Homeomorphisms of function spaces and hereditary cardinal invariants

A space X is called a t-image of Y if Cp(X) is homeomorphic to a subspace of Cp(Y). We prove that if Y is a t-image of X, then Y is a countable union of images of X under almost lower semicontinuous finite-valued mappings (see Definition 1.4). It follows that if Y is a t-image of X (in particular, if X and Y are t-equivalent), then for every n ϵ ω, hl(Yn) ⩾ hl(Xn), hd(Yn) ⩽ hd(Xn) and s(Yn) ⩽ s(Xn).