In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter in a topological space is the collection of all neighbourhoods

Definitions

Neighbourhood of a point or set

An of a point (or subset is any open subset A that contains if and only if. in its topological interior.

Importantly, a "neighbourhood" does have to be an open set; those neighbourhoods that also happen to be open sets are known as "open neighbourhoods." Similarly, a neighbourhood that is also a closed (respectively, compact, connected, etc.) set is called a (respectively, , , etc.). There are many other types of neighbourhoods that are used in topology and related fields like functional analysis. The family of all neighbourhoods having a certain "useful" property often forms a neighbourhood basis, although many times, these neighbourhoods are not necessarily open. Locally compact spaces, for example, are those spaces that, at every point, have a neighbourhood basis consisting entirely of compact sets.

Neighbourhood filter

The neighbourhood system for a point (or non-empty is a filter called the is the same as the neighbourhood filter of the singleton set

Neighbourhood basis

A or (or or is a filter base

in the sense that the following equality holds: is a cofinal subset with respect to the partial order (importantly, this partial order is the superset relation and not the subset relation).

Neighbourhood subbasis

A such that the collection of all possible finite intersections

Examples

has its usual Euclidean topology for which there exists some real number denotes the rational numbers.

is an open subset of a topological space denotes the topological interior is

Neighbourhood bases

in a metric space, the sequence of open balls form a countable. This means every metric space is first-countable.

with the indiscrete topology.

In the weak topology are continuous are positive real numbers.

Seminormed spaces and topological groups

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

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.

Properties

into a directed set by partially ordering is -indexed net).

See also

References

Bibliography