Neighbourhood system

In topology and related areas of mathematics, the neighbourhood system or neighbourhood filter $\mathcal{V}(x)$ 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

$\mathcal{B}(x) \subset \mathcal{V}(x)$

such that

$\forall V \in \mathcal{V}(x) \quad \exists B \in \mathcal{B}(x) \mbox{ with } B \subset V$ .

That is, for any neighbourhood $V$ we can find a neighbourhood $B$ in the neighbourhood basis which is contained in $V$ .

Conversely, as with any filter base, the local basis allows to get back the corresponding neighbourhood filter as $\mathcal{V}(x) =\left\{ V \supset B~:~ B \in \mathcal{B}(x)\right\}$ .^[1]

1 Examples
2 Properties
3 See also
4 References

Examples [edit]

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 only contains the whole space, $\mathcal{V}(x) = \{ X \}$

In a metric space, for any point x, the sequence of open balls around x with radius 1/n form a countable neighbourhood basis $\mathcal{B}(x) = \{ B_{1/n}(x) ; n \in \mathbb N^* \}$ . 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 $\nu$ is given by

$\left \{ \mu \in \mathcal{M}(E) : | \mu f_i - \nu f_i | < \varepsilon_i , i=1,\ldots, n\right \}$

where $f_i$ are continuous bounded functions from E to the real numbers.

Properties [edit]

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,

$\mathcal{V}(x) = \mathcal{V}(0) + x .$

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.