On finite-to-one maps

الأصل · EN

Let f X→ Y be a σ-perfect k-dimensional surjective map of metrizable spaces such that Y≤ m. It is shown that, for every positive integer p≥ 1 there exists a dense Gδ-subset H(k,m,p) of C(X,ᵏ⁺ᵖ) with the source limitation topology such that if g∈H(k,m,p), then each fiber of f g contains at most {m+k-p+2,1} points.This result provides a proof of two hypotheses of S. Bogatyi, V. Fedorchuk and J. van Mill.