Multistage n-dimensional universal spaces and extensions

Let I τ be the Tychonoff cube of weight τ ⩾ ω with a fixed point, σ τ and Σ τ be the correspondent σ- and Σ-products in I τ and σ τ ⊂ ( Σ σ τ = ( σ τ ) ω ) ⊂ Σ τ . Then for any n ∈ { 0 , 1 , 2 , … } , there exists a compactum U n τ ⊂ I τ of dimension n such that for any Z ⊂ I τ of dimension ⩽ n , there exists a topological embedding of Z in U n τ that maps the intersections of Z with σ τ , Σ σ τ and Σ τ to the intersections U σ n τ , U Σ σ n τ and U Σ n τ of U n τ with σ τ , Σ σ τ and Σ τ , respectively; U σ n τ , U Σ σ n τ and U Σ n τ are n-dimensional and U σ n τ is σ-compact, U Σ σ n τ is a Lindelöf Σ-space and U Σ n τ 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.