This paper, authored by Karl Menger, delves into the general theory of curves, focusing on the concept of the order of curve points and the properties of curves. The author introduces the notion of a curve as a compact, connected space where every point has arbitrarily small neighborhoods with connected boundaries. Points are classified as regular if they have neighborhoods with finite boundaries, and their order is defined by the maximum number of such neighborhoods. Points with infinite order are those where the boundaries of neighborhoods grow without bound. The paper discusses several key aspects: 1. **Order of Curve Points**: It explores the relationship between the order of a point in a regular curve and the number of intersecting and foreign arcs at that point. The author proves that for regular dendritic curves, each point of order \( n \) has \( n \) intersecting foreign arcs, and for points of infinite order, there are countably many such arcs with converging diameters. 2. **Extensive Curves**: The paper examines the existence of extensive curves, which contain every other curve as a topological subset. An example of such a curve is constructed using a recursive process of dividing a cube into smaller cubes and removing certain subcubes. 3. **Points of Infinite Order**: The author defines the concept of "gender" for half-regular points, which are points with neighborhoods that have empty higher-order derivatives. The paper discusses the structure of these points and their classification into types based on their gender and the number of points in their neighborhoods. The paper concludes with a discussion on the structure of spaces and the existence of condensed sets of points with high gender, providing foundational results for the theory of curves and dimensions.This paper, authored by Karl Menger, delves into the general theory of curves, focusing on the concept of the order of curve points and the properties of curves. The author introduces the notion of a curve as a compact, connected space where every point has arbitrarily small neighborhoods with connected boundaries. Points are classified as regular if they have neighborhoods with finite boundaries, and their order is defined by the maximum number of such neighborhoods. Points with infinite order are those where the boundaries of neighborhoods grow without bound. The paper discusses several key aspects: 1. **Order of Curve Points**: It explores the relationship between the order of a point in a regular curve and the number of intersecting and foreign arcs at that point. The author proves that for regular dendritic curves, each point of order \( n \) has \( n \) intersecting foreign arcs, and for points of infinite order, there are countably many such arcs with converging diameters. 2. **Extensive Curves**: The paper examines the existence of extensive curves, which contain every other curve as a topological subset. An example of such a curve is constructed using a recursive process of dividing a cube into smaller cubes and removing certain subcubes. 3. **Points of Infinite Order**: The author defines the concept of "gender" for half-regular points, which are points with neighborhoods that have empty higher-order derivatives. The paper discusses the structure of these points and their classification into types based on their gender and the number of points in their neighborhoods. The paper concludes with a discussion on the structure of spaces and the existence of condensed sets of points with high gender, providing foundational results for the theory of curves and dimensions.