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Local propertyFrom Wikipedia, the free encyclopedia

In mathematics, a phenomenon is sometimes said to occur locallyif, roughly speaking, it occurs on sufficiently small

or arbitrarily smallneighborhoodsof points.

Contents

1 Properties of a single space

1.1 Examples

2 Properties of a pair of spaces

3 Properties of infinite groups

4 Properties of finite groups

5 Properties of commutative rings

Properties of a single space [ edit ]

A topological spaceis sometimes said to exhibit a property locallyif the property is exhibited "near" each point in oneof the following differentsenses:

1. Each point has a neighborhoodexhibiting the property;

2. Each point has a neighborhood baseof sets exhibiting the property.

Sense (2) is in general stronger than sense (1), and caution must be taken to distinguish between the two senses. For

example, some variation in the definition of locally compactarises from different senses of the term locally.

Examples [ edit ]

Locally compacttopological spaces

Locally connectedand Locally path-connectedtopological spaces

Locally Hausdorff, Locally regular, Locally normaletc...

Locally metrizable

Properties of a pair of spaces [ edit ]

Given some notion of equivalence (e.g., homeomorphism, diffeomorphism, isometry) between topological spaces, two

spaces are 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 circleand 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 sphereand the plane are locally equivalent. A small enough observer standing on the surfaceof a sphere

(e.g., a person and the Earth) would find it indistinguishable from a plane.

Properties of infinite groups [ edit ]

For an infinite group, a "small neighborhood" is taken to be a finitely generatedsubgroup. An infinite group is said to be

locally Pif every finitely generated subgroup is P. For instance, a group is locally finiteif every finitely generated

subgroup is finite. A group is locally soluble if every finitely generated subgroup is soluble.

Properties of finite groups [ edit ]

For finite groups, a "small neighborhood" is taken to be a subgroup defined in terms of a prime numberp, usually the

local subgroups, the normalizersof the nontrivialp-subgroups. 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 offinite simple groupsdone during the 1960s.

Properties of commutative rings [ edit ]

Main article: local ring

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For commutative rings, ideas of algebraic geometrymake it natural to take a "small neighborhood" of a ring to be the

localizationat a prime ideal. A property is said to be local if it can be detected from the local rings. For instance, being a

flat moduleover a commutative ring is a local property, but being a free moduleis not. See also Localization of a

module.

Categories: General topology Homeomorphisms

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