strong deformation retract, of a perforated plane?

Question

Let $x_0 \in R^2$. How do I find a circle in $R^2$ which is a strong deformation retract of $R^2-\{x_0\}$? Is it just a unit circle with the center $x_0$?

Answer 1

Yes it is. If $x_0$ is the origin, then the desired deformation retract is $$ H(x, t) = t\dfrac{x}{\|x\|} + (1-t)x. $$

To visualize it, imagine all points on the plane moving radially. Those inside the unit circle move away from the origin towards the unit circle. Those outside move in the opposite direction, towards the circle too.

I'll let you adapt it to the general case.