ABSTRACT
Holomorphic mappings in several variables are even locally considerably more diverse and complicated than the familiar mappings defined by holomorphic functions of one variable. But there is a special class of general holomorphic mappings having many of the same local geometric properties as the mappings defined by holomorphic functions of one variable. It is perhaps clearest and most convenient to begin the discussion of this class of mappings by considering their purely topological properties. The some of the topological regularity conditions introduced in the discussion of general finite branched coverings are automatically fulfilled for all finite branched holomorphic coverings; so in particular it will always be possible to speak of the germ of a finite branched holomorphic covering and of its branching order.
