Winding-number proof of the no-retraction theorem
= Winding-number proof of the no-retraction theorem
If a continuous retraction from a closed disc to its boundary existed, applying it to a contraction of the boundary circle inside the disc would give a homotopy in $\mathbb C\setminus\{0\}$ from a loop of winding number one to a constant loop of winding number zero. <Homotopy invariance of winding number> rules this out.