TITLE: Prove that the unit circle cannot be a retract of $\mathbb{R}^2$-Munkres sec 35 exercise 4
QUESTION [5 upvotes]: Munkres topology section 35(Tietze Extension Theorem) exercise 4-(c). The question is

Can you conjecture whether or not $S^1$ is a retract of $\mathbb{R}^2$?

I've read the answers in Is the unit circle $S^1$ a retract of $\mathbb{R}^2$?, but I don't know about algebra(algebraic topology) or Brouwer Fixed Point Theorem. Is there any way of proving this using at most calculus, analysis, elementary complex analysis and elementary topology(for example, things that appear in the first half of Munkres topology)? I already know from the exercise that $S^1$ is a retract of $\mathbb{R}^2 - \{0\}$.

REPLY [4 votes]: A retract of a contractible space is contractible.
But $S^1$ is not contractible.
If you don't know about the fundamental group, or Brouwer's fixed point theorem, I still think there's an equivalent formulation in terms of extensions of maps from $S^1$ to the disc.  It's equivalent to Brouwer, but says there's no extension of the identity function of $S^1$.