Jordan Curve

A Jordan curve is a simple, closed curve in a plane that does not cross itself. Imagine drawing a loop with a pencil without lifting it and without crossings—such as a circle, square, or any smoothly closed shape with no overlaps. The key idea is that the curve divides the plane into two parts: inside and outside, with the curve as the boundary. Jordan's theorem states that every such simple closed curve indeed separates the plane into exactly two regions, highlighting fundamental properties of shapes and their boundaries in mathematics.