On Finite-to-One Maps

Abstract

AbstractLet f : X → Y be a σ-perfect k-dimensional surjective map of metrizable spaces such that dim Y ≤ m. It is shown that for every positive integer p with p ≤ m + k + 1 there exists a dense Gδ-subset (k,m, p) of C(X, k+p) with the source limitation topology such that each fiber of f△g, g ∈ (k, m, p), contains at most max ﹛k + m – p + 2, 1﹜ points. This result provides a proof the following conjectures of S. Bogatyi, V. Fedorchuk and J. van Mill. Let f : X → Y be a k-dimensional map between compact metric spaces with dimY ≤ m. Then: (1) there exists a map h: X → m+2k such that f△h: X → Y × m+2k is 2-to-one provided k ≥ 1; (2) there exists a map h: X → m+k+1 such that f△h: X →Y × m+k+1 is (k + 1)-to-one.