Suppose that I have an unbounded subset $X \subset \mathbb{R}^n$, definable in the $o$-minimal structure $\mathbb{R}\_{an, exp}$. Is it possible to find an unbounded, analytic and definable curve (i.e. real-analytic set of dimension one) contained in $X$?



Definable curves in definable sets
Clearly you can find an unbounded definable curve $C\subset X$. Any set in $\mathbb{R}\_{an, exp}$ admits a *finite* stratification by real analytic submanifolds. Pick one such stratification of $C$. One of the strata of this stratification is an unbounded real analytic curve contained in $C$ and, a fortiori, in $X$.


