# Nowhere dense covering number of a connected $T_2$ space

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 $\nu(X) = \kappa$?

• This is just a remark: if instead of nowhere dense sets we consider coverings by meager sets, we obtain the classical covering number of $X$, usually written as Cov$(X)$. But you are probably aware of that. – Mathieu Baillif Jul 17 '17 at 15:15
• What about $\ \kappa=2\$ ? – Wlod AA Jul 25 '17 at 11:29
• I don't think your older question is asking what you claim it is. Any space has $v(X)>2$. – Ramiro de la Vega Jul 25 '17 at 11:46
• (@RamirodelaVega, I was joking. Sorry. PS. Ok, Taras has answered the question fully in his comment to this question and to the follow up question). – Wlod AA Jul 25 '17 at 11:49
• @MathieuBaillif Note that $v(X)=cov(X)$ except for meager spaces where $v(X)=\aleph_0$ and $Cov(X)=1$. – Ramiro de la Vega Jul 25 '17 at 11:52