Fréchet manifold
In mathematics, in particular in nonlinear analysis, a Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space. More precisely, a Fréchet manifold consists of a Hausdorff space formula_1 with an atlas of coordinate charts over Fréchet spaces whose transitions are smooth mappings. Thus formula_1 has an open cover formula_3 and a collection of homeomorphisms formula_4 onto their images, where formula_5 are Fréchet spaces, such that formula_6 is smooth for all pairs of indices formula_7 Classification up to homeomorphism. It is by no  means true that a finite-dimensional manifold of dimension formula_8 is homeomorphic to formula_9 or even an open subset of formula_10 However, in an infinite-dimensional setting, it is possible to classify "well-behaved" Fréchet manifolds up to homeomorphism quite nicely. A 1969 theorem of David Henderson states that every infinite-dimensional, separable, metric Fréchet manifold formula_1 can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, formula_12 (up to linear isomorphism, there is only one such space). The embedding homeomorphism can be used as a global chart for formula_13 Thus, in the infinite-dimensional, separable , metric case, up to homeomorphism, the "only" topological Fréchet manifolds are the open subsets of the separable infinite-dimensional Hilbert space. But in the case of or Fréchet manifolds (up to the appropriate notion of diffeomorphism) this fails.