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Inmathematics, a structure on a set (or on some sets) refers to providing it (or them) with certain additional features (e.g. anoperation,relation,metric, ortopology). 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 aremeasures,algebraic structures (groups,fields, etc.),topologies,metric structures (geometries),orders,graphs,events,equivalence relations,differential structures, andcategories.
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 atopological group.[1]
Map between two sets with the same type of structure, which preserve this structure [morphism: structure in thedomain is mapped properly to the (same type) structure in thecodomain] is of special interest in many fields of mathematics. Examples arehomomorphisms, which preserve algebraic structures;continuous functions, which preserve topological structures; anddifferentiable functions, which preserve differential structures.
In 1939, the French group with the pseudonymNicolas Bourbaki saw structures as the root of mathematics. They first mentioned them in their "Fascicule" ofTheory of Sets and expanded it into Chapter IV of the 1957 edition.[2] They identifiedthreemother structures: algebraic, topological, and order.[2][3]
The set ofreal numbers has several standard structures:
There are interfaces among these:
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