This paper will be
concerned with the local structure of open, continuous functions f : M2→ N1 from a
2-manifold into the real line or the circle. The two main results are:
Theorem 1. Let f : M2→ N1 be an open mapping, and suppose that f has
isolated branch points. Then for each point p in M2 there are a neighborhood U of p
and a positive integer d (depending on p) such that f|U is topologically equivalent to
the real analytic mapping z → Re(zd). THEOREM 2. If f : M2→ N1 is an open,
real analytic mapping, then f has isolated branch points, and (hence) the conclusion
of Theorem 1 holds.
