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Neighbourhood system

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

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

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 $V$ .

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, $\mathcal{V}(x) = \{ X \}$

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 if first-countable.

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,

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

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.

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