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 $$X$$ with an atlas of coordinate charts over Fréchet spaces whose transitions are smooth mappings. Thus $$X$$ has an open cover $$\left\{ U_{\alpha} \right\}_{\alpha \in I},$$ and a collection of homeomorphisms $$\phi_{\alpha} : U_{\alpha} \to F_{\alpha}$$ onto their images, where $$F_{\alpha}$$ are Fréchet spaces, such that $$\phi_{\alpha\beta} := \phi_\alpha \circ \phi_\beta^{-1}|_{\phi_\beta\left(U_\beta\cap U_\alpha\right)}$$ is smooth for all pairs of indices $$\alpha, \beta.$$

Classification up to homeomorphism
It is by no means true that a finite-dimensional manifold of dimension $$n$$ is homeomorphic to $$\R^n$$ or even an open subset of $$\R^n.$$  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 $$X$$ can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, $$H$$ (up to linear isomorphism, there is only one such space).

The embedding homeomorphism can be used as a global chart for $$X.$$ 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.