Infinite beatitude of non-existence: a journey into Nothingland

In the novella Flatland by Edwin A. Abbott, the Sphere leads the Square “downward to the lowest depth of existence, even to the realm of Pointland, the Abyss of No dimensions”:

I caught these words, “Infinite beatitude of existence! It is; and there is nothing else beside It.” […] “It fills all Space,” continued the little soliloquizing Creature, “and what It fills, It is. What It thinks, that It utters; and what It utters, that It hears; and It itself is Thinker, Utterer, Hearer, Thought, Word, Audition; it is the One, and yet the All in All. Ah, the happiness, ah, the happiness of Being!”

Indeed, Pointland (a one-point space) is zero-dimensional by every concept of dimension that I know of. Yet there is something smaller: Nothingland — empty space, {\varnothing} — whose non-existent inhabitants must be perpetually enjoying the happiness of Non-Being.

What is the dimension of Nothingland?

In topology, the empty set has dimension {-1}. This fits the inductive definition of topological dimension, which is the smallest number {d} such that the space can be minced by removing a subset of dimension {\le d-1}. (Let’s say a space has been minced if what’s left has no connected subsets other than points.)

Thus, a nonempty finite (or countable) set has dimension {0}: it’s minced already, so we remove nothing, a set of dimension {-1}. A line or a curve is one-dimensional: they can be minced by removing a zero-dimensional subset, like rational numbers.

A curve minced by removing a zero-dimensional set
One-dimensional curve minced by removing zero-dimensional points.

The Flatland itself can be minced by removing a one-dimensional subset (e.g., circles with rational radius and rational coordinates of the center), so it is two-dimensional. And so on.

Flatland minced by removing a one-dimensional subset
Flatland minced by removing a one-dimensional subset

The convention {\mathrm{dim}\,\varnothing = -1}, helpful in the definition, gets in the way later. For example, the topological dimension is subadditive under products: {\mathrm{dim}\,(A\times B)\le \mathrm{dim}\,A + \mathrm{dim}\,B} … unless both {A} and {B} are empty, because then {-1\le -2} is false. So the case {A=B=\varnothing} must be excluded from the product theorem. We would not have to do this if {\mathrm{dim}\,\varnothing } was defined to be {-\infty}.

Next, consider the Hausdorff dimension. Its definition is not inductive, but one has to introduce other concepts first. First, define the {d}-dimensional premeasure on scale {\delta>0}:

\displaystyle    \mathcal H^d_\delta (X) = \inf \sum_j (\mathrm{diam}\,{U_j})^{d}

where the infimum is taken over all covers of {X} by nonempty subsets {U_j} with {\mathrm{diam}\,{U_j}\le \delta}. Requiring {U_j} to be nonempty avoids the need to define the diameter of Nothingland, which would be another story. The empty space can be covered by empty family of nonempty subsets. The sum of empty set of numbers is {0}, and so {\mathcal H^d_\delta (\varnothing) = 0}.

Then we define the {d}-dimensional Hausdorff measure:

\displaystyle    \mathcal H^d (X) = \lim_{\delta\rightarrow0} \mathcal H^d_\delta (X)

and finally,

\displaystyle    \mathrm{dim}_H (X) = \inf \{ d \colon \mathcal H^d (X)=0\}

If in this last infimum we require {d>0}, the result is {\mathrm{dim}_H (\varnothing) =0}. But why make this restriction? The {d}-dimensional pre-measures and measures make sense for all real {d}. It’s just that for nonempty {X}, we are raising some small (or even zero) numbers to negative power, getting something large as a result. Consequently, every nonempty space has {\mathcal H^d = \infty} for all {d < 0}.

But {\mathcal H^d_\delta (\varnothing) = 0}, from the sum of empty collection of numbers being zero. Hence, {\mathcal H^d (\varnothing) = 0} for all real {d}, and this leads to {\mathrm{dim}_H\,\varnothing = -\infty}.

To have {\mathrm{dim}_H\,\varnothing = -\infty} is also convenient because the Hausdorff dimension is superadditive under products: {\mathrm{dim}_H\,(A\times B)\ge \mathrm{dim}_H\,A + \mathrm{dim}_H\,B}. This inequality was proved for general metric spaces as recently as 1995, by John Howroyd. If we don’t have {\mathrm{dim}_H\,\varnothing = -\infty}, then both factors {A} and {B} must be assumed nonempty.

So… should Nothingland have topological dimension {-1} and Hausdorff dimension {-\infty}? But that would violate the inequality {\mathrm{dim} (X)\le \mathrm{dim}_H (X)} which holds for every other separable metric space. In fact, for such spaces the topological dimension is simply the infimum of the Hausdorff dimension over all metrics compatible with the topology.

I am inclined to let the dimension of Nothingland be {-\infty} for every concept of dimension.

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