I'm trying to characterise the connected subsets of the Moore plane. I already know the whole space is connected because it is the closure of the open superior half-plane, which has the Euclidean topology and is, thus, connected. I think the connected subsets of this space are the ones contained in the closure of a connected subset of the open superior half-plane. However, I'm not able to prove this.
In fact, it is a well-known theoretic result that all of this sets are connected. If I'm given a set whose intersecion with the open superior half-plane is connected but that contains points that are not in its closure, I think i'm able to prove it's not connected. However, I don't know what to do when the intersection with the open half-plane is not connected. It should be possible to prove the whole set cannot be connected , but I don't see how.
EDIT: OK, so this characterisation doesn't work. Could anyone give a good one?
If $A$ has the property that $A \cap \Bbb H$ (the intersection with the upper half plane) is connected, then $A$ is connected iff $A \cap (\Bbb R \times \{0\}) \subseteq \overline{A \cap \Bbb H}$.
If $A = \Bbb H \setminus \{(0,y): y >0\} \cup \{(0,0)\}$, then $A \cap \Bbb H$ has two open connected components with a common point in their closures, so $A$ is connected. So the characterisation is not yet complete.