Functions $f:X\to X$ with no fixed points, for $X$ a punctured disk or a sphere.

Question

$X$ is the punctured closed unit disc $D^2-\{0\} = \{(x, y) \in \mathbb{R}^2: 0 \lt x^2+y^2 \le 1\}$

Is the answer that $f$ maps all $(x,y)$ to $0$ which is not included in the unit disk and so there is no fixed point?

What if $X$ is unit sphere $S^2 = \{(x, y, z) \in \mathbb{R}^3: x^2+y^2+z^2=1\}$?

Answer 1

$f$ can't map anything to something that does not exist.

For the punctured disk, any rotation about the missing point is either a self map with no fixed points, or it is the identity. This gives you plenty of possibilities.

For the sphere $S^2$, the antipodal map, $f(p)=-p$, has no fixed points.