I'm familiar with this concept and it makes sense, but the proof for it is eluding me.
Let $p \in C$ and consider the set:
$\mathcal{U}=\{\operatorname{ext}(a,b)\mid p\in (a,b)\}$
Therefore no finite subset of $\mathcal{U}$ covers $C \setminus \{p\}$.
It makes sense that a finite number of exteriors will never cover the continuum $C$ ($C$ being nonempty, having no first or last point, ordered ($a