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A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, 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 ordering imposes a rigid form, shape, or topology on the set. As another example, if a set has both a topology and is a group, and these 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 source or domain are mapped to equivalent structures in the destination or 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.
In 1939, the French group with the pseudonym Nicolas Bourbaki saw structures as the root of mathematics. They first mentioned them in their "Fascicule" of Theory of Sets and expanded it into Chapter IV of the 1957 edition. They identified three mother structures: algebraic, topological, and order.
Example: the real numbers
The set of real numbers has several standard 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 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.
- Abstract structure
- Equivalent definitions of mathematical structures
- Intuitionistic type theory
- Space (mathematics)
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