On cohesive almost zero-dimensional spaces

We investigate C-sets in almost zero-dimensional spaces, showing that closed sigma C-sets are C-sets. As corollaries, we prove that every rim-sigma-compact almost zero-dimensional space is zero-dimensional and that each cohesive almost zero-dimensional space is nowhere rational. To show that these results are sharp, we construct a rim-discrete connected set with an explosion point. We also show that every cohesive almost zero-dimensional subspace of (Cantor set) x R is nowhere dense.