Let $(X,\tau)$ be a topological space such that $\tau$ contains no singleton. We say that a map $c:X\to \kappa$, where $\kappa$ is a cardinal, is a *coloring* for $(X,\tau)$, if for every $U\in \tau\setminus \{\emptyset\}$ the restriction $c|_U$ is non-constant. (Note that this coloring notion comes from hypergraph coloring.)

The *chromatic number* $\chi(X,\tau)$ of a space $(X,\tau)$ is the smallest cardinal $\kappa$ such that there is a coloring $c:X\to \kappa$.

We have $\chi(\mathbb{R})=2$ when $\mathbb{R}$ is endowed with the Euclidean topology: color $\mathbb{Q}$ with $0$ and $\mathbb{R}\setminus\mathbb{Q}$ with $1$. This works for all spaces having a dense set $D$ such that $X\setminus D$ is also dense. Note that not all connected $T_2$-spaces contain a dense subset with this property.

Given an integer $n>2$ is there a connected Hausdorff space $(X,\tau)$ such that $\chi(X,\tau) = n$?