I know that the fundamental group of $X = \mathbb R^2 \setminus \{(0,0)\}$ is the same as the fundamental group of the circle $Y = S^1$, namely $\mathbb Z$.
However, $X$ and $Y$ are not homotopic, i.e. we can't find continuous maps $f:X\to Y, g : Y \to X$ such that $f \circ g$ is homotopic to $id_Y$ and $g \circ f$ is homotopic to $id_X$.
I would like to prove it, but I don't really know how to do it. If such $f$ and $g$ existed, then it would be something like $f : x \mapsto x/\|x\|$, and $g(Y)$ has to be compact... I don't know how to continue.
Any hint would be appreciated. I apologize if this has already been asked.