This text presents a comprehensive study on the general theory of curves, focusing on the concept of the order of curve points, the most extensive curves, and points of infinite order. The author, Karl Menger, introduces the notion of a curve as a compact, connected space where every point has arbitrarily small neighborhoods with disconnected boundaries. Regular points, which have finite order, are defined based on the number of boundary points in these neighborhoods. Points of infinite order, or points of increasing order, are those where the number of boundary points grows without bound as the neighborhoods shrink. The paper discusses the implications of these definitions, particularly in relation to the dimension theory, which provides a foundation for the curve theory. It explores the properties of regular curves and their order, showing that a regular curve of order n contains n distinct branches ending at a point. It also addresses the existence of curves that contain infinitely many branches meeting at a point, and the implications of these properties for the structure of curves. The text then delves into the concept of "most extensive curves," which are curves that can contain all other curves as subsets. It references the Sierpiński curve and the Hilbert space as examples of such curves. The paper also discusses the construction of such curves and their properties, including their ability to contain all planar curves. Finally, the study examines points of infinite order, introducing the concept of "genus" for such points, which is an ordinal number from the first or second number class. It explores the classification of these points into different types based on their order and the properties of their neighborhoods. The paper concludes with a theorem that establishes the existence of certain types of curves and points, and highlights the importance of these concepts in the broader context of dimension and curve theory.This text presents a comprehensive study on the general theory of curves, focusing on the concept of the order of curve points, the most extensive curves, and points of infinite order. The author, Karl Menger, introduces the notion of a curve as a compact, connected space where every point has arbitrarily small neighborhoods with disconnected boundaries. Regular points, which have finite order, are defined based on the number of boundary points in these neighborhoods. Points of infinite order, or points of increasing order, are those where the number of boundary points grows without bound as the neighborhoods shrink. The paper discusses the implications of these definitions, particularly in relation to the dimension theory, which provides a foundation for the curve theory. It explores the properties of regular curves and their order, showing that a regular curve of order n contains n distinct branches ending at a point. It also addresses the existence of curves that contain infinitely many branches meeting at a point, and the implications of these properties for the structure of curves. The text then delves into the concept of "most extensive curves," which are curves that can contain all other curves as subsets. It references the Sierpiński curve and the Hilbert space as examples of such curves. The paper also discusses the construction of such curves and their properties, including their ability to contain all planar curves. Finally, the study examines points of infinite order, introducing the concept of "genus" for such points, which is an ordinal number from the first or second number class. It explores the classification of these points into different types based on their order and the properties of their neighborhoods. The paper concludes with a theorem that establishes the existence of certain types of curves and points, and highlights the importance of these concepts in the broader context of dimension and curve theory.