Local property

Inmathematics,a mathematical object is said to satisfy a propertylocally,if the property is satisfied on some limited, immediate portions of the object (e.g., on somesufficiently smallorarbitrarily smallneighborhoodsof points).

Properties of a point on a function

Perhaps the best-known example of the idea of locality lies in the concept oflocal minimum(orlocal maximum), which is a point in a function whose functional value is the smallest (resp., largest) within an immediateneighborhoodof 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 spaceis sometimes said to exhibit a propertylocally,if the property is exhibited "near" each point in one of the following ways:

1. Each point has aneighborhoodexhibiting the property;
2. Each point has aneighborhood baseof 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 oflocally compactcan arise as a result of the different choices of these conditions.

Examples

• Locally compacttopological spaces
• Locally connectedandLocally path-connectedtopological spaces
• Locally Hausdorff,Locally regular,Locally normaletc...
• Locally metrizable

Properties of a pair of spaces

Given some notion of equivalence (e.g.,homeomorphism,diffeomorphism,isometry) betweentopological spaces,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, thecircleand 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, thesphereand the plane are locally equivalent. A small enough observer standing on thesurfaceof a sphere (e.g., a person and the Earth) would find it indistinguishable from a plane.

Properties of infinite groups

For aninfinite group,a "small neighborhood" is taken to be afinitely generatedsubgroup.An infinite group is said to belocallyPif every finitely generated subgroup isP.For instance, a group islocally finiteif every finitely generated subgroup is finite, and a group is locally soluble if every finitely generated subgroup issoluble.

Properties of finite groups

Forfinite groups,a "small neighborhood" is taken to be a subgroup defined in terms of aprime numberp,usually thelocal subgroups,thenormalizersof the nontrivialp-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 theclassification of finite simple groups,which was carried out during the 1960s.

Properties of commutative rings

Main article:local ring

For commutative rings, ideas ofalgebraic geometrymake it natural to take a "small neighborhood" of a ring to be thelocalizationat aprime ideal.In which case, a property is said to be local if it can be detected from thelocal rings.For instance, being aflat moduleover a commutative ring is a local property, but being afree moduleis not. For more, seeLocalization of a module.

See also

• Local path connectedness
• Local-global principle