Bouquet of circles and sphere with k punctures homotopy equivalence

A sphere with k points removed is homotopy equivalent to a bouquet (or a wedge sum) of k-1 circles.
One may also notice, that previous statement easily follows from current statement, since k-punctured sphere can be deformed to a (k-1)-punctured plane through a stereographic projection.
More generally, Sⁿ without k points is homotopy equivalent to a wedge sum of (k-1) copies of Sⁿ⁻¹.

[igorkanzakhidov]

Wow, that's incredible! Thank you so much! I wish I'd seen it before I tried to show R^3\{union of 2 axes} ~ wedge sum of 3 circles. :(

[SachchidanandPrasad]

Hi! Your videos are nice. Could you tell which software are you using to make these animations? 

Thanks!