Mathematical structure

This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (April 2016) (Learn how and when to remove this message)

In Mathematics, a structure on a set (or on some sets) refers to providing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.

A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, graphs, events, equivalence relations, differential structures, and categories.

Sometimes, a set is endowed with more than one feature 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 feature and a group feature, such that these two features are related in a certain way, then the structure becomes a topological group.^[1]

Map between two sets with the same type of structure, which preserve this structure [morphism: structure in the domain is mapped properly to the (same type) structure in the codomain] is of special interest in many fields of mathematics. Examples are homomorphisms, which preserve algebraic structures; continuous functions, which preserve topological structures; and differentiable functions, which preserve differential structures.