On dimensions modulo a compact metric ANR and modulo a simplicial complex

V.V. Fedorchuk has recently introduced dimension functions K-dim⩽K-Ind and L-dim⩽L-Ind, where K is a simplicial complex and L is a compact metric ANR. For each complex K with a non-contractible join |K|⁎|K| (we write |K| for the geometric realisation of K), he has constructed first countable, separable compact spaces with K-dim• a compact Fréchet space XC,α such that , , and each component of XC,α is homeomorphic to C. If L⁎L is non-contractible, or n=1 and L is non-contractible, then C can be a cube [0,1]m for a certain natural number m=m(n,L).