A remark about critical sets in $R^3$

TL;DR

This paper establishes a necessary condition for a closed subset of R^3 to be a critical set of a smooth function, ruling out certain complex sets like the Whitehead continuum and p-adic solenoid.

Contribution

It provides a new criterion for identifying which closed sets in R^3 can serve as critical point sets of smooth functions, advancing understanding in differential topology.

Findings

Whitehead continuum is not a critical set

p-adic solenoid is not a critical set

Provides a necessary condition for critical sets in R^3

Abstract

We give a necessary condition for a closed subset of $R^3$ to be the set of critical points of some smooth function. In particular we obtain that for example neither the Whitehead continuum nor the p-adic solenoid are such a critical sets.