On the ideal $(v^0)$

TL;DR

This paper investigates the properties of the $\sigma$-ideal $(v^0)$ related to Silver forcing, introducing new topologies and confirming a conjecture under specific set-theoretic hypotheses.

Contribution

It introduces segments and $*$-segments topologies for $(v^0)$, confirms Halbeisen's conjecture under certain hypotheses, and describes the ideal's structure using a proof of the Base Matrix Lemma.

Findings

Confirmed Halbeisen's conjecture $cov(v^0) = add(v^0)$ under specific hypotheses.

Established the ideal type of $(v^0)$ as $(rak c, \omega_1, rak c)$ under $h= ext{omega}_1$.

Introduced new topologies to analyze the structure of $(v^0)$.

Abstract

The $\sigma$-ideal $(v^0)$ is associated with the Silver forcing, see \cite{bre}. Also, it constitutes the family of all completely doughnut null sets, see \cite{hal}. We introduce segments and $*$-segments topologies, to state some resemblances of $(v^0)$ to the family of Ramsey null sets. To describe $add(v^0)$ we adopt a proof of Base Matrix Lemma. Consistent results are stated, too. Halbeisen's conjecture $cov(v^0) = add(v^0)$ is confirmed under the hypothesis $t= \min \{\cf (\frak c), r\} $. The hypothesis $h=\omega_1$ implies that $(v^0)$ has the ideal type $(\frak c, \omega_1,\frak c)$.