Small infinite partitions and other features of the Nowhere Centered ideal

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.