Local property

In mathematics, a phenomenon is sometimes said to occur locally if, roughly speaking, it occurs on sufficiently small or arbitrarily small neighborhoods of points.

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:

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.

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.

For an infinite group, a "small neighborhood" is taken to be a 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 if every finitely generated subgroup is finite. A group is locally soluble if every finitely generated subgroup is soluble.

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