A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, equivalence relations, and differential structures.
Sometimes, a set is endowed with more than one structure simultaneously; this enables mathematicians to study it more richly. For example, an order induces a topology. As another example, if a set both has a topology and is a group, and the two structures are related in a certain way, the set becomes a topological group.
Mappings between sets which preserve structures (so that 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.
There are interfaces among these:
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