On a problem of Steve Kalikow

Volume 166 / 2000

Saharon Shelah Fundamenta Mathematicae 166 (2000), 137-151 DOI: 10.4064/fm-166-1-2-137-151


The Kalikow problem for a pair (λ,κ) of cardinal numbers,λ > κ (in particular κ = 2) is whether we can map the family of ω-sequences from λ to the family of ω-sequences from κ in a very continuous manner. Namely, we demand that for η,ν ∈ ω we have: η, ν are almost equal if and only if their images are. We show consistency of the negative answer, e.g., for $ℵ_ω$ but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants.


  • Saharon Shelah

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