski, Topology, Vol. 1, Academic Press, 1966.
9. [9] K. Kuratowski, Sur une famille d'ensembles singuliers, Fund. Math. 21 (1933), 127-128. Zbl0008.24801
10. [10] N. Lusin, Sur l'existence d'un ensemble non dénombrable qui est de première catégorie dans tout ensemble parfait, Fund. Math. 2 (1921), 155-157. Zbl48.0275.05
11. [11] N. Lusin, Sur les ensembles toujours de première catégorie, Fund. Math. 21 (1933), 114-126. Zbl0008.24704
12. [12] A. W. Miller, On generating the category algebra and the Baire order problem, Bull. Acad. Polon. Sci. 27 (1979), 751-755. Zbl0461.54032
13. [13] A. W. Miller, Special subsets of the real line, in: The Handbook of Set-Theoretic Topology, North-Holland, 1984, 201-223.
14. [14] F. Rothberger, Eine Verschärfung der Eigenschaft C, Fund. Math. 30 (1938), 50-55. Zbl64.0622.01
15. [15] F. Rothberger, On some problems of Hausdorff and of Sierpiński, Fund. Math. 35 (1948), 29-46. Zbl0032.33702
16. [16] W. Sierpiński, Sur l’hypothese du continu $\left({2}_{0}^{\aleph }={\aleph }_{1}\right)$, Fund. Math. 5 (1924), 177-187.
17. [17] W. Sierpiński, Sur deux consequences d'un théorème de Hausdorff, Fund. Math. 26 (1945), 269-272. Zbl0060.12715
18. [18] L. A. Steen and J. A. Seebach, Jr., Counterexamples in Topology, 2nd ed., Springer, 1978. Zbl0386.54001
19. [19] J. Steprāns, Combinatorial consequences of adding Cohen reals, in: Israel Math. Conf. Proc. 6, Bar-Ilan Univ., Ramat Gan, 1993, 583-617. Zbl0839.03037
20. [20] E. Szpilrajn, Sur un problème de M. Banach, Fund. Math. 15 (1930), 212-214.
21. [21] E. Szpilrajn, Sur une classe de fonctions de M. Sierpiński et la classe correspondante d'ensembles, Fund. Math. 24 (1934), 17-34. Zbl61.0229.01

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.