In

General topology
Homeomorphisms

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods
A neighbourhood (British English
British English (BrE) is the standard dialect
A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...

of points).
Properties of a point on a function

Perhaps the best-known example of the idea of locality lies in the concept oflocal minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function (mathematics), function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, e ...

(or local maximum
In mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (math ...

), which is a point in a function whose functional value is the smallest (resp., largest) within an immediate neighborhood
A neighbourhood (British English
British English (BrE) is the standard dialect
A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...

of points. This is to be contrasted with the idea of global minimum (or global maximum), which corresponds to the minimum (resp., maximum) of the function across its entire domain.
Properties of a single space

Atopological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

is sometimes said to exhibit a property locally, if the property is exhibited "near" each point in one of the following ways:
# Each point has a neighborhood
A neighbourhood (British English
British English (BrE) is the standard dialect
A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...

exhibiting the property;
# Each point has a neighborhood base In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...

of sets exhibiting the property.
Here, note that condition (2) is for the most part stronger than condition (1), and that extra caution should be taken to distinguish between the two. For example, some variation in the definition of locally compact In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

can arise as a result of the different choices of these conditions.
Examples

*Locally compact In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

topological spaces
* Locally connected
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), a ...

and Locally path-connected
In topology and other branches of mathematics, a topological space ''X'' is
locally connected if every point admits a neighbourhood basis consisting entirely of Open set, open, Connected set, connected sets.
Background
Throughout the history of ...

topological spaces
* Locally Hausdorff, Locally regular, Locally normal etc...
* Locally metrizable
Properties of a pair of spaces

Given some notion of equivalence (e.g.,homeomorphism
In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...

, diffeomorphism
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, isometry
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

) between topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s, two spaces are said to be locally equivalent if every point of the first space has a neighborhood which is equivalent to a neighborhood of the second space.
For instance, the circle
A circle is a shape
A shape or figure is the form of an object or its external boundary, outline, or external surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

and the line are very different objects. One cannot stretch the circle to look like the line, nor compress the line to fit on the circle without gaps or overlaps. However, a small piece of the circle can be stretched and flattened out to look like a small piece of the line. For this reason, one may say that the circle and the line are locally equivalent.
Similarly, the sphere
A sphere (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

and the plane are locally equivalent. A small enough observer standing on the surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight
Visual perce ...

of a sphere (e.g., a person and the Earth) would find it indistinguishable from a plane.
Properties of infinite groups

For aninfinite group
In group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...

, a "small neighborhood" is taken to be a finitely generated subgroup
In group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...

. An infinite group is said to be locally ''P'' if every finitely generated subgroup is ''P''. For instance, a group is locally finite if every finitely generated subgroup is finite, and a group is locally soluble if every finitely generated subgroup is soluble
In chemistry
Chemistry is the scientific
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity or awareness, of someone or something, such as facts
A fact is an occurrence i ...

.
Properties of finite groups

Forfinite group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...

s, a "small neighborhood" is taken to be a subgroup defined in terms of a prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

''p'', usually the local subgroups, the normalizer
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s of the nontrivial ''p''-subgroups. In which case, a property is said to be local if it can be detected from the local subgroups. Global and local properties formed a significant portion of the early work on the classification of finite simple groups
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, which was carried out during the 1960s.
Properties of commutative rings

{{main, local ring For commutative rings, ideas ofalgebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

make it natural to take a "small neighborhood" of a ring to be the localization
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...

at a prime ideal
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...

. In which case, a property is said to be local if it can be detected from the local ring In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...

s. For instance, being a flat module
In algebra, a flat module over a ring (mathematics), ring ''R'' is an ''R''-module (mathematics), module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor pro ...

over a commutative ring is a local property, but being a free module
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

is not. For more, see Localization of a module
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring (mathematics), ring or module (mathematics), module. That is, it introduces a new ring/module out of an existing ring/module ...

.
See also

* Local path connectednessReferences