TITLE: Is the plane minus a line segment homeomorphic with punctured plane?
QUESTION [18 upvotes]: Is $\mathbb R^2$ minus a line segment i.e. $\mathbb R^2 \setminus ([0,1]\times \{0\}) $ homeomorphic with a punctured plane $\mathbb R^2\setminus \{(0,0)\}$ ?

REPLY [4 votes]: This map should do:
$$(x,y)\mapsto\begin{cases}(x+\operatorname{sgn}(x)(|y|-1),y),&|x|>1\\(xy,y),&|x|\le1\end{cases}$$