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

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

• Both is correct. Some people define zero in $\Bbb N$, other people say it isn't in $\Bbb N$. Oct 17, 2020 at 16:47
• Oct 17, 2020 at 16:47

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").