In mathematics, a structure on a set is some additional mathematical objects that, loosely speaking, attach to the set, making it easier to visualize or work with.

A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, and equivalence relations. In mathematics, a measure is a function that assigns a number, e. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... Topology (Greek topos = place and logos = word) is a branch of mathematics concerned with the study of topological spaces. ... In mathematics, a metric space is a set (or space) where a distance between points is defined. ... Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...

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. In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ...

Example: the real numbers

The set of real numbers has several standard structures: Please refer to Real vs. ...

• 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: In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... In mathematics a metric or distance is a function which assigns a distance to elements of a set. ... The intuitive idea of flatness is important in several fields. ...

• 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.