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.It was established in [1] that either a continuous mapping of a closed domain onto an n-dimensional manifold that has an odd degree of mapping at the boundary of the domain is a homeomorphism or there exists a point in the image of the domain whose preimage contains at least three points. 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 having at least three preimages contains a subset of total dimension n.In the present paper, we show 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. Definition 1. If X and Y are locally compact spaces, then the mapping f : X → Y is called proper if the preimage of an arbitrary compact set belonging to Y is a compact set in X.
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