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 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.
|
Topics in mathematics related to structure | Edit |
| Abstract algebra | Universal algebra | Graph theory | Category theory | Order theory | Model theory | Structural proof theory | |
| Geometry | Topology | General topology | Algebraic geometry | Algebraic topology | Differential geometry and topology | |
| Analysis | Measure theory | Functional analysis | Harmonic analysis |