Neighbourhood system

In topology and related areas of mathematics, the neighbourhood system or neighbourhood filter Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{V}(x)

for a point x is the collection of all neighbourhoods for the point x.

A neighbourhood basis or local basis for a point x is a filter base of the neighbourhood filter, i.e. a subset

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{B}(x) \subset \mathcal{V}(x)

such that

Failed to parse (Missing texvc executable; please see math/README to configure.): \forall V \in \mathcal{V}(x) \quad \exists B \in \mathcal{B}(x) \mbox{ with } B \subset V

. That is, for any neighbourhood Failed to parse (Missing texvc executable; please see math/README to configure.): V

we can find a neighbourhood Failed to parse (Missing texvc executable; please see math/README to configure.): B
in the neighbourhood basis which is contained in Failed to parse (Missing texvc executable; please see math/README to configure.): V

.

Conversely, as with any filter base, the local basis allows to get back the corresponding neighbourhood filter as Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{V}(x) =\left\{ V \supset B~:~ B \in \mathcal{B}(x)\right\} .

Examples

• Trivially the neighbourhood system for a point is also a neighbourhood basis for the point.

• Given a space X with the indiscrete topology the neighbourhood system for any point x is the whole space, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{V}(x) = \{ X \}

• In a metric space, for any point x, the sequence of open balls around x with radius 1/n form a countable neighbourhood basis Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{B}(x) = \{ B_{1/n}(x) ; n \in \mathbb N^* \}

. This means every metric space is first-countable.

• In the weak topology on the space of measures on a space E, a neighbourhood base about Failed to parse (Missing texvc executable; please see math/README to configure.): \nu

is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \left \{ \mu \in \mathcal{M}(E) : | \mu f_i - \nu f_i | < \varepsilon_i , i=1,\ldots, n\right \}

where Failed to parse (Missing texvc executable; please see math/README to configure.): f_i

are continuous bounded functions from E to the real numbers.

Properties

In a semi normed space, that is a vector space with the topology induced by a semi norm, all neighbourhood systems can be constructed by translation of the neighbourhood system for the point 0,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{V}(x) = \mathcal{V}(0) + x .

This is because, by assumption, vector addition is separate continuous in the induced topology. Therefore the topology is determined by its neighbourhood system at the origin. More generally, this remains true whenever the topology is defined by a translation invariant metric or pseudometric.

Every neighbourhood system for a non empty set A is a filter called the neighbourhood filter for A.

The union of local bases for all points x are a base for the topology.

See also

• neighbourhood
• Base (topology)
• Locally convex topological vector space
• Filter (mathematics)eo:Vikipedio:Projekto matematiko/Najbareca sistemo

es:Base de entornos pl:Baza otoczeń