Topological spaces are of analytic nature. Open sets, given in a topological space by definition, lead to such notions as continuous functions, paths, maps; convergent sequences, limits; interior, boundary, exterior. However, uniform continuity, bounded sets, Cauchy sequences, differentiable functions (paths, maps) remain undefined. Isomorphisms between topological spaces are traditionally called homeomorphisms; these are one-to-one correspondences continuous in both directions. The open interval (0,1) is homeomorphic to the whole real line (-∞,∞) but not homeomorphic to the closed interval [0,1], nor to a circle. The surface of a cube is homeomorphic to a sphere (the surface of a ball) but not homeomorphic to a torus. Euclidean spaces of different dimensions are not homeomorphic, which seems evident, but is not easy to prove. The dimension of a topological space is difficult to define; inductive dimension (based on the observation that the dimension of the boundary of a geometric figure is usually one less than the dimension of the figure itself) and Lebesgue covering dimension can be used. In the case of a Euclidean space, both topological dimensions are equal to "n".