Can I define the $S^1$ and $S^2$ manifolds as follows? (by using equivalence relations $\sim$)
For $S^1$, let us define in $\mathbb R$ the equivalence class $\sim^1$ by
$x \sim^1 y \iff \exists k\in\mathbb{Z} : x-y = 2 k\pi$
Then, $S^1 = \mathbb R/\sim^1 $?
For, $S^2$, let us define $M = S^1 \times [0,\pi]$ and the equivalence relation $\sim^2$ defined by
$(x,y) \sim^2 (x', y') \iff \{y=y'=0 \, \, \rm{ or} \,\, y=y'=\pi\}$
Then, $S^2 = M/\sim^2$?
And, I guess, something similar for $S^n$.