Absorbing spaces for C-compacta

For every countable ordinal α we construct a C-compactum universal to the class of all compacta with dimC X ⩽ α. Using this result we prove that for uncountable many ordinals β there exist noncountable-dimensional pre-Hilbert spaces Dβ which are absorbing spaces for the class of compacta with dimC less than β where dimC is the transfinite extension of the covering dimension (Borst, preprint) which classifies C-compacta.