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Inmathematics, astructure on aset (or on some sets) refers to providing or endowing 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 ismeasures,algebraic structures (groups,fields, etc.),topologies,metric structures (geometries),orders,graphs,events,differential structures,categories,setoids, andequivalence relations.
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]
Amap between two similarly-structured sets that preserves their structure is known as amorphism, and such maps are of special interest in many fields of mathematics. Examples includehomomorphisms, which preservealgebraic structures;continuous functions, which preserve topological structures; anddifferentiable functions, which preserve differential structures.
In 1939, the French group with the pseudonym "Nicolas 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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