In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, 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 . 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.
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
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
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:
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 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
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
Seminormed spaces and topological groups
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 ).
- Base (topology) – Collection of open sets used to define a topology
- Filter (set theory) – Family of sets representing "large" sets
- Filters in topology – Use of filters to describe and characterize all basic topological notions and results.
- Locally convex topological vector space – A vector space with a topology defined by convex open sets
- Neighbourhood (mathematics) – Open set containing a given point
- Subbase – Collection of subsets that generate a topology
- Tubular neighborhood
- 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 "
- 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".
- Bourbaki, Nicolas (1989) . General Topology: Chapters 1–4 [Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129.
- Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York: Springer-Verlag. ISBN 978-0-387-90972-1. OCLC 10277303.
- Willard, Stephen (2004) . General Topology (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.