A generalization of Gruenhage's example

We construct, assuming the continuum hypothesis, an example of nonmetrizable n-dimensional Cantor manifold Xn (n∈N) with the following properties: 1) is hereditarily separable for all k∈N; 2) is perfectly normal for every k∈N; 3) the space F(Xn) is hereditarily normal for every seminormal functor F that preserves weights and one-to-one points and such that sp(F)={1,k}; in particular, and λ3Xn are hereditarily normal. This example is a generalization of famous Gruenhage's example given in Gruenhage and Nyikos (1993) [4].