# 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**

### Group (mathematics)

**groupgroupsgroup operation**

### Field (mathematics)

**fieldfieldsfield theory**

### Topology

**topologicaltopologicallytopologist**

### Metric space

**metricmetric spacesmetric geometry**

### Geometry

**geometricgeometricalgeometries**

### Order theory

**orderorder relationordering**

### Equivalence relation

**equivalenceequivalentmodulo**

### Category (mathematics)

**categorycategoriesobject**

### 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.