The sphere \( \mathbb{S}^n \) is connected since it is a quotient space of \( \mathbb{R}^{n+1} \setminus \{0\} \) ((0x684a8ab2) ) and \( \mathbb{R}^{n+1} \setminus \{0\} \) is connected (since it is path-connected). See (0x681c2fdb) .
The sphere \( \mathbb{S}^n \) is connected since it is a quotient space of \( \mathbb{R}^{n+1} \setminus \{0\} \) ((0x684a8ab2) ) and \( \mathbb{R}^{n+1} \setminus \{0\} \) is connected (since it is path-connected). See (0x681c2fdb) .