Carathéodoryvillkoret

Carathéodory's villkor is a set of conditions for the existence and uniqueness of a multivalued function. It is a result of the work of mathematicians Constantin Carathéodory and Ludwig Otto Hesse. The conditions are necessary and sufficient for a function to be a scalar field in the context of differential forms.

The first condition states that for any point in the domain, the function has at least one value in the image space. The second condition requires that for each point in the domain there exists a neighborhood, which is the set of all points to infinity, such that the function is extremely close to its value. The third condition is that in each neighborhood, the set of values of the function is countable. This is a weak form of continuity.

In more detail, the conditions can be stated as follows. If we regard a function as a multivalued mapping, then the first condition ensures that for each x in the domain, there exists a real number y such that y is a member of the set of values of the function at x. The second condition means that if y' is another value of the function at x, then there is a neighborhood U of x, and there is an y" which is in the image space and which equals y' in U. The third condition states that for any real number t, the set of all such neighborhoods U must be countable.

The significance of Carathéodory's villkor lies in its application to abstract differential geometry, where they are used to establish existence and uniqueness results for certain types of differential forms. The theorem is named after Constantin and many credit him for this important result.