Why do we think of ${\mathbb{R}}^2$ as a plane?

Question

It seems like every one assumes there's an obvious answer to this question but I just can't see it. ${\mathbb{R}}^2$ is just defined as the set of all 2-tuples of real numbers i.e.

${\mathbb{R}}^2 := \{(a,b)| a,b \in\mathbb{R} \}$

The $xy$ - plane clearly has a lot more structure in it than just being a set of points. There is a definitive order to the arrangement of points. We can define curves on it (maybe even closed curves with some enclosed area). We can talk about distances between 2 points & so on. So where does this additional structure come from & how does that relate to the set ${\mathbb{R}}^2$ .

Answer 1

There is a lot of additional structure, yes. However, we usually "construct" this structure on top of the "raw plane" $\mathbb R^2$

• The notion of distance is a metric
• Closeness is a topology
• Curves / solutions of polynomials are a variety
• ...

However, we always start with $\mathbb{R}^2$ and add structure on top of it. The choice of structure depends on what you want to study currently!

Answer 2

The answer to this question

So where does this additional structure come from & how does that relate to the set $\mathbb{R}^2$?

is that it comes from $\mathbb{R}$. You use the arithmetic and the order in $\mathbb{R}$ to define the extra things in $\mathbb{R}^2$ that let you think of it as the (familiar, geometric) Euclidean plane.

In general, if sets $A$ and $B$ have some structure (they may be ordered, or may be groups or manifolds or ...) then you may be able to define useful structure on the set of ordered pairs $A \times B$.