The sphere \(\mathbb{S}^n\) is a quotient space of \(\mathbb{R}^{n+1}\setminus \{0\}\). To see this consider the map \(q\colon \mathbb{R}^{n+1}\setminus \{0\}\to \mathbb{S}^n\) defined by \(q(x)=x/\lvert x\rvert\). This is a quotient map because it takes closed saturated subsets to to closed subsets (see (0x67a4b655) ).