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## Fundamenta Mathematicae

2014 | 227 | 1 | 35-68

## Rothberger gaps in fragmented ideals

EN

### Abstrakty

EN
The Rothberger number 𝔟(ℐ) of a definable ideal ℐ on ω is the least cardinal κ such that there exists a Rothberger gap of type (ω,κ) in the quotient algebra 𝓟(ω)/ℐ. We investigate 𝔟(ℐ) for a class of $F_{σ}$ ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is ℵ₁, 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.

35-68

wydano
2014

### Twórcy

autor
• Graduate School of System Informatics, Kobe University, 1-1 Rokkodai, Nada-ku, 657-8501 Kobe, Japan
• Graduate School of System Informatics, Kobe University, Kobe, Japan