Does notation $\mathbb{R}^0$ means $\{0\}$

Question

In the book "Riemannian Geometry" - Gallot, Hulin, Lafontaine, the definition of submanifold of Euclidean space as follows:

Definition. A subset $M\subset \mathbf{R}^{n+k}$ is an $n$-dimensional submanifold of class $C^p$ of $\mathbf{R}^{n+k}$ if, for any $x\in M$, there exists a neighborhood of $U$ of $x$ in $\mathbf{R}^{n+k}$ and a $C^p$ submersion $f : U \to \mathbf{R}^{k}$ such that $U\cap M = f^{-1}(0)$ (we recall that $f$ is a submersion if its differential map is surjective at each point).

In the case $k=0$ we have $\mathbb{R}^0$. In order to the definition works in this case, I think $\mathbb{R}^0$ must be $\{0\}$. I search for notation $\mathbb{R}^0$ with keywords \mathbb{R}^0, R^0, and not see what I want. So I ask here for sure.

Answer 1

Their definition is intended to apply only when $k>0$, but it does work (formally) even when $k=0$. An $n$-dimensional submanifold of $\Bbb R^n$ is an open subset, and an open set is only the preimage of $0$ when you take a constant function. This meets the letter of the definition, as a constant function will be a submersion to $\{0\} = T_0\{0\}$, but no one thinks about things this way.

Answer 2

In general if $A$ and $B$ are sets then $A^B$ is a notation for the set of functions $f:B\to A$.

You can interpret the notation $\mathbb R^n$ where $n$ is a non-negative integer as the a notation for the set of functions $f:n\to\mathbb R$ where $n:=\{0,1,\dots,n-1\}$ (so is looked at as a set).

In that context $\mathbb R^0$ stands for the set of functions $f:0\to\mathbb R$ where $0:=\varnothing$.

This set only contains one element which is the so-called empty function.

So if functions are identified with their graphs then $\mathbb R^0=\{0\}=1$.