An end of a connected, noncompact surface is a distinct way to leave every compact subset of
the surface. More precisely, consider continuous paths 
that eventually
leave each compact subset of 
. Two such paths represent
the same end iff, for every compact
subset 
of 
, their sufficiently late portions lie in the same connected component of 
(Meeks and Pérez 2012).
For a surface of finite topology without boundary, represented as a compact surface with finitely many points removed, each removed point corresponds to an end. These are missing points in a topological compactification, rather than boundary points of the original surface. The plane has one end, the catenoid has two, and the Costa surface has three. A catenoidal end of a minimal surface is asymptotic to an end of a catenoid. A planar end is asymptotic to a plane (Meeks and Pérez 2012).
