A surjective mapping between compact spaces is
said to be (see [59, Chapter 3 and 4, p. 12-28]):
- a local homeomorphism provided that for each point
there
exists a an open neighborhood U of x such that f(U) is an open neighborhood of f(x)
and that f restricted to U is a homeomorphism between U and f(U).
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih 2001-02-21
between compact spaces is
said to be (see [59, Chapter 3 and 4, p. 12-28]): 
there
exists a an open neighborhood U of x such that f(U) is an open neighborhood of f(x)
and that f restricted to U is a homeomorphism between U and f(U).