# Multiplicity of Continuous Mappings of Domains

### Abstract

We prove that either the proper mapping of a domain of an*n*-dimensional manifold onto a domain of another

*n*-dimensional manifold of degree

*k*is an interior mapping or there exists a point in the image that has at least |

*k*|+2 preimages. If the restriction of

*f*to the interior of the domain is a zero-dimensional mapping, then, in the second case, the set of points of the image that have at least |

*k*|+2 preimages contains a subset of total dimension

*n*. In addition, we construct an example of a mapping of a two-dimensional domain that is homeomorphic at the boundary and zero-dimensional, has infinite multiplicity, and is such that its restriction to a sufficiently large part of the branch set is a homeomorphism.

Published

25.04.2005

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 57, no. 4, Apr. 2005, pp. 554–558, http://umj.imath.kiev.ua/index.php/umj/article/view/3620.

Issue

Section

Short communications