An example of a retract that is not a deformation retract.

Question

I was told that the retraction of $S^2$ into a point is a retract but not a deformation retract , could someone explain this to me please?

If I understood this wrongly, could someone give me an example of a retract that is not a deformation retract?

Answer 1

Any point $p\in S^2$ is a retract of $S^2$, because the constant map $f:S^2\to\{p\}$ given by $f(x)=p$ is continuous and fixes $p$.

Nonetheless, no point is a deformation retract of $S^2$, because that would mean $S^2$ would be contractible, and that is false.

Intuitively, you can map $S^2$ to $p$ continuously (namely, by sending all $S^2$ to $p$), but you cannot continuously shrink the sphere to a point while staying in the sphere itself: there is always a hole inside.