TITLE: Is there an axiomatic approach of the notion of dimension ?
QUESTION [13 upvotes]: 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

REPLY [4 votes]: There is a notion of "Krull dimension" valid for arbitrary topological spaces, whose definition is totally analogous to that of algebraic varieties, but using lattices of closed subsets instead of rings of functions. 
As far as I know, it is the only dimension function $\dim$ defined for arbitrary topological spaces, with the following properties:

If $Y $ is a subsapce of $X$, then $ \dim Y \leq \dim X $.
$\dim (X \times Y) \leq \dim X + \dim Y $.
It coincides with Grothendieck's combinatorial dimension on noetherian spaces, and with the standard dimensions (cover and ind) on separable metric spaces.

This beautiful idea goes back to the 60's. You can check the following papers and the references therein.
Section 2 of:

Sancho de Salas, J.B. and M.T.: "Dimension of dense subalgebras of $C(X)$", in Proceedings of the American Mathematical Society, 105 (1989)

or the introduction of:

Sancho de Salas, J.B. and M.T.: "Dimension of distributive lattices and universal spaces", in Topology and its Applications, 42 (1991), 25-36