The category of all zero-dimensional realcompact spaces is not simple

For every zero-dimensional space E of non-measurable cardinality we construct a zero-dimensional, hereditarily realcompact, locally compact and locally countable space which cannot be embedded as a closed subspace into any topological power of the space E. Under the assumption that all cardinals are non-measurable it gives the result stated in the title.

This is an answer for a question raised by H. Herrlich