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Rothberger gaps in fragmented ideals

Volume 227 / 2014

Jörg Brendle, Diego Alejandro Mejía Fundamenta Mathematicae 227 (2014), 35-68 MSC: Primary 03E05; Secondary 03E15, 03E17, 03E35. DOI: 10.4064/fm227-1-4

Abstract

The Rothberger number $\mathfrak {b}(\mathcal {I})$ of a definable ideal $\mathcal {I}$ on $\omega $ is the least cardinal $\kappa $ such that there exists a Rothberger gap of type $(\omega ,\kappa )$ in the quotient algebra $\mathcal {P}(\omega ) / \mathcal {I}$. We investigate $\mathfrak {b}(\mathcal {I})$ for a class of $F_\sigma $ ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is $\aleph _1$, while for others, like the polynomial growth ideal, it is above the additivity of measure. We also show that it is consistent that there are infinitely many (even continuum many) different Rothberger numbers associated with fragmented ideals.

Authors

  • Jörg BrendleGraduate School of System Informatics
    Kobe University
    1-1 Rokkodai, Nada-ku
    657-8501 Kobe, Japan
    e-mail
  • Diego Alejandro MejíaGraduate School of System Informatics
    Kobe University
    Kobe, Japan
    e-mail

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