This is a special case of a [question](https://mathoverflow.net/questions/275489/nowhere-dense-covering-number-of-a-connected-t-2-space) that has not been answered so far.


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](https://en.wikipedia.org/wiki/Partition_of_a_set) of $X$ into [nowhere dense subsets](https://en.wikipedia.org/wiki/Nowhere_dense_set) of $X$ can have.


**Question.** Is there a connected Hausdorff space $(X,\tau)$ such that $\nu(X) = 3$?



Connected $T_2$-spaces with nowhere dense covering number $3$
Since the union of two nowhere dense sets is nowhere dense, the number $\nu(X)$ is always infinite. For meager spaces it is countable. But it cannot be equal to 3.


