Possible definition of infinite-dimensional manifold. Upon a quick google search for a definition of an infinite-dimensional manifold, we find that there are some different choices for definitions available. One definition using Banach spaces was discussed on MathOverflow here, for example.
Given that a $d$-dimensional manifold is a Hausdorff space which is locally homeomorphic to $\mathbb{R}^d = \bigcup_{n=1}^d \mathbb{R}^n $, a n alternative possible definition that came to mind - but that I can't see anywhere - is that an infinite-dimensional manifold is a Hausdorff topological space which is locally homeomorphic to $\bigcup_{n=1}^{\infty} \mathbb{R}^n$.
My question is, is this a definition which has been used anywhere, and if so, where?
On the other hand, is there anything that I'm overlooking which means that this definition doesn't make sense somehow?
 A: What you need to define manifolds is a "model" TVS, a topological vector space $E$, usually assumed to be Hausdorff. Then you define an $E$-manifold as a Hausdorff topological space which is locally homeomorphic to $E$. Which space (or a class of spaces) $E$ to use, depends on the objective: What are you planning to do with the $E$-manifolds, which properties do you find desirable, etc. Your example of $E= {\mathbb R}^\infty$, equal to the inductive limit of ${\mathbb R}^n$'s, is a perfectly reasonable TVS. However, it is lacking one property that might be desirable for applications, namely, the Baire Property: Each nonempty ${\mathbb R}^\infty$-manifold will contain a sequence of nowhere dense  (closed) subsets whose union has nonempty interior. This is because each ${\mathbb R}^n\subset {\mathbb R}^\infty$ has empty interior, but the union of these finite-dimensional subspaces equals the entire ${\mathbb R}^\infty$.
Edit. I did not see this class of manifolds anywhere in the literature, but, given vastness of the literature on infinite-dimensional spaces, it would not surprise me if somebody did use them for something. On the other hand, lack of the Baire Property suggests that the number of such references is likely to be small.
