More examples of metric spaces where the inductive dimensions disagree

We give three examples of metric spaces where the inductive dimensions disagree. The two main examples are both N-compact. The first has closed sets which are not clopen Borel (Borel in the σ-algebra generated by the clopen sets). The second has weight ω1 and, assuming all sets of cardinality ω1 in the interval are Q-sets, contrasts the first by having all closed sets clopen Borel. The third example provides, for each α with ω1⩽α⩽c, a metric space of weight α with noncoinciding dimensions for which all subsets of weight less than α are strongly zero-dimensional. Each example answers a question posed by Mrowka.