A local homeomorphism is a map 
of topological spaces such that every point 
has an open neighborhood 
for which 
is open in 
and the restriction 
is a homeomorphism.
Every local homeomorphism is a continuous map and
an open map, and each of its fibers
is a discrete set.
Every covering map is a surjective local homeomorphism, but the converse fails. For example, let 
be the upper half-plane.
The map
 |
is a surjective local homeomorphism. Its fiber over 1 consists only of 
, while its fiber over 
has two points. Since the target
is connected, this change in fiber
size rules out a covering map (McMullen 2026, p. 22).
