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)

References

[a1]	G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10
[a2]	G.T. Whyburn, "Topological analysis" , Princeton Univ. Press (1964)