Question about closed curves and surfaces

Question

Let $\mathbb{r}:[a,b]\to\mathbb{R}^2$ be a continuous non-intersecting loop (i.e. $\mathbb{r}(a)=\mathbb{r}(b)$ and $\mathbb{r}(x)\neq\mathbb{r}(y),\ \forall \{x,y\}\neq\{a,b\}$). Denote by $D$ the open domain enclosed by $\mathbb{r}$. Prove or disprove that there is a continuous function $F:\mathbb{R}^2\to\mathbb{R}^2$ such that $F(x,y)=0$ iff $(x,y)\in \partial D$ and $F(x,y)<0$ iff $(x,y)\in D$.

Answer 1

The "difficult part" in this connection is the fact that $K:=r\bigl([a,b]\bigr)$ separates ${\Bbb R}^2$ into two disjoint open sets $D$ and $\Omega$, one of them (named $D$) bounded. This is the Jordan curve theorem.

In order to construct a function $F: \>{\Bbb R}^2\to{\Bbb R}$ of the required kind we make use of the following fact: Given a compact set $K\subset {\Bbb R}^2$ the function $$x\mapsto d(x,K):=\inf_{y\in K} |y-x|$$ ("distance from $x$ to $K$") is continuous. Given this, put $$F(x):=\left\{\eqalign{-d(x,K)\qquad&(x\in D) \cr d(x,K)\qquad&(x\in\Omega)\ .\cr}\right.$$