Function spaces in dimension theory and factorization theorems

AbstractThe main result of this paper in the following theorem: given a mapping f:X→Z of a Tychonoff space X into a metrizable space Z of weight τ, for almost every (in the sense of Baire category) mapping g:X → J(τ)ω into the countable power of the hedgehog space of spininess τ we have 1.(i) dim g(X) ⩽ dim X and2.(ii) there exists a mapping h:g(X) → Z satisfying f = h ∘ g. This gives a new approach to factorization theorems of Pasynkov. We show also that, given a mapping f:X → X of a metrizable space X of weight τ into itself, for almost every mapping h:X → J(τ)ω there exists u:J(τ)ω → J(τ)ω such that u ∘ h = h ∘ ƒ