Fréchet manifold

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

Classification up to homeomorphism

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

The embedding homeomorphism can be used as a global chart for ${\displaystyle X.}$ 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 differentiable or smooth Fréchet manifolds (up to the appropriate notion of diffeomorphism) this fails.