Carathéodory's extension theorem

Carathéodore's Extension Theorem is a theorem that helps us understand how certain shapes are formed. To understand it, you first need to know what a "shape" is. A shape is a collection of points that can be connected in some way. For example, a square is a shape that has four points that are connected to form a square.

Carathéodore's Extension Theorem tells us that if we start with a shape that is made up of some points, we can add more points to it, and still create a shape. In other words, we can extend the shape without changing the shape itself.

To explain this in simpler terms, imagine you have a square on a piece of paper. The square is made up of four points that are all connected to form the shape. With Carathéodore's Extension Theorem, you can add four more points around the existing four points, and they will form a new shape (like an octagon). But even though you added four new points, the original shape (the square) still remains the same - all eight points are still connected and form the square shape.