Discrete open mappings on manifolds
Jussi Väısälä · Annales Academiae Scientiarum Fennicae Series A I Mathematica · 1967
Discrete open mappings on manifolds l.Introiluction.The pu{pose of this paper is to give a new proof for the following recent result of Cernavskii [2], [3]: If / is an open mapping of an z-manifold into an z-manifold such that each point-inverse consists of isolated points, then / is a local homeomorphism except for a set whose topological dimension is at most n -2.Our proof is more direct and elementary than that of öernavskii, who obtained.this result as a corollary of a more general result, concerning rnappings whose range space was not, required to be a manifold.In particular, we avoid completely the use of the rather deep theory of Smith on periodic homeomorphisms of manifolds' Instead, we will make use of the topological index (: degree) of a mapping.We will also give an application concerning light open mappings.2. Terru,inol,ogy, notation anil, preliminary results.All topological spaces considered in this paper are assumed to be Hausdorff.All manifolds are åssumed to be connected and to have a countable base.AII mappings are assumedto becontinuous.If X isaspaceandif EcAcX, welet 018 denote the boundary of E v'ith respect to A, and we abbreviate 0*E to 08.Let' f be a mapping of a space X into a space Y.It is open, (closed,) if the image of every open (closed) subset of X is open (closed) in I. It,istight if for each y e Y, the inverse-image l-'(y) i* totally disconnected.It, is d,iscrete if each point-inverse is discrete, i.e. consists of isolated points. It is proper if the inverse-image of each compact, subset of I is compact.If X and Y arelocallycompact,thenamapping f :X---> Y isproperif and only if / is closed and each point-inverse is compact.The branch set B, of / is the set of all points in X at which I fails to be a local homeo- morphism.Them,ultipl,icity I{(r,J) of f at a point r €X is the number of points in f-tf(r), and we set I/(/) : suP Ii(* ,f) over all r € X.The set of all points re X for which N(*,f) ( i is denoted by K,(/).It is well known that if / is open, then I{(r,J) is a lower semicontinuous function of r.In other words, for each a e X and for each integer k I N(q,f) thereexistsaneighborhood U of asuchlhab N(r,f)>k forall re U.In particular,if N(a,l){ *, then N(r,f)2N(a,/) in some' neighborhood of a. X'rom this it follows that the sets 1(,(/) are closed"