TITLE: Producing non-trivial homotopy classes in $\infty$-dimensional vector spaces by removing points
QUESTION [7 upvotes]: It is renowned that removing a point from an $\infty-$dimensional vector space $H$ preserves its contractibility (by Kuiper's theorem).
Is there any hope that $H\setminus A$ is not contractible, where $A$ is a disjoint union of points (possibly infinitely many, but still countable)?

REPLY [9 votes]: In comments it is stated that the space of most interest to the OP is the 
space of $C^k$ Riemannian metrics on a compact manifold. 
If you look at the space of $C^k$ Riemannian metrics with $C^k$ topology, where $k$ is finite or infinite, then the space is Fréchet and separable. In any separable Fréchet space $X$ the complement of any countable union of compact sets is homeomorphic to $X$ (this is proved in Bessaga-Pełczyński book, see definition IV.5.1 and section V.6).
Perhaps, the OP is interested in the space of $C^\infty$ Riemannian metrics with $C^k$ topology, where $k$ is finite. Theorem 1.1 in https://arxiv.org/abs/1510.07269 determines the homeomorphism type of the space. Namely, if $\Sigma$ denotes the linear span of the Hilbert cube in $l^2$, then the space of the $C^\infty$ Riemannian metrics with $C^k$ compact-open topology is homeomorphic to $\Sigma^\omega$, the product of countably many copies of $\Sigma$. I do not know whether any countable subset of $\Sigma^\omega$ has a contractible complement.