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 , R p ) of all bounded maps from X into R p with the source limitation topology contains a dense G δ -subset consisting of f-regularly branched maps. Here, a map g : X → R p is f-regularly branched if, for every n ⩾ 1 , the dimension of the set { z ∈ Y × R p : | ( 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.