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?
2026-03-30 02:53:34.1774839214
If $\dim X=\infty$, can there be a compact $C$ such that $X\setminus C$ has two components?
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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.