How to describe the 2D region bounded by a closed curve?

Question

Let C be some closed curve in $\mathbb{R}^2$. How would you describe the set $D$ which represents the region bounded by $C$ as an expression of $C$ using set notation. For instance, if $C$ were a circle of radius 1 centered at the origin, it could be written in set theory notation as $C=\{(x,y)\in\mathbb{R}^2\mid{}x^2+y^2=1\}$. Ultimately, I'm hoping for a rigorous way to define what the set $D$ in Green's theorem ($\oint_C{\vec{F}\cdot{}d\vec{r}}=\iint_D{(\nabla\cdot{}\vec{F})}{dA}$) is as some set theory expression of $C$.

Answer 1

This doesn't answer your question since it's a bit of a cheat. Let $B$ be a simple closed curve and $D$ be the set of points in "inside" $B$. How do we explicitly define $D$. Well one cheat is to first find a point $d \in D$. Then say that any curve that contains $d$ and doesn't contain any point in $B$ is itself in $D$. There's your explicit definition. Of course, this is cheating since I never proved that $d \in D$.

Answer 2

Maybe

$$\{x\in \mathbb R^2\setminus C:\text{there is no continous}\,\gamma:[0,1]\to \mathbb R^2\setminus C\,\text{with}\,\gamma(0)=x\, \text{and}\,\|\gamma(1)\|>\max_{c\in C}\|c\|\}$$

If you want the boundary to belong to your set just add $C$.

This definition captures the intuitive fact, that a point inside the curve cannot be joint to a point outside the curve without crossing the curve.