the topological type of M depends not only on these two numbers but also on how the infinite one(s) approach the space of ends. In general the topological type of M is determined by the four subspaces of E(M) that are limit points of infinitely many handles and infinitely many projective planes, limit points of only handles, limit points of only projective planes, and limit points of neither. Assumption of second-countability If one removes the assumption of second-countability from the definition of a surface, there exist (necessarily non-compact) topological surfaces having no countable base for their topology. Perhaps the simplest example is the Cartesian product of the long line with the space of real numbers. Another surface having no countable base for its topology, but not requiring the Axiom of Choice to prove its existence, is the Prüfer manifold, which can be described by simple equations that show it to be a real-analytic surface. The Prüfer manifold may be thought of as the upper half plane together with one additional "tongue" Tx hanging down from it directly below the point (x,0), for each real x. In 1925, Tibor Radó proved that all Riemann surfaces (i.e., one-dimensional complex manifolds) are necessarily second-countable (Radó's theorem). By contrast, if one replaces the real numbers in the construction of the Prüfer surface by the complex numbers, one obtains a two-dimensional complex manifold (which is necessarily a 4-dimensional real manifold) with no countable base. Surfaces in geometry Polyhedra, such as the boundary of a cube, are among the first surfaces encountered in geometry. It is also possible to define smooth surfaces, in which each point has a neighborhood diffeomorphic to some open set in E2. This elaboration allows calculus to be applied to surfaces to prove many results. Two smooth surfaces are diffeomorphic if and only if they are homeomorphic. (The analogous result does not hold for higher-dimensional manifolds.) Thus closed surfaces are classified up to diffeomorphism by their Euler characteristic and orientability. Smooth surfaces equipped with Riemannian metrics are of foundational importance in differential geometry. A Riemannian metric endows a surface with notions of geodesic, distance, angle, and area. It also gives rise to Gaussian curvature, which describes how curved or bent the surface is at each point. Curvature is a rigid, geometric property, in that it is not preserved by general diffeomorphisms of the surface. However, the famous Gauss–Bonnet theorem for closed surfaces