# Environment base

In topology , a branch of mathematics, the **environment** base is a special set system. Special classes of topological spaces such as locally compact spaces and locally convex spaces can be defined via the properties of surrounding bases . In addition, the first axiom of countability draws on the thickness of the surrounding base and thus implies fundamental structural topological properties. An important special case of neighborhood **bases** are **null neighborhood bases** .

## definition

Given a topological space and in it a .

Then a family is called

of neighborhoods of an *neighborhood basis of* , if each neighborhood of supersets is one of the sets for at least one .
* *

## Examples

If one considers the , provided with an arbitrary norm , then is

the open sphere with radius around the point . An environment basis with regard to the standard topology is then formed by

- .

In this case, a countable environment base can also be defined by

- .

Analogously, a (countable) environment base can be created in every metric space with regard to the topology generated by the metric over the open spheres

define.

## Special case zero environment base

If there is a topological vector space , an environment base consisting of environments of is also referred to as a *zero environment* base. For every point and every such zero neighborhood basis one obtains a neighborhood basis of by translation:

## Related terms

The set of all environments is referred to as the *environment filter* or *environment system* of . Hence, the environment filter of is the largest possible environment base of and is a filter according to its name .

## properties

If a topological space has at most a countable environment base, then it is said that it satisfies the first countability axiom . From a mathematical point of view, such spaces are "small" and easier to handle.

## literature

- Boto von Querenburg : Set theoretical topology . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2001, ISBN 978-3-540-67790-1 , doi : 10.1007 / 978-3-642-56860-2 .
- Horst Schubert : Topology: An Introduction (= mathematical guidelines ). 4th edition. BG Teubner , Stuttgart 1975, ISBN 3-519-12200-6 .
- Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21016-7 , doi : 10.1007 / 978-3-642-21017-4 .