# Closed and bounded intervals of definably complete ordered groups

True or false?

All closed and bounded intervals of definably complete ordered groups are definably compact.

Let $$G$$ be an ordered abelian group. Then, a definable subset $$D ⊆ G$$ is said to be definably compact if for every definable open cover $$\{ϕ(a, G)\}_{a∈G}$$ of $$D$$, there exists a pseudo-finite sub-cover $$\{ϕ(a, G)\}_{a∈P}$$ for some pseudo-finite set $$P = ψ(t_0, G)$$.

• I think it’d be clearer to say: “...there exists a sub-cover $\{\phi(a,G)\}_{a\in P}$ indexed by a definable closed bounded and discrete set $P$”. (Alternatively, if I’m not understanding pseudo-finiteness correctly, that’s worth clarifying in the question!) Mar 14, 2018 at 3:37