In topology, can someone please describe why the sphere $S^2$ is not contactable? Surely it can just 'shrink' to a point?
The issue with your logic is that you must shrink your space while staying within your original space. For example, your idea is roughly to consider the balls of radius $t$ around $0$ and let $t\to 0$. Now, the definition of the homotopy is a map $H:[0,1]\times S^2\to S^2$--your spaces (maps rigorously) don't lie on (map into) $S^2$ after $t=1$.