Interior mapping

A mapping from a topological space into a topological space such that the image of any set open in is also open in , while the inverse image of any point is totally disconnected (i.e. does not contain connected components other than points).

Let map some Riemann surface into the sphere ; a homeomorphism from an oriented surface will then induce a mapping

which is topologically equivalent with . For an analytic function and some mapping to be topologically equivalent it is necessary and sufficient for to be an interior mapping (then there exists a homeomorphism such that ) (Stoilow's theorem).

The local structure of the interior mapping may be described as follows. For any point there exist a neighbourhood and homeomorphisms of the unit disc onto and such that .

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)