Let $f : [0, 1] \rightarrow \mathbb{R}^2$ be a Jordan curve, $\textit{i.e.}$, $f$ is an injective continuous function with $f(0) = f(1)$.
Let $C$ be the set $f([0,1])$.
Assume that $p$ is an arbitrary point in $\mathbb{R}^2 \setminus C$.
Let $S(p)$ denote the set $\{ \, |f(t) - p| \, : \, ~ 0 \leq t \leq 1 \, \}$, $\textit{i.e.}$, $S(p)$ is the collection of lengths from $p$ to any points in $C$.
Then it is clear that $S(p) \subset (0, \infty)$.
It is my question : Is there an elementary way to prove that $0 < \inf S(p)$?
It is equivalent that we can choose a small ball $B(p;r)$, the ball whose center is $p$ and radius is $r$, which is contained in $\mathbb{R}^2 \setminus C$.
In other words, if we can prove that $C$ is a closed set in $\mathbb{R}^2$, then this problem is proved.