A generalization of Gruenhageʹs example

We construct, assuming the continuum hypothesis, an example of nonmetrizable n-dimensional Cantor manifold X n ( n ∈ N ) with the following properties: 1) X n k is hereditarily separable for all k ∈ N ; 2) X n k ∖ Δ k is perfectly normal for every k ∈ N ; 3) the space F ( X n ) is hereditarily normal for every seminormal functor F that preserves weights and one-to-one points and such that s p ( F ) = { 1 , k } ; in particular, X n 2 and λ 3 X n are hereditarily normal. This example is a generalization of famous Gruenhageʹs example given in Gruenhage and Nyikos (1993) [4].