The text discusses the concept of "non-planar curves" in topology, focusing on characterizing such curves intrinsically. It begins with a mathematical framework involving certain properties of curves and their relationships with regions and boundaries. The main problem is to determine the intrinsic characteristics of non-planar curves, which cannot be homeomorphic to any curve on the plane. The text references important results by mathematicians like Ważewski and Ayres, who showed that non-planar curves cannot be dendrites or contain certain specific structures. The paper then delves into the properties of Peano continua (continuous curves) that contain only a finite number of simple closed curves. It proves that such continua must contain specific non-planar curves, such as those depicted in figures 1 and 2, which are composed of edges of a tetrahedron and additional segments. These curves are characterized by specific points of order, and they are not homeomorphic to any curve on the plane. The text also explores the topological properties of surfaces, particularly non-planar surfaces, and shows that they must contain specific non-planar curves. It uses a combination of topological separation theorems and properties of continua to establish these results. The paper concludes by reducing the problem of characterizing non-planar curves to a problem in the topology of the plane, demonstrating that such curves must have specific structural properties. The key theorems presented include the characterization of non-planar curves in terms of their topological structure and the existence of specific non-planar curves in certain types of continua and surfaces.The text discusses the concept of "non-planar curves" in topology, focusing on characterizing such curves intrinsically. It begins with a mathematical framework involving certain properties of curves and their relationships with regions and boundaries. The main problem is to determine the intrinsic characteristics of non-planar curves, which cannot be homeomorphic to any curve on the plane. The text references important results by mathematicians like Ważewski and Ayres, who showed that non-planar curves cannot be dendrites or contain certain specific structures. The paper then delves into the properties of Peano continua (continuous curves) that contain only a finite number of simple closed curves. It proves that such continua must contain specific non-planar curves, such as those depicted in figures 1 and 2, which are composed of edges of a tetrahedron and additional segments. These curves are characterized by specific points of order, and they are not homeomorphic to any curve on the plane. The text also explores the topological properties of surfaces, particularly non-planar surfaces, and shows that they must contain specific non-planar curves. It uses a combination of topological separation theorems and properties of continua to establish these results. The paper concludes by reducing the problem of characterizing non-planar curves to a problem in the topology of the plane, demonstrating that such curves must have specific structural properties. The key theorems presented include the characterization of non-planar curves in terms of their topological structure and the existence of specific non-planar curves in certain types of continua and surfaces.