The passage discusses the characterization of left curves in topology, focusing on the properties and characteristics of such curves. It begins by defining a left curve and stating that a left curve cannot be a dendrite. The author then presents key results, including the characterization of left curves in terms of specific geometric configurations, such as curves composed of edges of a tetrahedron and segments connecting the tetrahedron's vertices to its center of gravity. These curves are shown to be essential in understanding the structure of left curves, particularly in the context of Peano continua that contain only a finite number of simple closed curves. The text also delves into the topological properties of Peano continua, proving that any such continuum that contains a finite number of simple closed curves must contain either a curve of the first or second type mentioned. Additionally, it discusses the reduction of the problem of left curves to a planar topology problem, demonstrating that any Peano continuum that contains a finite number of simple closed curves and two non-conjugate points must contain a curve homomorphic to one of the specified types. Finally, the passage concludes with a theorem stating that any polyhedral left surface (except the sphere) contains a curve homomorphic to the first or second type mentioned. This theorem is derived from the properties of the surface's triangulation and the existence of a simple closed curve that is not a cut of the surface.The passage discusses the characterization of left curves in topology, focusing on the properties and characteristics of such curves. It begins by defining a left curve and stating that a left curve cannot be a dendrite. The author then presents key results, including the characterization of left curves in terms of specific geometric configurations, such as curves composed of edges of a tetrahedron and segments connecting the tetrahedron's vertices to its center of gravity. These curves are shown to be essential in understanding the structure of left curves, particularly in the context of Peano continua that contain only a finite number of simple closed curves. The text also delves into the topological properties of Peano continua, proving that any such continuum that contains a finite number of simple closed curves must contain either a curve of the first or second type mentioned. Additionally, it discusses the reduction of the problem of left curves to a planar topology problem, demonstrating that any Peano continuum that contains a finite number of simple closed curves and two non-conjugate points must contain a curve homomorphic to one of the specified types. Finally, the passage concludes with a theorem stating that any polyhedral left surface (except the sphere) contains a curve homomorphic to the first or second type mentioned. This theorem is derived from the properties of the surface's triangulation and the existence of a simple closed curve that is not a cut of the surface.