# 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

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

#### Translations

with respect to place

- Finnish: paikallisesti
- French: localement

# 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:

- Each point has a 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 topological spaces
- Locally connected and Locally path-connected topological spaces

## 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