TITLE: Are $\{(x,y)\in \mathbb{R}^2 : (x,y)\neq(0,0)\}$ and $\{(x,y)\in \mathbb{R}^2 : (x,y)\notin [0,1]\times\{0\}\}$ homeomorphic?
QUESTION [8 upvotes]: Let $X_1$ and $X_2$ be the spaces
\begin{align*}
X_1&=\{(x,y)\in \mathbb{R}^2 : (x,y)\neq(0,0)\}, \\
X_2&=\{(x,y)\in \mathbb{R}^2 : (x,y)\notin [0,1]\times\{0\}\}.
\end{align*}
Are these spaces homeomorphic? If these spaces are homeomorphic, what is the homeomorphism map? If these spaces are not homeomorphic, why?
I suspect these spaces are homeomorphic, but I can't construct homeomorphism.

REPLY [2 votes]: On $A=I\times([-1,1]\setminus\{0\})$ let $f:A\to\Bbb R^2$ be the map
$$f(x,y)=(x|y|,\,y)$$
on $B=[1,\infty)\times[-1,1]\setminus\{(1,0)\}$ let $f:B\to\Bbb R^2$ be the map 
$$f(x,y)=(x-(1-|y|),\,y)$$
and on $C=\Bbb R^2\setminus\left((0,\infty)\times(-1,1)\cup\{(0,0)\}\right)$ let $f:C\to\Bbb R^2$ be defined by 
$$f(x,y)=(x,y)$$
Can you show that $f:X_2\to X_1$ is continuous and find its inverse?