ON FINITE-DIMENSIONAL MAPS

Let f: X → Y be a perfect surjective map of paracompact spaces. It is shown that if Y is a C-space (resp., dim Y ≤ n and dim f ≤ m), then the function space C(X, I∞) (resp., C(X, I2n+1+ m)) equipped with the source limitation topology contains a dense Gδ-set K such that f × g embeds X into Y × I∞(resp., into Y × I2n+1+m) for every g ∊ K. Some applications of this result are also given.