Let $\mathcal{C}$ be a class of sets/spaces/structures among which we have a dimension. Namely a map $d:\mathcal{C}\rightarrow \mathbb{N}$ defined in a certain manner that motivated the appellative dimension.
What properties would you expect $d$ to have?
E.g. if $Y, X\in\mathcal{C}$ and $Y\subseteq X$ then $\dim Y\leq \dim X$.
Another way of phrasing this question is, what properties are shared by every standard notion of dimension in mathematics?