The Riemann mapping theorem states that for every non-empty, simply connected, open set $\Omega \subset \mathbb{C}$, which is not all of $\mathbb{C}$ there exists a function $f$ that maps $\Omega$ onto the open unit disc $D := \{ z \in \mathbb{C} : | z | < 1 \}$ and is biholomorphic.

I am currently reading a proof (about the construction of the Faber polynomials) that seems to assume that in the case that the complement of $\Omega$ is compact, the map $f$ fulfills the following property. For every open set $A \subseteq \mathbb{C}$ that contains the boundary of $\Omega$ there exists $\varepsilon > 0$ such that the set $S_\varepsilon := \{ z \in \mathbb{C} : | z | = 1 - \varepsilon \}$ fulfills $f^{-1}(S_\varepsilon) \subseteq A$.

In other words, $f^{-1}$ maps curves that are close to the boundary of $D$ to curves that are close to the boundary of $\Omega$.

If you prove the theorem by constructing a harmonic function that maps the boundary of $\Omega$ onto the boundary of $D$ (which requires that the boundary of $\Omega$ can be given by a smooth curve), the statement above is clear. Is the statement, however, also true in general? Is there a simple proof that shows that a biholomorphic function that maps $\Omega$ onto $D$ has the above property?

**Edit** Added the requirement that $\Omega^\textrm{c}$ should be a compact set.