Is $\mathbb C \setminus [0,\infty)$ simply connected? Is $\mathbb R^2 \setminus ( [0,\infty)\times \{0\}) $ simply connected ? My guess is it is , but I can only show it is path connected , apart from that I am stuck . Please help . Thanks in advance 

Hint.
$$ (x,y)\in\mathbb R^2 \mapsto -(e^x+iy)^2 \in \mathbb C\setminus[0,\infty)$$
is a homeomorphism.
