Mathematical structure

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).

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.

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.

These definitions are equivalent in the context of a given mathematical structure (Euclidean space, in this case).

In mathematics, a space is a set (sometimes called a universe) with some added 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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