  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 these
results are sharp, we construct a rim-discrete connected set with an explosion
point. We also show every cohesive almost zero-dimensional subspace of
$($Cantor set$)$$\times\mathbb R$ is nowhere dense.
