Mathematical structure

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a structure on a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> (or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an <a href="/facts/Operation_(mathematics)/Yplv602Q">operation</a>, <a href="/facts/Relation_(mathematics)/HjblDaMy">relation</a>, <a href="/facts/Metric_(mathematics)/RkUptSo8">metric</a>, or <a href="/facts/Topological_space/v2ut7CbK">topology</a>). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.
A partial list of possible structures are <a href="/facts/Measure_theory/yCq7zlI4">measures</a>, <a href="/facts/Algebraic_structure/TkuXca24">algebraic structures</a> (<a href="/facts/Group_(mathematics)/IQTYqINt">groups</a>, <a href="/facts/Field_(mathematics)/xAjAS4ko">fields</a>, etc.), <a href="/facts/Topology/2EqW47cU">topologies</a>, <a href="/facts/Metric_space/gnBBRXtc">metric structures</a> (<a href="/facts/Geometry/wTQf5ffQ">geometries</a>), <a href="/facts/Order_theory/gn6fM3oJ">orders</a>, <a href="/facts/Graph_theory/VVPCd4TX">graphs</a>, <a href="/facts/Event_structure/GdpmNhk4">events</a>, <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relations</a>, <a href="/facts/Differential_structure/nHHTJud2">differential structures</a>, and <a href="/facts/Category_(mathematics)/7WzxUThf">categories</a>.
Sometimes, a set is endowed with more than one feature 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 feature and a group feature, such that these two features are related in a certain way, then the structure becomes a <a href="/facts/Topological_group/cAgNMMRU">topological group</a>.
<a href="/facts/Map_(mathematics)/FABF1l2f">Map</a> between two sets with the same type of structure, which preserve this structure [<a href="/facts/Morphism/Kdpuyz3S">morphism</a>: structure in the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> is mapped properly to the (same type) structure in the <a href="/facts/Codomain/MGc43nwQ">codomain</a>] is of special interest in many fields of mathematics. Examples are <a href="/facts/Homomorphisms/0gyqdEse">homomorphisms</a>, which preserve algebraic structures; <a href="/facts/Continuous_functions/sKbl02pB">continuous functions</a>, which preserve topological structures; and <a href="/facts/Differentiable_functions/jQqTgLk1">differentiable functions</a>, which preserve differential structures.

Mathematical structure open-in-new

Mathematical structure