Mathematical Structure

In mathematics, a structure on a set, or on several sets, refers to endowing it with additional features such as an operation, a relation, a metric or a topology, features that attach additional meaning or significance to the underlying set. Structures of this kind include measures, algebraic structures such as groups and fields, topologies, metric structures such as geometries, orders, graphs, differential structures, categories and equivalence relations, and a single set can carry more than one such feature at once, allowing mathematicians to study how the different structures interact, as when a set with both a compatible topology and a group structure forms a topological group. A map between two similarly structured sets that preserves that structure is called a morphism, and morphisms that can be inverted are called isomorphisms, which describe when two sets carrying the same kind of structure share the same properties from that structure point of view; examples of structure-preserving maps include homomorphisms for algebraic structures, continuous functions for topological structures, and differentiable functions for differential structures.