In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods,[1] or neighbourhood filter for a point in a topological space is the collection of all neighbourhoods of



Neighbourhood of a point or set

An open neighbourhood of a point (or subset[note 1])   in a topological space   is any open subset   of   that contains   A neighbourhood of   in   is any subset   that contains some open neighbourhood of  ; explicitly,   is a neighbourhood of   in   if and only if there exists some open subset   with  .[2][3] Equivalently, a neighborhood of   is any set that contains   in its topological interior.

Importantly, a "neighbourhood" does not have to be an open set; those neighbourhoods that also happen to be open sets are known as "open neighbourhoods."[note 2] Similarly, a neighbourhood that is also a closed (respectively, compact, connected, etc.) set is called a closed neighbourhood (respectively, compact neighbourhood, connected neighbourhood, 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 subset)   is a filter called the neighbourhood filter for   The neighbourhood filter for a point   is the same as the neighbourhood filter of the singleton set  

Neighbourhood basis


A neighbourhood basis or local basis (or neighbourhood base or local base) for a point   is a filter base of the neighbourhood filter; this means that it is a subset

such that for all   there exists some   such that  [3] That is, for any neighbourhood   we can find a neighbourhood   in the neighbourhood basis that is contained in  

Equivalently,   is a local basis at   if and only if the neighbourhood filter   can be recovered from   in the sense that the following equality holds:[4]

A family   is a neighbourhood basis for   if and only if   is a cofinal subset of   with respect to the partial order   (importantly, this partial order is the superset relation and not the subset relation).

Neighbourhood subbasis


A neighbourhood subbasis at   is a family   of subsets of   each of which contains   such that the collection of all possible finite intersections of elements of   forms a neighbourhood basis at  



If   has its usual Euclidean topology then the neighborhoods of   are all those subsets   for which there exists some real number   such that   For example, all of the following sets are neighborhoods of   in  :

but none of the following sets are neighborhoods of  :
where   denotes the rational numbers.

If   is an open subset of a topological space   then for every     is a neighborhood of   in   More generally, if   is any set and   denotes the topological interior of   in   then   is a neighborhood (in  ) of every point   and moreover,   is not a neighborhood of any other point. Said differently,   is a neighborhood of a point   if and only if  

Neighbourhood bases

In any topological space, the neighbourhood system for a point is also a neighbourhood basis for the point. The set of all open neighbourhoods at a point forms a neighbourhood basis at that point. For any point   in a metric space, the sequence of open balls around   with radius   form a countable neighbourhood basis  . This means every metric space is first-countable.

Given a space   with the indiscrete topology the neighbourhood system for any point   only contains the whole space,  .

In the weak topology on the space of measures on a space   a neighbourhood base about   is given by

where   are continuous bounded functions from   to the real numbers and   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 of the neighbourhood system for the origin,


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.



Suppose   and let   be a neighbourhood basis for   in   Make   into a directed set by partially ordering it by superset inclusion   Then   is not a neighborhood of   in   if and only if there exists an  -indexed net   in   such that   for every   (which implies that   in  ).

See also



  1. ^ Usually, "neighbourhood" refers to a neighbourhood of a point and it will be clearly indicated if it instead refers to a neighborhood of a set. So for instance, a statement such as "a neighbourhood in  " that does not refer to any particular point or set should, unless somehow indicated otherwise, be taken to mean "a neighbourhood of some point in  "
  2. ^ Most authors do not require that neighborhoods be open sets because writing "open" in front of "neighborhood" when this property is needed is not overly onerous and because requiring that they always be open would also greatly limit the usefulness of terms such as "closed neighborhood" and "compact neighborhood".
  1. ^ Mendelson, Bert (1990) [1975]. Introduction to Topology (Third ed.). Dover. p. 41. ISBN 0-486-66352-3.
  2. ^ Bourbaki 1989, pp. 17–21.
  3. ^ a b Willard 2004, pp. 31–37.
  4. ^ Willard, Stephen (1970). General Topology. Addison-Wesley Publishing. ISBN 9780201087079. (See Chapter 2, Section 4)