Neighbourhood system
In topology and related areas of mathematics, the neighbourhood system or neighbourhood filter <math>\mathcal{V}(x)<math> 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
- <math>\mathcal{B}(x) \subset \mathcal{V}(x)<math>
such that
- <math>\forall V \in \mathcal{V}(x) \quad \exists B \in \mathcal{B}(x) \mbox{ with } B \subset V<math>.
That is for any neighbourhood <math>V<math> we can find a neighbourhood <math>B<math> in the neighbourhood basis which is contained in <math>V<math>.
Conversely, as with any filter base, the local basis allows to get back the corresponding neighbourhood filter as <math>\mathcal{V}(x) =\left\{ V \supset B~;~ B \in \mathcal{B}(x)\right\}<math>.
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, <math>\mathcal{V}(x) = \{ X \}<math>
- In a metric space, for any point x, the sequence of open balls around x with radius 1/n form a countable neighbourhood basis <math>\mathcal{B}(x) = \{ B_{1/n}(x) ; n \in \mathbb N^* \}<math>. 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,
- <math>\mathcal{V}(x) = \mathcal{V}(0) + x .<math>
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
Categories: General topology