In topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 and related areas of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the neighbourhood system or neighbourhood filter for a point x is the collection of all neighbourhood

Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...

s 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
such that.
That is, for any neighbourhood we can find a neighbourhood in the neighbourhood basis which is contained in .

Conversely, as with any filter base, the local basis allows to get back the corresponding neighbourhood filter as .

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 only contains the whole space,

• In a metric space
  Metric space
  In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean 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 is first-countable.

• In the weak topology
  Weak topology
  In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
  
   on the space of measures on a space E, a neighbourhood base about is given by

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

Properties

In a semi normed space, that is a vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 with the topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 induced by a semi norm, all neighbourhood systems can be constructed by translation

Translation (geometry)

In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...

 of the neighbourhood system for the point 0,
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

Pseudometric

Pseudometric may refer to:* Pseudo-Riemannian manifold* Pseudometric space...

.

Every neighbourhood system for a non empty set A is a filter

Filter (mathematics)

In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...

 called the neighbourhood filter for A.

See also

• neighbourhood
  Neighbourhood (mathematics)
  In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
  
  
• Base (topology)
  Base (topology)
  In mathematics, a base B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T...
  
  
• Locally convex topological vector space
  Locally convex topological vector space
  In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces which generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of ...
  
  
• Filter (mathematics)
  Filter (mathematics)
  In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...