> [!theorem] > > $ > \real^{n + 1} \setminus \bracs{0} \iso \mathbb{S}^n \times \real^+ > $ > *Proof*. Identify $\mathbb{S}^n$ with unit vectors on $\real^{n + 1}$, > $ > \pi: \mathbb{S}^n \times \real^+ \to \real \setminus \bracs{0} \quad (\hat x, r) \mapsto r\hat x > $ > which is continuous and has an inverse > $ > x \mapsto \paren{\frac{x}{\norm{x}}, \norm{x}} > $ > that is continuous on each coordinate.