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Let $G \subset \mathbb{C}$ be a simply connected bounded domain. By the Riemann mapping theorem there is a holomorphic bijection $f : G \to \mathbb{D}$.

An accessible point of $\partial G$ is an equivalence class of jordan half intervals $\gamma : [0,1) \to G$ where two jordan half intervals $\gamma_1, \gamma_2$ are said to be equal if $\lim_{t \to 1^{-}} \gamma_1(t) = \lim_{t \to 1^{-}} \gamma_2(t) = \zeta$ for some $\zeta \in \partial G$ and there is a jordan half interval $\gamma_3$ such that for any neighbourhood $N$ of $\zeta$ in $G$ $\gamma_3$ intersects $\gamma_1$ and $\gamma_2$ in $N$. Abusing notation I will identify an accessible point with its final point $\zeta$ ( note $\zeta$ may correspond to many (even uncountably many) accessible points).

The point of this is that $f$ admits a continuous extension at an accessible point. Precisely,

For every accessible point $\zeta$, $f$ admits a continuous extension to $\zeta$ such that $f(\zeta) \in \partial \mathbb{D}$ and distinct accessible points are sent to distinct points on $\partial \mathbb{D}$.

Given an accessible point $\zeta$ one defines $f(\zeta) = \lim_{t \to 1^{-}} f(\gamma(t))$ for any $\gamma$ in the accessible point. I have no problem showing that $f$ is well-defined and that distinct accessible points are sent to distinct boundary points, but am stuck on showing that $f$ is continuous at $\zeta$. To me it seems that for this theorem to even make sense one would have to provide a topology for $G$ together with its set of accessible points. Any ideas on how to proceed or a reference to a proof? Many thanks!

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