A branched covering map 
between surfaces is a continuous
map that is surjective and, in suitable coordinate
charts centered at each point 
and its image 
, has the form 
for a positive
integer 
.
The integer 
is the ramification index. When 
, the map is locally a homeomorphism.
Points with 
are ramification points, and their images
are branch points. The set of branch points is the
branch locus, and the map restricts to a covering
map away from it.
Every nonconstant holomorphic map between compact connected Riemann surfaces is a branched covering map of this form.
