Homotopy equivalence for $S^n$ with finite k punctures I need help determining to what $S^n/${k points}--the n-dimensional sphere missing a finite k number of points-- is homotopy equivalent.
I tried envisioning the above for n=2:
$S^2\setminus${1 point} is homotopy equivalent to a point.
$S^2\setminus${2 points} is homotopy equivalent to $S^1$.
$S^2\setminus${3 points} is homotopy equivalent to $\vee_2S^1$, or the union of 2 $S^1$ joined at 1 point. 
Assuming that the above are correct, I believe that I can generalize to $S^n\setminus${k points} is homotopy equivalent to $\vee_{k-1}S^{n-1}$. 
Thus, I have 2 main questions.


*

*Is my generalization correct?

*And if so, how do I show this homotopy equivalence rigorously? I came up with the specific examples for $S^2$ by actually envisioning a sphere with k punctures.

(1) Yes
(2) The starting point is observing that a once-punctured $n$-sphere is homeomorphic to $\mathbb{R}^n$, so if you remove $k$ points from $S^n$, you get $\mathbb{R}^n$ with $(k-1)$ points removed.
