I am looking for examples of topological spaces that are interesting from a dimension theoretical point of view - in particular I am looking for these three notions of dimension:
- Little inductive dimension,
- big inductive dimension,
- Lebesgue covering dimension.
I know that if a space is separable and metrizable, the three dimensions coincide but I am looking for simple cases where it does not.
I am also looking for examples of spaces (and their) where the monotonicity, sum and/or product theorem do not hold and various other interesting counterexamples where conditions are not satisfied (such as $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$ being 0-dimensional subspaces of 1-dimensional $\mathbb{R}$)