Prove that a point $(a,b)$ in $\mathbb{R^2}$ has the same homotopy type as $\mathbb{R^2}$.
If someone could verify my proof that would be great. I just started this learning this material and I want to make sure I understand the detail. Thanks in advance.
Proof: Define function $f:(a,b)\to\mathbb{R^2}$ such that $f(a,b)=(a,b)$. Define function $g:\mathbb{R^2}\to (a,b)$ such that $g(x,y)=(a,b)$. In order to show that $(a,b)$ has the same homotopy type as $\mathbb{R^2}$, we need to show that $g(f)\sim id_{(a,b)}$ and $f(g)\sim id_{\mathbb{R^2}}$. $g(f)=(a,b)=id_{(a,b)}$.
$f(g(x,y))=f(a,b)=(a,b)$. Define function $H:\mathbb{R}^2\times[0,1]\to \mathbb{R^2}$ such that $H((x,y),t)=((x-a)t+a,(y-b)t+b)$. Then $H((x,y),0)=(a,b)=f(g),$ and $H((x,y),1)=(x,y)=id_{\mathbb{R}^2}$.