TITLE: If $\dim X=\infty$, can there be a compact $C$ such that $X\setminus C$ has two components?
QUESTION [5 upvotes]: Let $X$ be an infinite-dimensional normed space. Can there be a compact $C$ such that $X\setminus C$ has two components?
My guess is no because compact sets are kind of "small" (their interiors are empty). But how can we prove it?

REPLY [3 votes]: The answer is "no". In fact, $X \setminus C$ is homeomorphic to $X$. See for example Corollary 5.1 in
Bessaga, Czesław, and Aleksander Pełczyński. Selected topics in infinite-dimensional topology. Vol. 58. Panstwowe wyd. naukowe, 1975.