TheInfoList

In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... and related areas of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter $\mathcal\left(x\right)$ for a point $x$ is the collection of all
neighbourhood A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographicall ...
s of the point $x.$

# Definitions

Neighbourhood of a point or set An of a point (or subset) $x$ of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
$X$ is any
open subset Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
$U$ of $X$ that contains $x.$ A is any subset $N \subseteq X$ that contains open neighbourhood of $x$ in $X$; explicitly, $N$ is a neighbourhood of $x$ in $X$ if and only if there exists some subset $U \subseteq N$ such that $U$ is an open subset of $X$ and $U$ contains $x.$ Equivalently, a neighborhood of $x$ is any set that contains $x$ in its
topological interior In mathematics, specifically in general topology, topology, the interior of a subset of a topological space is the Union (set theory), union of all subsets of that are Open set, open in . A point that is in the interior of is an interior point ...
. 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 Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
, 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 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
. The family of all neighbourhoods having a certain "useful" property often forms a
neighbourhood basisIn topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point is the collection of all Neighbourhood (mathematics), neighbourhoods of the point . Definit ...
, although many times, these neighbourhoods are not necessarily open.
Locally compact spaceIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
s, for example, are those spaces that, at every point, have a neighbourhood basis consisting entirely of compact sets. A given set is a neighbourhood of a point $x \in X$ if and only if it is a neighborhood of the
singleton set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
$\.$ Neighbourhood filter The neighbourhood system for a point (or non-empty subset) $x$ is a
filter Filter, filtering or filters may refer to: Science and technology Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass ** Filter (aquarium), critical ...
called the The neighbourhood filter for a point $x \in X$ is the same as the neighbourhood filter of the
singleton set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
$\.$

## Neighbourhood basis

A or (or or ) for a point $x$ is a
filter base In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of the neighbourhood filter; this means that it is a subset $\mathcal \subseteq \mathcal(x)$ such that for all $V \in \mathcal\left(x\right),$ there exists some $B \in \mathcal$ such that $B \subseteq V.$ That is, for any neighbourhood $V$ we can find a neighbourhood $B$ in the neighbourhood basis that is contained in $V.$ Equivalently, $\mathcal$ is a local basis at $x$ if and only if the neighbourhood filter $\mathcal$ can be recovered from $\mathcal$ in the sense that the following equality holds: (See Chapter 2, Section 4) $\mathcal(x) = \left\.$ A family $\mathcal \subseteq \mathcal\left(x\right)$ is a neighbourhood basis for $x$ if and only if $\mathcal$ is a
cofinal subset In mathematics, let ''A'' be a set and let be a binary relation on ''A''. Then a subset is said to be cofinal or frequent in ''A'' if it satisfies the following condition: :For every , there exists some such that . A subset that is not frequ ...
of $\left\left(\mathcal\left(x\right), \supseteq\right\right)$ with respect to the partial order $\,\supseteq\,$ (importantly, this partial order is the
superset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
relation and not the
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... relation).

## Neighbourhood subbasis

A at $x$ is a family $\mathcal$ of subsets of $X,$ each of which contains $x,$ such that the collection of all possible finite intersections of elements of $\mathcal$ forms a neighbourhood basis at $x.$

# Examples

If $\R$ has its usual
Euclidean topologyIn mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition In any metric space, the Ball (mathematics), ope ...
then the neighborhoods of $x := 0$ are all those subsets $N \subseteq \R$ for which there exists some real number $r > 0$ such that $\left(-r, r\right) \subseteq N.$ For example, all of the following sets are neighborhoods of $0$ in $\R:$ $(-2, 2), \;$
2,2 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
\; 2, \infty), \; [-2, 2) \cup \, \; [-2, 2\cup \Q, \; \R but none of the following sets are neighborhoods of $0$: $\, \; \Q, \; (0,2), \; [0, 2), \; [0, 2) \cup \Q, \; (-2, 2) \setminus \left\$ where $\Q$ denotes the rational numbers. If $U$ is an open subset of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
$X$ then for every $u \in U,$ $U$ is a neighborhood of $u$ in $X.$ More generally, if $N \subseteq X$ is any set and if $\operatorname_X N$ denote the
topological interior In mathematics, specifically in general topology, topology, the interior of a subset of a topological space is the Union (set theory), union of all subsets of that are Open set, open in . A point that is in the interior of is an interior point ...
of $N$ in $X,$ then $N$ is a neighborhood (in $X$) of every point $x \in \operatorname_X N$ and moreover, $N$ is a neighborhood of any other point. Said differently, $N$ is a neighborhood of a point $x \in X$ if and only if $x \in \operatorname_X N.$ Neighborhood 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. * Given a space $X$ with the
indiscrete topologyIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
the neighbourhood system for any point $x$ only contains the whole space, $\mathcal\left(x\right) = \$ * For any point $x$ in a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
, the sequence of
open ball In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s around $x$ with radius $1/n$ form a
countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
neighbourhood basis $\mathcal = \left\.$ This means every metric space is
first-countable In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
. In the
weak topology In mathematics, weak topology is an alternative term for certain initial topology, initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initia ...
on the space of measures on a space $E,$ a neighbourhood base about $\nu$ is given by $\left\$ where $f_i$ are continuous bounded functions from $E$ to the real numbers and $r_1, \ldots, r_n$ are positive real numbers.

# Properties

In a
seminormed space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
, that is a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
with the
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... induced by a
seminorm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
, all neighbourhood systems can be constructed by
translation Translation is the communication of the meaning Meaning most commonly refers to: * Meaning (linguistics), meaning which is communicated through the use of language * Meaning (philosophy), definition, elements, and types of meaning discusse ...
of the neighbourhood system for the origin, $\mathcal(x) = \mathcal(0) + x.$ 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 350px, The real numbers form a topological group under addition ">addition.html" ;"title="real numbers form a topological group under addition">real numbers form a topological group under addition In mathematics, a topological group is a group ...
or the topology is defined by a pseudometric.