# Curve selection lemma

There are many versions of the curve selection lemma.

In o-minimal structures, we have to consider definable curves, definable functions, deinable sets,… The definition of definable sets,… we can find in many documents.

Let $fr(A)$ be frontier of $A$, ie $fr(A) = \bar{A} - A$. We have:

Curve selection lemma: In the o-minimal structure $\mathcal{O}$. If $x \in fr(A)$, then there is a definable map $\gamma: [0, 1) \to \mathbb{R}^n$ such that $\gamma(0,1) \subset A$ and $\gamma(0) = x$.