I am looking for some examples of partitioning of a connected topological space $X$ by pairwise disjoint closed connected subspaces. Off course this partition would consist of infinitely many such connected closed sets otherwise the space $X$ would not be connected.
Also, I know that we can take singleton sets, but I am looking for subspaces which are not totally disconnected.
Thank you.
The plane ($\Bbb R^2$, standard topology) is a disjoint union of circles around the origin (one for each radius, including $0$, so $\{0\}$ as well). All of the circles are connected, and the plane is too. Parallel lines will work too.
By Sierpinski's theorem, we cannot have a countable partition of closed sets for a compact and connected Hausdorff $X$ (a continuum).