 Mathematical structure

For the notion of "structure" in mathematical logic, see Structure (mathematical logic).
In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach (or relate) 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, equivalence relations, differential structures, and categories.
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.
Mappings between sets which preserve structures (so that structures in the domain are mapped to equivalent structures in the codomain) are of special interest in many fields of mathematics. Examples are homomorphisms, which preserve algebraic structures; homeomorphisms, which preserve topological structures; and diffeomorphisms, which preserve differential structures.
N. Bourbaki suggested an explication of the concept "mathematical structure" in their book "Theory of Sets" (Chapter 4. Structures) and then defined on that base, in particular, a very general concept of isomorphism.
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, 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.
See also
References
 Structure on PlanetMath (provides a model theoretic definition.)
Categories: Type theory
 Set theory
 Mathematical structures
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