The length of some diagonalization games

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Abstract

For X a separable metric space and α an infinite ordinal, consider the following three games of length α: In Gα1 ONE chooses in inning γ an ω-cover Oγ of X; TWO responds with a Tγ ∈ Oγ. TWO wins if {Tγ : γ < α} is an ω-cover of X; ONE wins otherwise. In Gα2 ONE chooses in inning γ a subset Oγ of Cp(X) which has the zero function 0 in its closure, and TWO responds with a function Tγ ∈ Oγ. TWO wins if 0 is in the closure of {Tγ : γ < α}; otherwise, ONE wins. In Gα3 ONE chooses in inning γ a dense subset Oγ of Cp(X), and TWO responds with a Tγ ∈ Oγ. TWO wins if {Tγ : γ < α} is dense in Cp(X); otherwise, ONE wins. After a brief survey we prove: 1. If α is minimal such that TWO has a winning strategy in Gα1, then α is additively indecomposable (Theorem 4) 2. For α countable and minimal such that TWO has a winning strategy in Gα1 on X, the following statements are equivalent (Theorem 9): a) TWO has a winning strategy in Gα2 on Cp(X). b) TWO has a winning strategy in Gα3 on Cp(X). 3. The Continuum Hypothesis implies that there is an uncountable set X of real numbers such that TWO has a winning strategy in Gω21 on X (Theorem 10).

Original languageEnglish
Pages (from-to)103-122
Number of pages20
JournalArchive for Mathematical Logic
Volume38
Issue number2
DOIs
StatePublished - Feb 1999

Keywords

  • Density type
  • Infinite game
  • Point-open type
  • Strong-fan type
  • ω-concentrated
  • ω-type

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