Universal map of σ onto Σ and absorbing sets in the classes of absolute Borelian and projective finite-dimensional spaces

We construct a map of σ (the span of the usual orthonormal basis of the Hilbert space l2) onto Σ (the span of the standard Hilbert cube in l2) which is universal for the class of maps of strongly countable-dimensional σ-compact spaces into σ-compact spaces. As an application, we construct absorbing sets in the classes of finite-dimensional spaces of given absolute Borelian or projective class.