This is a generalization of an older question.

If $(X,\tau)$ is a connected $T_2$ space with more than 1 point, we define its nowhere dense covering number $\nu(X)$ by the smallest cardinality that a partition of $X$ into nowhere dense subsets of $X$ can have. (The older question can be reformulated as asking whether there are connected Hausdorff spaces $X$ with $\nu(X) > 2$; the answer is yes.)

Question. If $\kappa > 2$ is a cardinal, is there a connected Hausdorff space $(X,\tau)$ such that $\nu(X) = \kappa$?

(As an aside, it would be interesting to know if $\nu(\cdot)$ has been studied before, and what name that concept was given.)

This is a generalization of an older question.

If $(X,\tau)$ is a connected $T_2$ space with more than 1 point, we define its nowhere dense covering number $\nu(X)$ by the smallest cardinality that a partition of $X$ into nowhere dense subsets of $X$ can have. (The older question can be reformulated as asking whether there are connected Hausdorff spaces $X$ with $\nu(X) > 2$; the answer is yes.)

Question. If $\kappa > 2$ is a cardinal, is there a connected Hausdorff space $(X,\tau)$ such that $\nu(X) = \kappa$?