Connected subset of the circle path connected?

Question

Consider a connected subset $X$ of the circle $S^1.$ Is $X$ path connected?

Answer 1

Let $C \subseteq S^1$ be a connected set. Now if $C = S^1$, then $C$ is trivially path connected. Otherwise there exists $p \in S^1 \setminus C$. Now, take the stereographic projection $\varphi : S \setminus \{p\} \to \mathbb{R}$, which is a homeomorphism. Thus, $\varphi(C)$ is connected on $\mathbb{R}$, and so it is path connected: the only connected sets on the reals are the intervals. But then again $\varphi$ is a homeomorphism, so $C$ was path connected to begin with.