Archive for Mathematical Logic

, Volume 55, Issue 1–2, pp 85–104 | Cite as

Baire spaces and infinite games

Article

Abstract

It is well known that if the nonempty player of the Banach–Mazur game has a winning strategy on a space, then that space is Baire in all powers even in the box product topology. The converse of this implication may also be true: We know of no consistency result to the contrary. In this paper we establish the consistency of the converse relative to the consistency of the existence of a proper class of measurable cardinals.

Keywords

Baire space Infinite game Measurable cardinal 

Mathematics Subject Classification

03E55 03E60 03E65 54B10 54E52 91A44 91A46 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Davis M.: Infinite games of perfect information. Adv. Game Theory Ann. Math. Stud. 52, 85–101 (1964)MATHGoogle Scholar
  2. 2.
    Galvin F., Jech T., Magidor M.: An ideal game. J. Symb. Log. 43(2), 284–292 (1978)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Jech T.: Set theory. The third millennium edition. Springer, Berlin (2003)MATHGoogle Scholar
  4. 4.
    Jech T., Magidor M., Mitchell W., Prikry K.: Precipitous ideals. J. Symb. Log. 45(1), 1–8 (1980)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Juhász I.: Cardinal Functions in Topology. Mathematisch Centrum, Amsterdam (1971)MATHGoogle Scholar
  6. 6.
    Mauldin, R.D. (ed.): The Scottish Book: Mathematics from the Scottish Café. Birkhäuser, Boston (1981)Google Scholar
  7. 7.
    Schreier J.: Eine Eigenschaft abstrakter Mengen. Stud. Math. 7, 155–156 (1938)Google Scholar
  8. 8.
    Telgársky R.: Topological games: on the 50th anniversary of the Banach–Mazur Game. Rocky Mt. J. Math. 17, 227–276 (1987)CrossRefMATHGoogle Scholar
  9. 9.
    Ulam S.: Combinatorial analysis in infinite sets and some physical theories. SIAM Rev. 6, 343–355 (1964)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    White H.E. Jr.: Topological spaces that are α-favorable for a player with perfect information. Proc. Am. Math. Soc. 50, 477–482 (1975)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KansasLawrenceUSA
  2. 2.Department of MathematicsBoise State UniversityBoiseUSA

Personalised recommendations