Let $S^n$ the unit sphere in $\mathbb{R}^{n+1}$ and $I=[0,1]$ the unit interval.
By using polar and spherical coordinates, we can show that
$S^1\text{ is homeomorphic to }I/\partial I,\text{ and}$ $S^2\text{ is homeomorphic to }I^2/\partial (I^2)$ (with the quotient topology).
But what about higher dimensions ? Do we have $S^n\simeq I^n/\partial(I^n)$ ? The usage of trigonometric function can't be generalized, can it ?
Same question for $S^{n}\simeq \partial (I^{n+1})$: we can construct homeomorphisms in $\mathbb{R}^2$ and $\mathbb{R}^3$ through projections on the faces, but the argument seems to be very hard (or a lot of work) to generalize.