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In abstract algebra, a mathematical structure is a set, referred to as a carrier set, endowed with one or more additional features, such as operations, relations, orders, metrics, topologies. Almost always, these extra features, whether attached or just related to the carrier set, endow it with some additional meaning.
Mathematical structures include
measures
algebraic structures (groups, fields...)
topologies
metric structures (geometries)
orders
events
equivalence relations
differential structures
categories
Sometimes, a set is endowed with more than one structure 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 structure and a group structure, such that these two structures are related in a certain way, then the set becomes a topological group.
Mappings between sets which preserve structures (i.e. structures in the source or domain are mapped to equivalent structures in the destination or codomain) are of special interest in many fields of mathematics:
homomorphisms preserve algebraic structures
homeomorphisms preserve topological structures
diffeomorphisms preserve differential structures