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.

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.