# Hatcher 0.2

Hatcher, Algebraic Topology, Chapter 0

2. Construct an explicit deformation retraction of $\mathbb{R}^n\setminus\{0\}$ onto $S^{n-1}$.
Proof. This problem is essentially a problem from Calculus III. Note that for a vector $\mathbf{x}$ in $\mathbb{R}^n\setminus\{0\}$, the normalized vector $\mathbf{x}/\|\mathbf{x}\|$ lies on $S^{n-1}$. It suffices, then, to do the normalization process in a way that’s continuous for a time parameter $t\in I$, and one way to accomplish this is to define a family $f_t:\mathbb{R}^n\setminus\{0\}\to S^{n-1}$ so that, for each $\mathbf{x}=(x_1,x_2,\ldots,x_n)$ in the domain,

$f_t(x_1,x_2,\ldots,x_n)=\left(\dfrac{x_1}{t\|\mathbf{x}\|+(1-t)},\cdots,\dfrac{x_n}{t\|\mathbf{x}\|+(1-t)}\right).$

As noted in problem 1, the function $f_t$ is continuous for each $t\in I$. Moreover, $f_0(\mathbf{x})=\mathbf{x}$, $f_1(\mathbf{x})=\mathbf{x}/\|\mathbf{x}\|$, and $f_t(\mathbf{x})|_{S^{n-1}}=\mathbf{x}$ due to the fact that $\|\mathbf{x}\|=1$ for all $\mathbf{x}\in S^{n-1}$.    $\,\,\square$