Let $X$ be a manifold modeled on a Banach space. Let $E$ be a fixed Banach space, and for any point $x\in X$ there exists a local chart toplinear isomorphism $\psi_x:U_x\to\psi_x(U_x)$ where $\psi_x(U_x)$ is some Banach space.
Let $S_E$ be the set of points $x\in X$ such that there exists some nbhd $\tilde{U}$ of $x$ and a toplinear isomorphism $\tilde{\psi}:\tilde{U}\to E$.
How does one see that $S_E$ is clopen in $X$?
It is clear that is is open, from its very definition: for each $x\in S_E$, take that same $U$, and it will lie inside of $S_E$ automatically.
However, why is it closed? I'm not sure, because if $x\in\bar{S_E}-S_E$, then for any $U\in Nbhd_X(x)$, $U\cap S_E\neq\varnothing$ and $\nexists$ isomorphism $\psi:U\to E$. But how to go on from here?