# Connected $T_2$-spaces with nowhere dense covering number $3$

This is a special case of a question 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 of $X$ into nowhere dense subsets of $X$ can have.

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

• Since the union of two nowhere dense sets is nowhere dense, the number $v(X)$ is always infinite. For meager spaces it is countable. But it cannot be equal to 3. – Taras Banakh Jul 25 '17 at 11:17
• Note that Taras' comment doesn't assume that the space is connected or Hausdorff. – Ramiro de la Vega Jul 25 '17 at 11:44
• BTW, when space $\ X\$ has an isolated point than $\ \nu(X):=\infty\$ by definition (it'd be only natural). – Wlod AA Jul 25 '17 at 11:51
• Thanks @taras, can you post your comment here as an answer or in the more general question? – Dominic van der Zypen Jul 25 '17 at 16:28

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.