The aim is to demonstrate that every definable G set has a definable slice at each point.
Consider a definably compact definable group G and a definable G set X.
Prove the existence of definable slices at every point of X.
Show that X can be covered by finitely many definable G tubes.
Every point in the definable G set X has an associated definable slice.
The set X is sufficiently structured, being covered by finitely many definable G tubes.
Abstract
Let G be a definably compact definable group and X a definable G set.We prove that there exists a definable slice at every point of X and X is covered by finitely many definable G tubes.