if it is diffeomorphic to a compact surface minus a finite set. If is a closed surface of genus then the surface obtained by removing points from is usually denoted and its Teichmüller space by Teichmüller spaces and hyperbolic metrics Every finite type orientable surface other than the ones above admits complete Riemannian metrics of constant curvature . For a given surface of finite type there is a bijection between such metrics and complex structures as follows from the uniformisation theorem. Thus if the Teichmüller space can be realised as the set of marked hyperbolic surfaces of genus with cusps, that is the set of pairs where is a hyperbolic surface and is a diffeomorphism, modulo the equivalence relation where and are identified if is isotopic to an isometry. The topology on Teichmüller space In all cases computed above there is an obvious topology on Teichmüller space. In the general case there are many natural ways to topologise , perhaps the simplest is via hyperbolic metrics and length functions. If is a closed curve on and a marked hyperbolic surface then one is homotopic to a unique closed geodesic on (up to parametrisation). The value at of the length function associated to (the homotopy class of) is then: Let be the set of simple closed curves on . Then the map is an embedding. The space has the product topology and is endowed with the induced topology. With this topology is homeomorphic to In fact one can obtain an embedding with curves, and even . In both case one can use the embedding to give a geometric proof of the homeomorphism above. More examples of small Teichmüller spaces There is a unique complete hyperbolic metric of finite volume on the three-holed sphere and so the Teichmüller space of finite-volume complete metrics of constant curvature is a point (this also follows from the dimension formula of the previous paragraph). The Teichmüller spaces and are naturally realised as the upper half-plane, as can be seen using Fenchel–Nielsen coordinates. Teichmüller space and conformal structures Instead of complex structures of hyperbolic metrics one can define Teichmüller space using conformal structures. Indeed, conformal structures are the same as complex structures in two (real) dimensions. Moreover, the Uniformisation Theorem also implies that in each conformal class of Riemannian metrics on a surface there is a unique metric of constant curvature. Teichmüller spaces as representation spaces Yet