Neighbourhood system

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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

Definitions[edit]

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[edit]

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[edit]

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

Examples[edit]

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.

Properties[edit]

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[edit]

References[edit]

  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)

Bibliography[edit]