Proving Homeomorphism: D^n / S^{n-1} to S^n using n=1,2,3 cases

[latentcorpse]

If $D^n$ is the unit n ball in Euclidean n-space. i.e.
$D^n = \{ x \in \mathbb{R}^n : ||x|| \leq 1 \}$
and $S^n$ is an n-sphere.

how do i show that $D^n / S^{n-1}$ is homeomorphic to $S^n$?
there's a hint suggesting i first of all try the n=1,2,3 cases. where $X/Y= X \backslash Y \cup \{ t \}$ where $t \in X$ is a single distinguished point.

i'm not really sure how to start.
i can visualise for example the n=1 case where $D^1 \backslash S^0=[x-1,x) \cup (x,x+1]$ and $S^1$ is the unit circle so if we take t=x then $D^1/S^0=[x-1,x+1]$and i was thinking a homeomorphism here would be some sort of function along the lines of $f(y)= \left( \cos{(y+1) \pi},\sin{(y+1) \pi} \right)$. i think this works as a homeomorphism but do i have to run through the steps of showing it's bijective and continuous and that the inverse is continuous?

 

[Quantumpencil]

Follow the hint. Think about for example, the The circle minis one point being homeomorphic to the line . You should be able to be explicit about this homeomorphism. It makes sense right? take away one point, and then "fold"" the surface out. Show that R is homeomorphic to any open interval (this is also easy and explicit).

Think about this for a while.

 

[latentcorpse]

is the homeomorphism i gave above wrong and if so why? also, why am i considering the circle minus one point? surely i want the line minus one point unioned with one point to be homeomorphic to the enitre circle?