This article goes over examples of Mathematical Structures

Mathematical Structures - Example #1 On Real Numbers

The set of real numbers has several standard mathematical structures:

• an order: each number is either less or more than any other number.
• algebraic structure: there are operations of multiplication and addition that make it into a field.
• a measure: intervals along 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 induces its topology
• Its order and algebraic structure make it an ordered field
• Its algebraic structure and topology make it into a Lie group, a type of topological group