In topology and related areas of mathematics, the neighbourhood system 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

such that

.

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,

• In a metric space, for any point x, the sequence of open balls around x with radius 1/n form a countable neighbourhood basis . 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,

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)