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In topology, can someone please describe why the sphere $S^2$ is not contactable? Surely it can just 'shrink' to a point?

user53076
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    But imagine trying to shrink $S^2$ to a point while trying to stay inside the sphere itself. Of course you can let it shrink to a point if you let it move through $\mathbb{R}^3$. But the homotopy must be between the identity and a constant map and lie inside the sphere for all $t$. – Zach L. Dec 14 '12 at 12:10

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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 $t=1$.

Alex Youcis
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