In topology and related areas of mathematics, the neighbourhood system or neighbourhood filter for a point x is the collection of all neighbourhoods for the point x. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...

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 V we can find a neighbourhood B in the neighbourhood basis which is contained in V.

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 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.

In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. ... In mathematics, a metric space is a set (or space) where a distance between points is defined. ... A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ... In mathematics the term countable set is used to describe the size of a set, e. ... In topology, a first-countable space is a topological space satisfying the first axiom of countability. Specifically, a space X is said to be first-countable if each point has a countable local base. ...

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, The fundamental concept in linear algebra is that of a vector space or linear space. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... In functional analysis, a seminorm is a function on a vector space with certain properties characteristic of a measure of length. A space with such a seminorm is then known as a seminormed space. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...

More generally, this remains true whenever the topology is defined by a translation invariant metric or pseudometric. In mathematics, a metric space is a set (or space) where a distance between points is defined. ...

Every neighbourhood system for a non empty set A is a filter called the neighbourhood filter for A. In mathematics, a filter is a special subset of a partially ordered set. ...

The union of local bases for all points x are a base for the topology. In mathematics, a base (or basis) 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. Bases...

See also

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