# A strange application of the Heine-Borel lemma (Ahlfors)

In Ahlfors' Complex Analysis text, pages 289-290, the author discusses analytic continuation along arcs. Among other things, he proves the equivalence with analytic continuation using a chain of function elements $(f_k ,\Omega_k)$.

The fact that two function elements being related by a chain of direct analytic continuations implies that an appropriate analytic continuation along an arc is possible is clear to me. However, in page 290 Ahlfors is trying to prove the converse. Namely, that the arc from the analytic continuation can be "followed" by a chain of direct analytic continuations, from the initial point of the arc to its endpoint.

His words are:

Conversely, if $\bar{\gamma}$ is given, we can find a chain of direct analytic continuations which follows the arc $\gamma$ in the same way as in the preceding construction. In fact, by Heine-Borel's lemma the parametric interval $[a,b]$ can be subdivided into $[a,t_1], [t_1,t_2], \dots , [t_{n-1},b]$ such that $\bar{\gamma}(t)$ = $(f_k,\gamma(t))$ in $[t_{k-1},t_k]$ for suitably chosen function elements $(f_k,\Omega_k)$. Although $(f_k,\Omega_k)$ and $(f_{k+1},\Omega_{k+1})$ need not be direct analytic continuations of each other, they are at least direct continuations of their common restrictions to a neighborhood of $\gamma(t_k)$.

I'm having a hard time understanding this. As far as I know the Heine Borel lemma characterises compact subsets of Euclidean space alone as bounded and compact ones. The only set I see here which lies in Euclidean space is the parameter interval $[a,b]$ of the arc. But, even if that's where the lemma is applied, the outcome should be a finite open subcover of $[a,b]$, while the intervals $\{[t_{k-1},t_k]\}$ are only closed.

P.S. Here $\bar{\gamma}:[a,b] \to \mathfrak{S_0}(\mathbf{f})$ takes values on some Riemann surface, and $\gamma= \pi \circ \bar{\gamma}:[a,b] \to \mathbb C$ is its projection.