Question about $\mathbb R^n$ and mathematical space(s)

Question

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:

https://www.youtube.com/watch?v=lCsjJbZHhHU&t=5m40s

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:

http://www.youtube.com/watch?v=2WH6NTciV2Q&t=3m0s

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

Both is correct. Some people define zero in $\Bbb N$, other people say it isn't in $\Bbb N$. — Dietrich Burde, Oct 17 '20 at 16:47
See Has the notion of having a complex amount of dimensions ever been described? And what about negative dimensionality? — Dave L. Renfro, Oct 17 '20 at 16:47

Milo Brandt · Accepted Answer · 2020-10-17T17:02:38.853

Sure! The notation $\mathbb R^n$ really just means "the set of $n$-tuples of real numbers" - that is, ordered lists $(a_1,a_2,\ldots,a_n)$ where each $a_i$ is a real number. By this reasoning $\mathbb R^0$ is just the set of ordered lists of $0$ real numbers - and there is exactly one such list of zero real numbers: $()$. So, $\mathbb R^0$ is just a single point and it happens to be a vector space of dimension zero.

More generally, if you want to write $\mathbb R^n$, all that $n$ needs to do is specify the size of a set* - and zero is a perfectly acceptable value here. The exponent doesn't even need to be a natural number - you can happily talk about $\mathbb R^{\mathbb N}$ as the set of sequences $(a_1,a_2,a_3,\ldots)$ with countably many terms (or, more formally, of functions $\mathbb N\rightarrow\mathbb R$) or even do this with larger sets in the exponent (then meaning "an sequence of real numbers indexed by that set").

(*This idea is known as cardinality in general; the purpose of the natural numbers in this context is that they count how big finite collections are, which is a good hint that zero is a valid value, since collections can be empty)

Question about $\mathbb R^n$ and mathematical space(s)

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