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Local propertyIn mathematics, a phenomenon is sometimes said to occur '''locally''' if, roughly speaking, it occurs on ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points.
Properties of a single space
A topological space is sometimes said to exhibit a property '''locally''' if the property is exhibited "near" each point in one of the following '''different''' senses:
- Each point has a neighbourhood (mathematics)|neighborhood exhibiting the property;
- Each point has a neighborhood base of 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 compact arises from different senses of the term ''locally''.
Examples
- Locally compact space|Locally compact topological spaces
- Locally connected and Locally path-connected topological spaces
- Locally Hausdorff space|Locally Hausdorff, Locally regular space|Locally regular, Locally normal space|Locally normal etc...
- Locally metrizable space|Locally metrizable
Properties of a pair of spaces
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 circle 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 and the plane are locally equivalent. A small enough observer standing on the surface of a sphere (e.g., a person and the Earth) would find it indistinguishable from a plane.
Properties of infinite groups
For an infinite group, a "small neighborhood" is taken to be a finitely generated group|finitely generated subgroup. An infinite group is said to be '''locally P''' if every finitely generated subgroup is P. For instance, a group is locally finite group|locally finite if every finitely generated subgroup is finite. A group is locally soluble if every finitely generated subgroup is soluble group|soluble.
Properties of finite groups
For finite groups, a "small neighborhood" is taken to be a subgroup defined in terms of a prime number ''p'', usually the '''local subgroups''', the normalizers of the nontrivial p-group|''p''-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 of finite simple groups done during the 1960s.
Properties of commutative rings
For commutative rings, ideas of algebraic geometry make it natural to take a "small neighborhood" of a ring to be the localization at a prime ideal. A property is said to be local if it can be detected from the local rings. For instance, being a flat module over a commutative ring is a local property, but being a free module is not.
Category:General topology
Category:Homeomorphisms
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