On finite-dimensional maps II

Let f :X→Y be a perfect n-dimensional surjective map of paracompact spaces and Y a C-space. We consider the following property of continuous maps g :X→Ik=[0,1]k, where 1⩽k⩽ω: each g(f−1(y)), y∈Y, is at most n-dimensional. It is shown that all maps g∈C(X,In+1) with the above property form a dense Gδ-set in the function space C(X,In+1) equipped with the source limitation topology. Moreover, for every n+1⩽m⩽ω the space C(X,Im) contains a dense Gδ-set of maps having this property.