The set of real numbers has several standard structures:
- An order: each number is either less than or greater than any other number.
- Algebraic structure: there are operations of addition and multiplication, the first of which makes it into a group and the pair of which together make it into a field.
- A measure: intervals of the real line have a specific length, which can be extended to the Lebesgue measure on many of its subsets.
- A metric: there is a notion of distance between points.
- A geometry: it is equipped with a metric and is flat.
- A topology: there is a notion of open sets.
There are interfaces among these:
- Its order and, independently, its metric structure induce its topology.
- Its order and algebraic structure make it into an ordered field.
- Its algebraic structure and topology make it into a Lie group, a type of topological group.
不必了解这些结构也能做数学（应用数学），因为学习数学不是学习reality,而是模型，脑子里面的模型去解释真实！
如果实数已经拥有这么些好的性质，能够很好地描述现实，that's enough!