TITLE: CW structure on infinite-dimensional manifolds
QUESTION [6 upvotes]: It is well-known (due to this work of Palais, I believe) that Banach manifolds are dominated by countable CW complexes. It then follows (due Whitehead, as indicated by Milnor in this work) that they have the homotopy type of CW complexes. 

Are there conditions ensuring that a Banach or Fréchet manifold admits a genuine CW complex structure?
What are the main examples of that?

Actually, I'm working with Banach fiber bundles, which are Serre fibrations. I would like to regard them as Hurewicz fibrations to use HLP (homotopy lifting property). Due to this answer of Peter May (and the references cited by him), I know that it is enough to get a CW structure.
Any help is welcome.

REPLY [6 votes]: Let $M$ be a Hilbert manifold. Then there exists a Riemannian metric $g$ on $M$ such that the induced metric $d$ is complete (See https://arxiv.org/pdf/1610.01527.pdf by Biliotti and Mercuri). Thus $M$ has the topology of a complete metric space. By the Baire category theorem $(M,d)$ is a Baire space: It cannot be written as a countable union of closed sets with empty interior. 
If $M$ had the structure of a CW complex, it does. It must clearly be an infinite dimensional $CW$-complex, but such a thing is the countable union of its $n$-skeleta, which have empty interior.
With this argument you can treat also other infinite dimensional manifolds: using the fact that seperable Frechet spaces are homeomorphic to seperable Hilbert space, one can show that a Frechet manifold is homeomorphic to a Hilbert manifold.