Suppose $F$ is a closed set in $\mathbb{R}^n$ with $n>1$. Are there some known conditions that must be imposed on $F$ so that its complement in $\mathbb{R}^n$ has a finite number of components? Equivalently, that the zeroth homology group $H_0(\mathbb{R}^n\setminus F)$ is isomorphic to $\oplus_{k=1}^m\mathbb{Z}$.
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1$\begingroup$ There's a problem with your last formula. $\endgroup$– Greg FriedmanCommented Feb 7, 2022 at 5:49
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4$\begingroup$ Certainly relevant: en.wikipedia.org/wiki/Alexander_duality $\endgroup$– Connor MalinCommented Feb 7, 2022 at 6:10
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