Is there a common notion of $\mathbb{R}^n$, for non-integer $n$?

Question

This is not a very well-defined question.

Are there any standard constructions of metric spaces, parameterized by real-valued $n \ge 1$, such that:

1. When $n$ is an integer, the metric space is precisely $\mathbb{R}^n$.
2. When $n$ is non-integer, the metric space can be seen as a reasonable generalization of $\mathbb{R}^n$. For example, perhaps it has Hausdorff dimension of $n$.

Alternately, a non-existence result that you can't maintain some of the important properties of $\mathbb{R}^n$ in a generalization like this would be interesting to me.

Answer 1

There is no topological space $X$ such that $X\times X\cong\mathbb{R}^n$ if $n$ is an odd integer. You can prove this using homology; see, for instance, this answer on MathOverflow. In particular, this seems like pretty good evidence that there is no reasonable notion of "$\mathbb{R}^{n/2}$" when $n$ is an odd integer. By similar homology arguments you can show that if $n$ is not divisible by $m$ then there is no space $X$ such that $X^m\cong\mathbb{R}^n$, so there is no good topological candidate for $\mathbb{R}^{n/m}$.

These topological obstructions aside, I can say that if there is a "common notion" of $\mathbb{R}^n$ for non-integer $n$, it can't be too common, because I've never heard of it.