Rothberger gaps in fragmented ideals

Jörg Brendle ¹   ; Diego Alejandro Mejía ²

¹ Graduate School of System Informatics Kobe University 1-1 Rokkodai, Nada-ku 657-8501 Kobe, Japan
² Graduate School of System Informatics Kobe University Kobe, Japan

Fundamenta Mathematicae, Tome 227 (2014) no. 1, pp. 35-68

Résumé

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.

Détail

DOI : 10.4064/fm227-1-4

Keywords: rothberger number mathfrak mathcal definable ideal mathcal omega least cardinal kappa there exists rothberger gap type omega kappa quotient algebra mathcal omega mathcal investigate mathfrak mathcal class sigma ideals fragmented ideals prove these ideals linear growth ideal rothberger number aleph while others polynomial growth ideal above additivity measure consistent there infinitely many even continuum many different rothberger numbers associated fragmented ideals