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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 ${\mathbb{R}}^2$ .

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    $\begingroup$ $\mathbb R^2$ isn't "just a set of points". It is equipped with many kinds of structures - topology, vector space structure, one can even view it as a field. $\endgroup$ – Wojowu Dec 9 '16 at 12:09
  • $\begingroup$ What is "a plane" to you? For example, do you mean a "Euclidean plane", i.e. a mathematical object that satisfies the axioms and theorems of Euclid's planar geometry? $\endgroup$ – Lee Mosher Dec 9 '16 at 14:00
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    $\begingroup$ Why the downvote with no comment? $\endgroup$ – Ethan Bolker Dec 9 '16 at 14:08
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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!

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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 $A \times B$.

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