Definition

1. Noun:
   • A simple closed curve in a plane: A "Jordan curve" is a non-self-intersecting, continuous loop in a plane. It divides the plane into exactly two distinct regions: an interior and an exterior.

Usage Examples

• Noun:
  • A circle is a classic example of a Jordan curve.
  • The theorem states that any Jordan curve divides the plane into an inside and an outside region.

Advanced Usage

• "Jordan curve theorem": A fundamental theorem in topology stating that every Jordan curve divides the plane into an interior region (which is bounded) and an exterior region (which is unbounded), and any continuous path connecting a point in the interior to a point in the exterior must intersect the curve.
  • The proof of the Jordan curve theorem is more complex than its intuitive statement suggests.

Variants and Related Words

• Simple closed curve (n): A synonym for a Jordan curve; a continuous loop that does not cross itself.

  • In topology, a simple closed curve is often called a Jordan curve.

• Jordan arc (n): A simple curve that is not closed; it does not intersect itself but has two distinct endpoints.

  • Extending a Jordan arc by connecting its endpoints creates a Jordan curve.

Synonyms

• Simple closed curve: A non-self-intersecting continuous loop.
• Simple loop: An informal term for a simple closed curve.

Related Concepts

• Planar curve: A curve that lies in a single plane.

  • A Jordan curve is a specific type of planar curve.

• Topology: The branch of mathematics concerned with the properties of geometric objects that are preserved under continuous deformations.

  • The concept of a Jordan curve is central to planar topology.

Noun

1. a closed curve that does not intersect itself