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# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Rothberger gaps in fragmented ideals

### Tom 227 / 2014

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

#### Streszczenie

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.

#### Autorzy

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

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