Producing non-trivial homotopy classes in $\infty$-dimensional vector spaces by removing points 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)?
 A: 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. 
A: No, at least if $H$ is Banach. The argument is similar to this MO post that proves $\Bbb R^3 \setminus S$, where $S$ is some countable set, is simply connected.
Let $S \subset H$ be a countable set. Fix a map $i: S^n \to H \setminus S$; let $X = F(D^n,\partial S^n; H)$ be the space of null-homotopies of $i$: the space of continuous maps $D^n \to H$ that restrict to $i$ on the boundary, equipped with the compact-open topology. As long as $H$ is at least an F-space, $X$ is completely metrizable, so the Baire category theorem applies. Enumerating $S$, let $X_k$ be the space of null-homotopies whose image does not contain $k$. if $H$ is Banach, we can use smooth approximation and the Sard-Smale transversality theorem to see that the $X_k$ are open, dense, nonempty sets, and because Baire applies to $X$ the intersection $\bigcap_k X_k$ is nonempty, providing a null-homotopy of $i$. More generally we can delete the union of countably many finite-dimensional submanifolds of $X$ and preserve (weak!) contractibility.
It seems like an interesting question to ask which topological vector spaces always have $H \setminus S$ weakly contractible.
A: Such things are discussed in 
C. Bessaga and A. Pełczyński
Selected topics in infinite-dimensional topology. 
I don't have access to the book now, but if I remember correctly the principle is that removing a compact set of a Banach manifold does not affect the homotopy --and even the diffeomorphism-- type of the manifold. You should have a look at this book for precise statements. 
