On strong measure zero subsets of $^{\kappa}2$

Volume 170 / 2001

Aapo Halko, Saharon Shelah Fundamenta Mathematicae 170 (2001), 219-229 MSC: Primary 03E15, 03E05. DOI: 10.4064/fm170-3-1


We study the generalized Cantor space $^\kappa 2$ and the generalized Baire space $^\kappa \kappa $ as analogues of the classical Cantor and Baire spaces. We equip ${}^\kappa \kappa $ with the topology where a basic neighborhood of a point $\eta $ is the set $\{\nu:(\forall j< i)(\nu(j)=\eta(j))\}$, where $i< \kappa$.

We define the concept of a strong measure zero set of ${}^\kappa 2$. We prove for successor $\kappa =\kappa ^{<\kappa }$ that the ideal of strong measure zero sets of ${}^\kappa 2$ is ${\frak b}_\kappa$-additive, where ${\frak b}_\kappa $ is the size of the smallest unbounded family in ${}^\kappa \kappa $, and that the generalized Borel conjecture for ${}^\kappa 2$ is false. Moreover, for regular uncountable $\kappa $, the family of subsets of ${}^\kappa 2$ with the property of Baire is not closed under the Suslin operation.

These results answer problems posed in [2] .


  • Aapo HalkoDepartment of Mathematics
    P.O. Box 4
    FIN-00014 University of Helsinki
    Helsinki, Finland
  • Saharon ShelahInstitute of Mathematics
    Hebrew University
    Jerusalem, Israel

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