AskDefine | Define locally

Dictionary Definition

locally adv
1 by a particular locality; "it was locally decided"
2 to a restricted area of the body; "apply this medicine topically" [syn: topically]

User Contributed Dictionary

English

Adverb

  1. With respect to place; in place; as, to be locally separated or distant

Translations

with respect to place

Extensive Definition

In 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:
  1. Each point has a neighborhood exhibiting the property;
  2. 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

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.
locally in German: Lokal (Topologie)
locally in French: Propriété locale
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