The Rothberger number (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 (ω)/ℐ. We investigate (ℐ) for a class of 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm227-1-4,
author = {J\"org Brendle and Diego Alejandro Mej\'\i a},
title = {Rothberger gaps in fragmented ideals},
journal = {Fundamenta Mathematicae},
volume = {227},
year = {2014},
pages = {35-68},
zbl = {06330486},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm227-1-4}
}
Jörg Brendle; Diego Alejandro Mejía. Rothberger gaps in fragmented ideals. Fundamenta Mathematicae, Tome 227 (2014) pp. 35-68. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm227-1-4/