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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."

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The Jordan curve theorem tells us that the plane is divided into two disjoint regions. The interior is a bounded region. The exterior is unbounded. Saying much more than that is probably fairly difficult -- as difficult as, say, proving the Jordan curve theorem. – Bill Cook Nov 8 '11 at 14:29
I 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? – analysisj Nov 8 '11 at 15:00
up vote 3 down vote accepted

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 $C$.

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So if it was odd or even would it be inside C? – analysisj Nov 8 '11 at 16:04
Hint: Think about a very simple example to get the parity right. – Mark Bennet Nov 8 '11 at 16:30
So if the parity was odd, then it would be inside C, correct? – analysisj Nov 8 '11 at 17:01

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