On regularly branched maps

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⋅(dimf+dimY)−(n−1)⋅(p+dimY). This is a parametric version of the Hurewicz theorem on regularly branched maps.