Neighbourhood system

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

${\displaystyle {\mathcal {B}}(x)\subseteq {\mathcal {V}}(x)}$

such that

${\displaystyle \forall V\in {\mathcal {V}}(x)\quad \exists B\in {\mathcal {B}}(x){\text{ with }}B\subseteq V.}$

That is, for any neighbourhood ${\displaystyle V}$ we can find a neighbourhood ${\displaystyle B}$ in the neighbourhood basis that is contained in ${\displaystyle V}$.

Conversely, as with any filter base, the local basis allows the corresponding neighbourhood filter to be recovered as ${\displaystyle {\mathcal {V}}(x)=\left\{V\supseteq B~:~B\in {\mathcal {B}}(x)\right\}}$.[2]

A neighbourhood subbasis at x is a collection 𝒮 of subsets of X, each of which contains x, such that the collection of all possible finite intersections of elements of 𝒮 forms a neighborhood basis at x.

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

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

where ${\displaystyle f_{i}}$ are continuous bounded functions from E to the real numbers.

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

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

This is because, by assumption, vector addition is separately continuous in the induced topology. Therefore, the topology is determined by its neighbourhood system at the origin. More generally, this remains true whenever the space is a topological group or the topology is defined by a pseudometric.

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

• Base (topology)
• Filter (mathematics)
• Locally convex topological vector space
• Neighbourhood
• Subbase