Mathematical structure
structurestructuresmathematical structuresstructuralstructured sets
In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.wikipedia
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Mathematics
mathematicalmathmathematician
In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (mathematical analysis).








Mathematical object
mathematical objectsobjectsgeometric object
In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.
Algebraic structure
algebraic structuresunderlying setalgebraic system
A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.
Equivalent definitions of mathematical structures
more than one definition
These definitions are equivalent in the context of a given mathematical structure (Euclidean space, in this case).
Space (mathematics)
spacemathematical spacespaces
In mathematics, a space is a set (sometimes called a universe) with some added structure.
Event structure
events
A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.
Differential structure
differentiable structuresmooth structure
A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.
Abstract structure
abstract abstractabstract form
Set (mathematics)
setsetsmathematical set
In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.

Measure (mathematics)
measuremeasure theorymeasurable
A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.

Group (mathematics)
groupgroupsgroup operation
A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.

Field (mathematics)
fieldfieldsfield theory
A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.

Topology
topologicaltopologicallytopologist
A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.



Metric space
metricmetric spacesmetric geometry
A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.
Geometry
geometricgeometricalgeometries
A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.









Order theory
orderorder relationordering
A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.
Equivalence relation
equivalenceequivalentmodulo
A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.
Category (mathematics)
categorycategoriesobject
A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.
Topological group
continuous groupclosed subgrouptopological groups
As another example, if a set has both a topology and is a group, and these two structures are related in a certain way, the set becomes a topological group.
Map (mathematics)
mappingmapmaps
Mappings between sets which preserve structures (so that 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.

Domain of a function
domaindomainsdomain of definition
Mappings between sets which preserve structures (so that 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.
Codomain
valuesimagetarget
Mappings between sets which preserve structures (so that 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.
Homomorphism
homomorphichomomorphismse-free homomorphism
Examples are homomorphisms, which preserve algebraic structures; homeomorphisms, which preserve topological structures; and diffeomorphisms, which preserve differential structures.
Homeomorphism
homeomorphichomeomorphismstopologically equivalent
Examples are homomorphisms, which preserve algebraic structures; homeomorphisms, which preserve topological structures; and diffeomorphisms, which preserve differential structures.


Diffeomorphism
diffeomorphicdiffeomorphismsdiffeomorphism group
Examples are homomorphisms, which preserve algebraic structures; homeomorphisms, which preserve topological structures; and diffeomorphisms, which preserve differential structures.