A relation between spaces implied by their t-equivalence

We prove that if X and Y are t-equivalent spaces (that is, if C p ( X ) and C p ( Y ) are homeomorphic), then there are spaces Z n , locally closed subspaces B n of Z n , and locally closed subspaces Y n of Y, n ∈ N + , such that each Z n admits a perfect finite-to-one mapping onto a closed subspace of X n , Y n is an image under a perfect mapping of B n , and Y = ⋃ { Y n : 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.