Given any Jordan curve, how does one describe the set of all points entirely contained within the interior portion of the curve? I would like a formal explanation and definition of those points other than the "interior of the curve."
Jordan Curve Explanation
-
0I understand the Jordan curve theorem but I am just curious how one would refer to the set of all points on the interior of the curve in a formal way? – 2011-11-08
1 Answers
The formal way to go about it would be, let $\mathbb R^2$ be the Euclidean plane. Let $C \subset \mathbb R^2$ be the Jordan curve, this is a continuous 1-1 image of a circle into the plane.
Then consider the path components of $\mathbb R^2 \setminus C$. Let $P$ be a path component, then the closure $\overline{P}$ is either compact or non-compact. Moreover, there are precisely two path components of $\mathbb R^2 \setminus C$, one has compact closure, and the other not.
Heuristically (and with some work this can be turned into a proof if $C$ is PL or smooth), given any point $p \in \mathbb R^2 \setminus C$, consider $\{ p + tv : t \geq 0 \}$ where $v$ is a unit vector. "Generically" this ray intersects $C$ in finitely many points. The parity of this number tells you whether or not $p$ is inside or outside of $C$.
-
0So if the parity was odd, the$n$ it would be i$n$$s$ide C, correct? – 2011-11-08