# Sto\"ilow's theorem revisited

## TL;DR

This paper revisits Stoilow's theorem, providing a detailed proof that continuous, open, and light maps between surfaces are locally modeled by power maps and admit holomorphic factorization.

## Contribution

It offers an accessible proof of Stoilow's theorem, clarifying the conditions under which such maps are discrete and can be factored holomorphically.

## Key findings

- Maps are locally modeled by power maps z↦z^k
- Such maps are discrete with a discrete branch set
- They admit a holomorphic factorization

## Abstract

Sto\"ilow's theorem from 1928 states that a continuous, open, and light map between surfaces is a discrete map with a discrete branch set. This result implies that such maps between orientable surfaces are locally modeled by power maps $z\mapsto z^k$ and admit a holomorphic factorization. The purpose of this expository article is to give a proof of this classical theorem having readers in mind that are interested in continuous, open and discrete maps.