On regularly branched maps

Abstract

Let f: X → Y be a perfect map between finite-dimensional metrizable spaces and p ≥ 1. It is shown that the space C* (X, ℝp) of all bounded maps from X into ℝp with the source limitation topology contains a dense Gδ-subset consisting of f-regularly branched maps. Here, a map g : X → ℝp is f-regularly branched if, for every n ≥ 1, the dimension of the set {z ∈ Y × ℝp: (f × g)-1 (z) ≥ n} is ≤ n. (dim f + dim Y) - (n - 1) · (p + dim Y). This is a parametric version of the Hurewicz theorem on regularly branched maps. © 2004 Elsevier B.V. All rights reserved.