Let $X$ be a connected, locally compact space and let $K$ be a compact subset of $X$. Is the number of connected components of the complement $X\setminus K$ finite? If it is not, are there some hypotheses over $X$ for finiteness of connected components?
2026-04-17 17:26:04.1776446764
Number of connected components of the complement of a compact
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Choose $X = \mathbb{R}$ and let $K$ to be the Cantor set. Then $X \setminus K$ has infinitely many connected components.