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A partial list of possible structures are Measures , Algebraic structures (groups, fields, etc.), Topologies , Metric Structures ( Geometries ), Orders , and Equivalence Relation s.

Sometimes, a set is endowed with more than one structure simultaneously; this enables mathematicians to study it more richly. For example, an order induces a topology. As another example, if a set both has a topology and is a group, and the two structures are related in a certain way, the set becomes a Topological Group .


EXAMPLE: THE REAL NUMBERS

The set of Real Number s has several standard structures:
  • an order: each number is either less or more than every 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 certain Length .

  • a geometry: it is equipped with a Metric and is Flat .

  • a topology: numbers are close to or far from one another.

    • 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 .



REFERENCES

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