Critical sets in 3-space
The paper is devoted to the problem when a nonempty compact set \(C\) in \(\mathbb{R}^ 3\) is the set of critical points for a proper, smooth function on \(\mathbb{R}^ 3\). It is proved that this is true if \(C\) is Antoine's necklace in \(\mathbb{R}^ 3\). In addition one establishes that any tame link in \(\mathbb{R}^ 3\) is the set of critical points for some smooth functions but generally not for a proper function.
scientific article; zbMATH DE number 508544
scientific article