PARTIAL PROPERNESS AND REAL PLANAR MAPS

If a local homeomorphism of the real plane, h = (f,g) : R(2) --> R(2), is not one-to-one, then both f and g have fibers with connected compo nents whose images under h are disjoint. Global homeomorphisms h are c haracterized in terms of properness along fibers of, and simplicity of , component maps. Two known counterexamples to univalence conjectures (for Samuelson maps and polynomial maps) are used to illustrate the re sults.