TITLE: Is $\mathbb{R}^1$ really the same as $\mathbb{R}$?
QUESTION [13 upvotes]: We write $\mathbb{R}^n$ to be the set $\{(x_1,\dots,x_n): x_i \in \mathbb{R} \mbox{ for } i=1,\dots,n\}$, so $\mathbb{R}^n$ is the set of $n$-tuples of real numbers. Is then $\mathbb{R}^1$ a set of 1-tuples or is $\mathbb{R}^1$ just equal to $\mathbb{R}$? Is there any subtle difference between the two?

REPLY [8 votes]: Here is a somewhat different perspective.
It is quite common when we talk about $\mathbb{R}$ to think of the complete ordered field of $\left(\mathbb{R},+,\cdot,<\right)$. But we also have to  consider the context in which the symbol $\mathbb{R}$ is used and this can make a big difference.

Here are some examples of text blocks we can sometimes read:

(1) Let's consider the $\color{blue}{\text{additive group}}$ of real numbers $\mathbb{R}$ ...
(2) Let's consider the $\color{blue}{\text{topological vector space}}$ $\mathbb{R}$ together with the usual euclidean topology ...
(3) Let's consider the $\color{blue}{\text{ring}}$ $(\mathbb{R},+,\cdot)$ of real numbers where we use the usual addition and multiplication of real numbers ...
(4) Let's consider the real numbers $\mathbb{R}$ ...
(5) Let's consider $\color{blue}{\text{the set}}$ of real numbers $\mathbb{R}$.


In the first three examples we have additionally to the set $\mathbb{R}$ also some kind of mathematical structure given. Each of these three examples provides us with some algebraic, or topological, or whatsoever mathematical context in which the set $\mathbb{R}$ should be considered.
Usually we do not have any problems to immediately switch into the right context (given we are already familiar with it). We then consider the real numbers precisely within this specific and restricted environment.
The example (4) is somewhat special, since here we do not have any additional context given. In this case it is common to consider $\mathbb{R}$ as complete ordered field $\left(\mathbb{R},+,\cdot,<\right)$ and often this is the correct view.

The example (5) is different to (4) since it explicitly puts the focus on the plain set structure of the real numbers: Let's consider the set of real numbers ... In this case the indicated context is rather the plain set structure and nothing else.
We have to keep in mind that a set per se is just a set without any additional mathematical structures.

When looking at OP's question, the OP is addressing explicitly the set structure of $\mathbb{R}$ and $\mathbb{R}^1$.
So, what is the difference when restricting (naively, without formal considerations) on set structures? Let's exemplarily consider a finite set $X=\{a,b,c\}$. We have for $n\geq 1$:
\begin{align*}
X^n=\{(x_1,\ldots,x_n)|x_j\in X, 1\leq j\leq n\}
\end{align*}
and as special case $n=1$ we have
\begin{align*}
X^1=\{(a),(b),(c)\}
\end{align*}
clearly showing that $X^1$ and $X$ are the same set with only some minor notational differences.
Conclusion: Considering the set-structure of $\mathbb{R}$ and $\mathbb{R}^1$ the difference is solely notational.