Small infinite partitions and other features of the Nowhere Centered ideal
Abstract
The \textit{nowhere dense ideal} $\mathcal{NC}$ is introduced. It is a coanalytic ideal of $\omega\times\omega$ whose defining characteristic is that the sets of the form $X\times Y$, where $X,Y$ are infinite subsets of $\omega$, are dense in the quotient $\mathcal{P}(\omega\times\omega)\slash\mathcal{NC}$. This quotient has countable partitions and consistently has partitions of size $\omega_1$ while $\mathfrak{a}>\omega_1$. This represents a huge contrast with other definable quotients. Other combinatorial features of this ideal are presented, as well as some results on a family of similar, higher dimensional ideals.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2022
- DOI:
- 10.48550/arXiv.2203.00234
- arXiv:
- arXiv:2203.00234
- Bibcode:
- 2022arXiv220300234J
- Keywords:
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- Mathematics - Logic