A relation between spaces implied by their t-equivalence

We prove that if X and Y are t-equivalent spaces (that is, if Cp(X) and Cp(Y) are homeomorphic), then there are spaces Zn, locally closed subspaces Bn of Zn, and locally closed subspaces Yn of Y, n∈N+, such that each Zn admits a perfect finite-to-one mapping onto a closed subspace of Xn, Yn is an image under a perfect mapping of Bn, and Y=⋃{Yn:n∈N+}. It is deduced that some classes of spaces, which for metric spaces coincide with absolute Borelian classes, are preserved by t-equivalence. Also some limitations on the complexity of spaces t-equivalent to “nice” spaces are obtained.