Caratheodory's extension theorem is an important theorem in mathematics, which has an important influence on geometry. The proof method of this theorem is to construct a sphere, then define an inner product on this sphere, and finally prove that this inner product satisfies some properties, thus proving Caratheodory's extension theorem.

Caratheodory's continuation theorem is widely used in geometry, for example, in differential geometry, it can be used to study the properties of curves and surfaces; In topology, it can be used to study the properties of topological space; In algebraic geometry, it can be used to study the properties of algebraic clusters and so on.