A surface has finite topology if it is homeomorphic to a compact surface with finitely many points removed. For a connected orientable surface without boundary, this means finite genus and finitely many ends (Meeks and Pérez 2012). Each removed point corresponds to an end of the surface.
The plane and helicoid each have genus 0 and one end. The catenoid has genus 0 and two ends, while the Costa surface has genus 1 and three ends. Finite topology is a topological condition and does not by itself imply conformal equivalence to a punctured compact Riemann surface.
