In the context of Euclidean and real coordinate spaces ($\mathbb R^n$), does n (or $\mathbb N$) include 0?
$\mathbb R^1$ is the 1-dimensional real number line
$\mathbb R^2$ is the 2-dimensional coordinate plane
$\mathbb R^3$ is the 3-dimensional coordinate space
And so on…
There are of course higher dimensional spaces, for example it's mentioned in this video from Khan Academy:
But is it possible to have lower dimensional spaces? As in $\mathbb R^0$. I am unsure because not all authors include 0 in the set of natural numbers.
Professor Norman J. Wildberger briefly mentions 0-dimensional spaces here, but this is in the context of a “theory of mathematical space which doesn’t involve the infinities that are usually associated with a real number treatment” as he puts it:
And to mention a literary source, there’s “Pointland” in the novella Flatland: A Romance of Many Dimensions by Edwin A. Abbott.
Any recommendations as to further reading would also be greatly appreciated. Thank you