An Open Mapping Approach to Hurwitz's Theorem

It is a consequence of known results^) that any compact light open mapping / of one two-dimensional manifold A into another one B has a definite finite degree k(f), that is, there is an integer k(f) such that for each y£j(A),f~l(y) contains exactly k(f) points provided the points of/_l(y) are counted with multiplicity in an obvious manner. Further, considered as a function on the subset O of compact light strongly open mappings in the mapping space BA, k(f) can be proven lower semi-continuous for very much broader classes of spaces A and B. Upper semi-continuity of k(f) presents considerably more difficulty and requires restrictions on A and B as indicated. However, results will be established below from which it follows that k(f) is indeed continuous on the set O of compact light strongly open mappings of A into B where strongly open means that the image of every open set in A is open in B and not merely open in/(.4) (or, equivalently,/is an open mapping and f(A) is an open set in B). As a consequence of our main theorem, we obtain a conclusion concerning the equality of the number of "y-places" in a given region of two "sufficiently close" light open mappings which is closely related to the classical theorems of Rouché and Hurwitz on the zeros of analytic functions and indeed which includes the Hurwitz theorem in its full strength as a special case. Thus, not only do we have a purely topological approach to this basic result of analysis, but one which leans entirely on the properties and techniques of open mappings and does not even involve any homotopy results or methods. As would be expected, the Hurwitz theorem can also be obtained by means of homotopy results and by using the concept of degree of a mapping defined in terms of homology or cohomology group mappings. The major difficulty in this latter is encountered in showing that, in the cases under consideration, the two types of degree are numerically the same. This and other related questions will be considered in a later paper. Usually our spaces A and B are assumed to be locally connected generalized continua, that is, separable, metric, locally compact, connected, and locally connected. Any different conditions on the spaces will be clearly