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## Fundamenta Mathematicae

1997 | 153 | 1 | 41-58
Tytuł artykułu

### Gδ -sets in topological spaces and games

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
41-58
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-07-02
poprawiono
1996-10-17
Twórcy
autor
• Department of Mathematics, Ohio University, Athens, Ohio 45701, U.S.A.
autor
• epartment of Mathematics, Boise State University, Boise, Idaho 83725, U.S.A.
autor
• Department of Mathematics, York University, North York, Ontario, Canada
autor
• Department of Mathematics, Ohio University, Athens, Ohio 45701, U.S.A.
Bibliografia
• [1] Z. Balogh, There is a Q-set space in ZFC, Proc. Amer. Math. Soc. 113 (1991), 557-561.
• [2] T. Bartoszyński and M. Scheepers, A-sets, Real Anal. Exchange 19 (1993-94), 521-528.
• [3] A. S. Besicovitch, Concentrated and rarified sets of points, Acta Math. 62 (1934), 289-300.
• [4] E. Čech, Sur la dimension des espaces parfaitement normaux, Bull. Internat. Acad. Bohême (Prague) 33 (1932), 38-55.
• [5] H F. Hausdorff, Dimension und äusseres Mass, Math. Ann. 79 (1919), 157-179.
• [6] W. Just, A. Miller, M. Scheepers and P. J. Szeptycki, The combinatorics of open covers II, Topology Appl. 73 (1996), 241-266.
• [7] K. Kunen, Set Theory: An Introduction to Independence Proofs, North-Holland, 1984.
• [8] K. Kuratowski, Topology, Vol. 1, Academic Press, 1966.
• [9] K. Kuratowski, Sur une famille d'ensembles singuliers, Fund. Math. 21 (1933), 127-128.
• [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.
• [11] N. Lusin, Sur les ensembles toujours de première catégorie, Fund. Math. 21 (1933), 114-126.
• [12] A. W. Miller, On generating the category algebra and the Baire order problem, Bull. Acad. Polon. Sci. 27 (1979), 751-755.
• [13] A. W. Miller, Special subsets of the real line, in: The Handbook of Set-Theoretic Topology, North-Holland, 1984, 201-223.
• [14] F. Rothberger, Eine Verschärfung der Eigenschaft C, Fund. Math. 30 (1938), 50-55.
• [15] F. Rothberger, On some problems of Hausdorff and of Sierpiński, Fund. Math. 35 (1948), 29-46.
• [16] W. Sierpiński, Sur l'hypothese du continu $(2^ℵ_0 = ℵ_1)$, Fund. Math. 5 (1924), 177-187.
• [17] W. Sierpiński, Sur deux consequences d'un théorème de Hausdorff, Fund. Math. 26 (1945), 269-272.
• [18] L. A. Steen and J. A. Seebach, Jr., Counterexamples in Topology, 2nd ed., Springer, 1978.
• [19] J. Steprāns, Combinatorial consequences of adding Cohen reals, in: Israel Math. Conf. Proc. 6, Bar-Ilan Univ., Ramat Gan, 1993, 583-617.
• [20] E. Szpilrajn, Sur un problème de M. Banach, Fund. Math. 15 (1930), 212-214.
• [21] E. Szpilrajn, Sur une classe de fonctions de M. Sierpiński et la classe correspondante d'ensembles, Fund. Math. 24 (1934), 17-34.
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