Rothberger gaps in fragmented ideals
Volume 227 / 2014
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.
