Mathematical structure

In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance.

A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, and equivalence relations.

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 numbers 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 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. (this is implied by the metric)

There are interfaces among these:

• Its order and, independently, its metrics 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

• "Structure". PlanetMath. (provides a categorical definition.)