Finally, we get to Step 3 of the proof of this theorem: establishing a theorem of the complement for the “right” collection of existentially definable sets. We start with a few exercises.
Exercises
Let $\tau_n:\RR^n \into (-1,1)^n$ be the semialgebraic diffeomorphism defined here.
- Show that, for every semialgebraic set $A \subseteq \RR^n$, the set $\tau_n(A)$ is a semi-$\C$-set.
- Show that, for every restricted $\C$-function $\bar f:\RR^n \into \RR$, the graph of $\tau_1 \circ \bar f \circ \tau_n^{-1}$ is a semi-$\C$-set. (Note that the graph of $\tau_1 \circ \bar f \circ \tau_n^{-1}$ is the image under $\tau_{n+1}$ of the graph of $\bar f$.)
Definition
A global semi-$\C$-set is a set $A \subseteq \RR^n$ such that $\tau_n(A)$ is a semi-$\C$-set. A global sub-$\C$-set is a projection of a global semi-$\C$-set.
Exercise
- Show that every global sub-$\C$-set is existentially definable in $\RR_\C$.
Lemma
The collection of all global sub-$\C$-sets is closed under taking finite unions, finite intersections, projections and cartesian products.
Proof. We leave it as an exercise to check that the collection of all semi-$\C$-sets is closed under taking finite unions, finite intersections and cartesian products. It follows that the collection of all global semi-$\C$-sets has the same closure properties. This in turn implies that the collection of all global sub-$\C$-sets is closed under taking finite unions, projections and cartesian products. For closure of the latter under taking finite intersections, note that the last two properties imply that we only need to show that the intersection of a global sub-$\C$-set $A \subseteq \RR^n$ with a semialgebraic set $B \subseteq \RR^n$ is again a sub-$\C$-set. However, $A = \Pi_n(C)$ for some global semi-$\C$-set $C \subseteq \RR^m$, where $m \ge n$; since $$A \cap B = \Pi_n(C \cap (B \times \RR^{m-n})),$$ closure under taking finite intersections also follows. $\qed$
In view of the above lemma and exercises, and since every global sub-$\C$-set has finitely many connected components, this theorem will follow once we prove that the complement of a global sub-$\C$-set is again a global sub-$\C$-set. We will show this in an axiomatic setting based on the properties of global sub-$\C$-sets, similar to, but distinct from, this theorem of the complement.