Let f : X→ Y be a perfect surjective map of paracompact spaces. It is shown that if Y is a C-space (resp., Y≤ n and 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 H such that f× g embeds X into Y× I∞ (resp., into Y× I²ⁿ⁺¹⁺ᵐ) for every g∈ H. Some applications of this result are also given.
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Tuncali et al. (2004) studied this question.