# Interior mapping

A mapping $f: X \rightarrow Y$ from a topological space $X$ into a topological space $Y$ such that the image of any set $U$ open in $X$ is also open in $Y$, while the inverse image $f ^ { - 1 } ( y)$ of any point $y \in Y$ is totally disconnected (i.e. does not contain connected components other than points).

Let $F$ map some Riemann surface $R$ into the sphere $S ^ {2}$; a homeomorphism $T: M \rightarrow R$ from an oriented surface $M$ will then induce a mapping

$$\widetilde{F} = F \circ T: M \rightarrow S ^ {2} ,$$

which is topologically equivalent with $F$. For an analytic function $F$ and some mapping $\widetilde{F}$ to be topologically equivalent it is necessary and sufficient for $\widetilde{F}$ to be an interior mapping (then there exists a homeomorphism $T$ such that $\widetilde{F} = F \circ T$) (Stoilow's theorem).

The local structure of the interior mapping $\widetilde{F} : M \rightarrow \mathbf R ^ {2}$ may be described as follows. For any point $a \in M$ there exist a neighbourhood $U( a)$ and homeomorphisms $T _ {1} : B \rightarrow U( a)$ of the unit disc $B = \{ {z \in \mathbf R ^ {2} } : {| z | < 1 } \}$ onto $U( a)$ and $T _ {2} : \widetilde{F} ( U( a)) \rightarrow B$ such that $T _ {2} \circ \widetilde{F} \circ T _ {1} = z ^ {n}$.

#### References

 [1] S. [S. Stoilov] Stoilow, "Leçons sur les principes topologiques de la théorie des fonctions analytiques" , Gauthier-Villars (1938)