Parametric bing and Krasinkiewicz maps

It is shown that if f : X → Y is a perfect map between metrizable spaces and Y is a C-space, then the function space C ( X , I ) with the source limitation topology contains a dense G δ -subset of maps g such that every restriction map g y = g | f −1 ( y ) , y ∈ Y , satisfies the following condition: all fibers of g y are hereditarily indecomposable and any continuum in f −1 ( y ) either contains a component of a fiber of g y or is contained in a fiber of g y .