Multistage n-dimensional universal spaces and extensions

Let Iτ be the Tychonoff cube of weight τ?ω with a fixed point, στ and Στ be the correspondent σ- and Σ-products in Iτ and στ⊂(Σστ=ω(στ))⊂Στ. Then for any n∈{0,1,2,…}, there exists a compactum Unτ⊂Iτ of dimension n such that for any Z⊂Iτ of dimension?n, there exists a topological embedding of Z in Unτ that maps the intersections of Z with στ, Σστ and Στ to the intersections , and of Unτ with στ, Σστ and Στ, respectively; , and are n-dimensional and is σ-compact, is a Lindelöf Σ-space and is a sequentially compact normal Fréchet-Urysohn space. This theorem (on multistage universal spaces of given dimension and weight) implies multistage extension theorems (in particular, theorems on Corson and Eberlein compactifications) for Tychonoff spaces.