Editing Jordan curve theorem (section)

== Definitions and the statement of the Jordan theorem ==

A ''Jordan curve'' or a ''simple closed curve'' in the plane <math>\mathbb{R}^2</math> is the [[image (mathematics)|image]] <math>C</math> of an [[injective]] [[continuous map]] of a [[circle]] into the plane, <math>\varphi: S^1 \to \mathbb{R}^2</math>. A '''Jordan arc''' in the plane is the image of an injective continuous map of a closed and bounded interval <math>[a, b]</math> into the plane. It is a [[plane curve]] that is not necessarily [[Curve#Differential curve|smooth]] nor [[algebraic curve|algebraic]].

Alternatively, a Jordan curve is the image of a continuous map <math>\varphi: [0,1] \to \mathbb{R}^2</math> such that <math>\varphi(0) = \varphi(1)</math> and the restriction of <math>\varphi</math> to <math>[0,1)</math> is injective. The first two conditions say that <math>C</math> is a continuous loop, whereas the last condition stipulates that <math>C</math> has no self-intersection points.

With these definitions, the Jordan curve theorem can be stated as follows:

{{math theorem|math_statement=
Let <math>C</math> be a Jordan curve in the plane <math>\mathbb{R}^2</math>. Then its [[complement (set theory)|complement]], <math>\mathbb{R}^2 \setminus C</math>, consists of exactly two [[connected component (topology)|connected component]]s. One of these components is [[bounded set|bounded]] (the '''interior''') and the other is unbounded (the '''exterior'''), and the curve <math>C</math> is the [[boundary (topology)|boundary]] of each component.
}}

In contrast, the complement of a Jordan ''arc'' in the plane is connected.