Dimension Notation for Topological Spaces Lots of families of topological spaces get superscripts denoting dimension: $\mathbb{R}^n$, $B^n$, $D^n$, $\Delta^n$, $S^n$, $\mathbb{R}P^n$, $T^n$. There's a niceness to this: $S^n$ is the boundary of $D^{n+1}$, which is a bit awkward at first, especially if you think about how you would define their dimensions as subsets of $\mathbb{R}^n$ as a vector space, but under any applicable topological definition of dimension, their superscripts represent their dimensions. For once, there is a nice continuity of notation. However, there are multiple topological notions of dimension: topological dimension, both inductive dimensions, dimension of CW-complexes, dimension of manifolds... So are there any explicit rules for what the superscript $n$ is actually telling us about a space, or is it just a guide that should wind up being equivalent under different definitions for sufficiently nice spaces?

I would say there are no explicit rules. For me the point is that you have a sequence of “similar” spaces naturally indexed by $n$. For the examples you mentioned it feels natural to align the indexes with the dimension (where for these spaces various notions of dimension coincide). If one wants to directly discuss some dimension of a space, I'd suggest being explicit: $\operatorname{somedim}(X) = n$ while $\operatorname{otherdim}(X) = n'$.
There are more examples of the general pattern not directly related to dimension: $ℤ_n$ for cyclic group od order $n$, $S_n$ and $A_n$ for the symmetric and alternating group; $\ell^p$ and $L^p$ (also denoted by $\ell_p$ and $L_p$) for the Lebesgue spaces.
Note that in $ℝ^n$ the superscript is also a genuine operation – the Cartesian power. (Maybe the similarity with $ℝ^n$ is the reason why $S^n$ instead of $S_n$ is used for spheres, etc, but that is just a speculation.)
