I am having some problems proving the following:
Let $D = \mathbb{R}^2 \setminus \{(x_0,y_0)\}$ for some point $(x_0,y_0)\in \mathbb{R}^2$. Show that $D$ cannot be written as $D = D_1 \cup D_2$ where $D_1, D_2$ are open and simply connected regions with $D_1 \cap D_2$ connected.
I have been told to show this by assuming it can be done and then use this to show that any vectorfield $\vec{F}(x,y) = (P(x,y),Q(x,y))$ with $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$ on $D$ is conservative, which would be a contradiction.