Mathematical structure

In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance.
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, which allows mathematicians to study the interaction between the different structures more richly. For example, an ordering imposes a rigid form, shape, or topology on the set, and if a set has both a topology structure and a group structure, such that these two structures are related in a certain way, then the set becomes a topological group.
Mappings between sets which preserve structures (i.e., 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.
History.
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:
There are interfaces among these: