Let $X$ and $Y$ be 2 disjoint connected subsets of $\mathbb{R}^{2}$, can $X \cup Y =\mathbb{R}^{2}$? Let $X$ and $Y$ be 2 disjoint connected subsets of $\mathbb{R}^{2}$. Then can $$X \cup Y =\mathbb{R}^{2}$$
I think this cannot be true, but I don't know of a formal proof. Any help would be nice.

consider $X:=\{(x,0) :x>0\}$ and $Y:=\mathbb{R^2}-X$ ,Both are connected and disjoint but $X\cup Y=\mathbb{R^2}$
