Dimension Notation for Topological Spaces

Question

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?

Answer 1

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.)

Answer 2

I normally use the dimension-number to describe the number of orthogonal lines supported at an instant of space. In other words, if one imagines 2-cloth being shaped into any thing, it's still 2D.

Not with standing, there are a number of issues here.

1. Spaces of more positive curvature are usually represented in euclidean spaces of n+1 or so dimensions. The most common of these is to represent S2 in E3 (ie the surface of the sphere against its volume). The reality is that any space will appear as a ball in a space of higher dimension and lesser curvature. Euclidean space E3 appears in hyperbolic space H4 as a surface bounding a horoglome (or horochoron).

The point in the PG is to distinguish between the fabric against what it is made into. For example a 'hedrix' is a 2d fabric, and one describes E2 as a horohedrix (horizon-centred 2d fabric), the sphere as a glomohedrix (round 2d space), and the hyperbolic space H2 as a bollohedrix (negative-curvature 2d fabric).

If one talks of 'polyhedron' or 'apeirohedron', then this refers to something sewn together from patches, a polyhedron is (many [closed] 2d patched), an apeirohedron is [without + space-boundary (fence) + 2d + patches] or a tiling in any 2d space. A {3,6} is a horohedron or (2d patches centred on the horizon).

2. The products of draught are rooted at -1 dimensions. This means that the exponents on the algebraic products defined by them, is equal to the number of vertices a simplex has, so 3P is the same as 2D. The products of repetition are rooted in 0 dimensions, so the usual dimension is the same as the repetitions.

Most mathematicians have not caught up with the drawn product, so it's less of a matter to them.

The explicit rule is that you should refer to the dimension of 'all-space', which is to say, that the surface of a sphere treated as spherical geometry is S2, and leaving such S2 is to go into 'hyperspace'. If one is looking at a sphere as a solid, then it is an object in E3 or S3 or H3. The space is normally referred to in terms of the repetitions of a line, rather than any drawn product. So a triangle is 2-dimensional, although it occurs as a³ in products of draught.

PG = polygloss at http://www.os2fan2.com/gloss/index.html