There are many notions of dimension : algebraic, topological, Hausdorff, Minkowski... (and others). While the topological one generalize the algebraic one, the last three need not coincide for every sets. Yet it is generally acknowledged that the Hausdorff dimension has "nice enough" properties to work with (the interest of the Minkowski dimension lies mainly in the fact that it's easier to compute).
So my main question is this : is there an axiomatic approach that would tidy up this mess ? For example, is there a result of the form : if you ask these axioms then the only map from "reasonnable sets" to the set of positive real integers is the Hausdorff dimension ? (or another one ?). If so what are they ?
Are there also a clearly identified list of properties that you would ask from any notion of dimension ? I give the following as an example :
- it should coincide with the algebraic dimension for finite dimensional vector spaces
- dim A $\leq$ dim B if $A \subset B$
- some sort of nice behaviour for cartesian products (at least for reasonnable sets)
- some sort of nice behaviour for infinite increasing unions and/or decreasing intersections