Every simple closed curve in the plane divides the plane into exactly two regions, an interior and an exterior, and the curve is the common boundary of both. Named for Camille Jordan, its statement is visually obvious but its rigorous proof is surprisingly delicate.
Facts
Disputed
Proof YearJordan published a claimed proof in 1887; for decades many mathematicians considered that proof flawed and credited Oswald Veblen with the first rigorous proof, but this assessment has since been challenged by Thomas Hales and others, so which proof was genuinely first and correct is not settled among historians of mathematics. StatementEvery simple closed curve in the plane divides the plane into exactly two regions, a bounded interior and an unbounded exterior, and any continuous path from a point in one region to a point in the other must cross the curve. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Proved By
Sources
1. Jordan Curve Theorem (Wikipedia)
Wikimedia FoundationLead section
For decades, mathematicians generally thought that this proof was flawed and that the first rigorous proof was carried out by Oswald Veblen; however, this notion has been overturned by Thomas C. Hales and others.
In Group: Geometric and Differential Topology, Lead paragraph
In topology, a discipline within mathematics, the Jordan curve theorem (JCT), formulated by Camille Jordan in 1887, asserts that every simple closed curve in the plane (known as a Jordan curve) divides the plane into two regions: the interior, bounded by the curve, and an unbounded exterior, containing all of the nearby and faraway exterior points.
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